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Simulation of Complex Systems in GIS

Simulation of Complex
Systems in GIS
Patrice Langlois

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Adapted and updated from Simulation des systèmes complexes en géographie : fondements théoriques et
applications published 2010 in France by Hermes Science/Lavoisier © LAVOISIER 2010
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,
or in the case of reprographic reproduction in accordance with the terms and licenses issued by the
CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the
undermentioned address:
ISTE Ltd John Wiley & Sons, Inc.
27-37 St George’s Road 111 River Street
London SW19 4EU Hoboken, NJ 07030
UK USA
www.iste.co.uk www.wiley.com
© ISTE Ltd 2011
The rights of Patrice Langlois to be identified as the author of this work have been asserted by him in
accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Langlois, Patrice.
Simulation of complex systems in GIS / Patrice Langlois.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-223-7
1. Geographic information systems. 2. Geography--Simulation methods. I. Title.
G70.212.L267 2010
910.285--dc22
2010042667
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-223-7
Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Table of Contents
General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
PART 1. THE STRUCTURE OF THE GEOGRAPHIC SPACE . . . . . . . . . . . . . 1
Part 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1. Structure and System Concepts . . . . . . . . . . . . . . . . . . . . 5
1.1. The notion of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1. In mathematics and in physics . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2. In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3. In human, social and life sciences . . . . . . . . . . . . . . . . . . . . 8
1.2. The systemic paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1. The systemic triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2. The whole is greater than the sum of its parts . . . . . . . . . . . . . 12
1.3. The notion of organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1. Structure and organization . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2. Sequential organizations. . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3. Organization in classes and partitions . . . . . . . . . . . . . . . . . . 17
1.3.4. Organizations in trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.5. Network organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.6. Hierarchical organizations. . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.7. The use of the graph theory for complex organizations . . . . . . . 23
1.3.8. Complexity of an organization, from determinism to chaos . . . . . 25
Chapter 2. Space and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1. Different theories of space . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1. Euclidian models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2. Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vi Simulation of Complex Systems in GIS
2.1.3. Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.4. Pseudo-Euclidian spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.5. Riemann’s spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.6. Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.7. About equality in a space . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2. Geometry and its data structures . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.1. Planes structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.2. The elevation model (2D½). . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.3. Non-Euclidian space, anamorphoses and gravitation field. . . . . . 56
2.2.4. Possible morphologies of a finite space without limits . . . . . . . . 58
2.3. “Neat” geometry and “fuzzy” geometry . . . . . . . . . . . . . . . . . . . 60
Chapter 3. Topological Structures: How Objects are Organized in
Spatial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2. Metrics and topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3. Calculated topology, structural topology . . . . . . . . . . . . . . . . . . 71
3.3.1. Square grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2. Hexagonal grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.3. Neighborhood structure for an irregular mesh . . . . . . . . . . . . . 73
3.3.4. Neighborhood operator for an irregular mesh . . . . . . . . . . . . . 76
3.3.5. “Vector-topological” model of a meshing of random zones. . . . . 76
3.3.6. Network topological model . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4. Hierarchization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter 4. Matter and Geographical Objects. . . . . . . . . . . . . . . . . . . 79
4.1. Geographic matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.1. The material field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.2. Hypothesis of spatial and temporal differentiation of matter . . . . 80
4.2. The notion of observation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3. The geographic object: Definitions and principles . . . . . . . . . . . . . 84
4.3.1. Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2. Spatial base of an object . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.3. Material content of an object . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.4. Material geographic object and layers of objects . . . . . . . . . . . 88
4.3.5. The principle of separation . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.6. The principle of mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.7. The principle of impenetrability . . . . . . . . . . . . . . . . . . . . . 90
4.3.8. The dimensionality of an object . . . . . . . . . . . . . . . . . . . . . 91
4.3.9. The principle of embedding. . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.10. Evaluated geographic object. . . . . . . . . . . . . . . . . . . . . . . 92
4.3.11. Description forms of the object . . . . . . . . . . . . . . . . . . . . . 94

Table of Contents vii
Chapter 5. Time and Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1. Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2. Temporalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1. Life interval T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.2. Minimum time step dt . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.3. Time base BT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.4. Activity support of a phenomenon σ . . . . . . . . . . . . . . . . . . 103
5.2.5. Phenomenon with discrete (or isolated) support. . . . . . . . . . . . 104
5.2.6. Phenomenon with continuous or piecewise continuous support . . 104
5.3. Events, processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.1. Morphological discretization of a phenomenon . . . . . . . . . . . . 109
5.3.2. Billiard balls example . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.3. Temporality of a spatial process . . . . . . . . . . . . . . . . . . . . . 113
5.4. Decomposition of a complex process . . . . . . . . . . . . . . . . . . . . 115
5.5. An epistemic choice: reciprocal dependency between the
complexity levels of a phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 117
Chapter 6. Spatial Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1. Presentation of the concept. . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2. Definition of macroscopic interaction . . . . . . . . . . . . . . . . . . . . 125
6.3. The four elementary (inter)actions . . . . . . . . . . . . . . . . . . . . . . 127
6.4. Microscopic interaction like a multigraph . . . . . . . . . . . . . . . . . . 128
6.5. Composition of successive interactions . . . . . . . . . . . . . . . . . . . 130
6.6. The configurations and the trajectories of a simulation are
categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.7. Intermediary level matrix representation . . . . . . . . . . . . . . . . . . 133
6.8. Examples of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8.1. Flux and transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8.2. Movement of an object in space . . . . . . . . . . . . . . . . . . . . . 135
6.8.3. Collision between two objects . . . . . . . . . . . . . . . . . . . . . . 135
6.8.4. Accumulation by confluence . . . . . . . . . . . . . . . . . . . . . . . 136
6.8.5. Centrifugal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.8.6. Equalization through communication vases . . . . . . . . . . . . . . 137
6.9. First definition of the notion of spatial system . . . . . . . . . . . . . . . 138
Part 1. Conclusion: Stages of the Ontogenesis . . . . . . . . . . . . . . . . . . 141
PART 2. MODELING THROUGH CELLULAR AUTOMATA . . . . . . . . . . . . . . 145
Chapter 7. Concept and Formalization of a CA . . . . . . . . . . . . . . . . . 147
7.1. Cellular automata paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . 148

viii Simulation of Complex Systems in GIS
7.2. Notion of finite-state automata . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3. Mealy and Moore automata . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.4. A simple example of CA: the game of life . . . . . . . . . . . . . . . . . 152
7.5. Different decompositions of the functions of a cell . . . . . . . . . . . . 153
7.6. Threshold automaton, window automaton. . . . . . . . . . . . . . . . . . 155
7.7. Micro level and Stochastic automaton . . . . . . . . . . . . . . . . . . . . 156
7.8. Macro level and deterministic automaton . . . . . . . . . . . . . . . . . . 156
7.9. General definition of a geographic cellular automaton . . . . . . . . . . 157
7.10. Different scheduling regimes of the internal tasks of the system. . . . 160
7.11. Ports, channels, encapsulation . . . . . . . . . . . . . . . . . . . . . . . . 162
7.12. Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.13. Space associated with a geographic cellular automaton . . . . . . . . . 168
7.14. Topology and neighborhood operator of a GCA . . . . . . . . . . . . . 168
7.15. The notion of cellular layer. . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.16. Hierarchized GCA models . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16.1. Spatial hierarchization . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16.2. Temporal hierarchization . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16.3. Hierarchization of the control . . . . . . . . . . . . . . . . . . . . . . 170
Chapter 8. Examples of Geographic Cellular Automaton Models . . . . . . 171
8.1. SpaCelle, multi-layer cellular automaton . . . . . . . . . . . . . . . . . . 172
8.1.1. Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.1.2. Choice of metrics and the notion of neighborhood in SpaCelle. . . 173
8.1.3. Universe Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.1.4. Definition of cellular behavior with SpaCelle . . . . . . . . . . . . . 174
8.1.5. General structure of a model . . . . . . . . . . . . . . . . . . . . . . . 175
8.1.6. Cellular behavior, birth, life and death, law of the most
pertinent rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.1.7. Deterministic or stochastic functioning . . . . . . . . . . . . . . . . . 176
8.1.8. Rule syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.1.9. Calculation of the pertinence of a transition rule . . . . . . . . . . . 177
8.1.10. Strict or fuzzy evaluation of a neighborhood . . . . . . . . . . . . . 178
8.1.11. The rule base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1.12. The SpaCelle meta-model . . . . . . . . . . . . . . . . . . . . . . . . 180
8.1.13. Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.2. Example: the evolution model of the Rouen agglomeration . . . . . . . 181
8.2.1. From the map to the cellular automaton. . . . . . . . . . . . . . . . . 181
8.2.2. The rule base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2.3. Evolution observed in the Rouen space between 1950
and 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.2.4. Current assessment of SpaCelle use . . . . . . . . . . . . . . . . . . . 188

Table of Contents ix
8.3. RuiCells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.3.1. Presentation of the model . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.3.2. Recognition of soil occupation and surface development . . . . . . 194
8.3.3. Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.3.4. The outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.4. GeoCells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.4.1. The generic GeoCells model . . . . . . . . . . . . . . . . . . . . . . . 207
8.4.2. The GeoCells-Europe model . . . . . . . . . . . . . . . . . . . . . . . 212
8.4.3. The GeoCells-Votes model . . . . . . . . . . . . . . . . . . . . . . . . 224
Part 2. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
PART 3. A GENERAL MODEL OF GEOGRAPHIC AGENT SYSTEMS . . . . . . . . 237
Part 3. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Chapter 9. Theoretical Approach of an Integrated Simulation
Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.1. For an integrated platform of simulation. . . . . . . . . . . . . . . . . . . 241
9.2. General specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Chapter 10. A Formal Ontology of Geographic Agent Systems. . . . . . . . 245
10.1. The conceptual framework . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.2. The notion of a geographic agent system . . . . . . . . . . . . . . . . . 247
10.3. A generalization of the notion of process . . . . . . . . . . . . . . . . . 249
10.4. The notion of a geographic agent . . . . . . . . . . . . . . . . . . . . . . 250
10.4.1. When the geographic object becomes an agent . . . . . . . . . . . 250
10.4.2. The agent dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.4.3. The agent-organization duality . . . . . . . . . . . . . . . . . . . . . 252
10.4.4. The formalization of the geographic agent . . . . . . . . . . . . . . 254
10.5. The formalization of the notion of organization . . . . . . . . . . . . . 258
10.5.1. Re-examining the concept of organization . . . . . . . . . . . . . . 258
10.5.2. Social organization, spatial organization . . . . . . . . . . . . . . . 260
10.5.3. Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.5.4. Two examples of organization . . . . . . . . . . . . . . . . . . . . . 263
10.5.5. Predefined spatial organization . . . . . . . . . . . . . . . . . . . . . 264
10.5.6. Predefined social organization . . . . . . . . . . . . . . . . . . . . . 268
10.6. The formalization of behavior . . . . . . . . . . . . . . . . . . . . . . . . 268
10.6.1. The evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.6.2. The decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.6.3. The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10.6.4. The formalization of a basic behavior . . . . . . . . . . . . . . . . . 272

x Simulation of Complex Systems in GIS
10.7. Formalization of a general AOC model . . . . . . . . . . . . . . . . . . 279
10.8. The Schelling model example . . . . . . . . . . . . . . . . . . . . . . . . 280
Part 3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
General Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

General Introduction
In [DAU 03], André Dauphiné describes geography as the core of the complexity
in human and social sciences. The information tools that permit us to enter the
paradigm of geography’s complexity were brought forth by Tobler and Hagerstand
through the use of cellular automatons. Then, multi-agent systems appeared near the
end of the 1980s thanks to the combined evolution of artificial intelligence, object-
oriented programming and distributed intelligence, which were later developed into
numerous fields such as physics, biology and computer science [WEI 89], [BRI 01].
Thus, numerous works have contributed to applying these computing and theoretical
tools specifically to geography. These studies continue to appear today, in the works
of different teams such as the geosimulation group, RIKS, CASA, Milan’s
politechnico and urban simulation (SIMBOGOTA) and city network studies from
the Universities of Paris and Strasbourg in France. We will not explain these in
detail.
Geography is essentially ingrained in space. The geographical map is its direct
expression. If we are interested in complex processes, we must consider the
interlocking organizational levels that are necessary to understand these phenomena.
Modeling adds to the temporal and fundamental dimensions of the expression of
dynamics. The multi-level representation in space forces us to address different
temporality levels of processes in play.
This work will attempt to contribute to the challenge that is geographic
complexity. Complexity is characterized as being a crossroads between physical and
human sciences, by its intermediary position in overlapping levels of reality, which
are spatial and temporal and finally, by its key position in the degrees of
organization complexity, which is the position of human kind in both the living and
mineral domains.

xii Simulation of Complex Systems in GIS
It took many years to accomplish this work in the area of geographic modeling.
It is a product of reflection and fulfillment in the area of cartography, spatial
analysis and geomatics. This study began at the beginning of the 1980s, and
coincided with the arrival of micro-computers. This decade witnessed the
construction of tools and concepts of solid and efficient representation of space. At
the end of the 1990s and at the beginning of the 21st century, necessity passed to the
next level: dynamic spatial simulation. It was imposed by the powerful level attained
by computers, by the mature development of cartography software and by spatial
analysis, through the development of complexity theories and associated simulation
tools, since developed in other areas, such as physics.
Through this work, our goal is to share our knowledge in the area of modeling
spatial dynamics, based on a systemic, individual-centered and distributed approach.
This work is also the continuation of diverse contributions on this theme in works
such as [GUE 08] and [AMB 06]. Here we will present a more personal analysis
through our theoretic reflections and by means of a few of our realizations which
were developed by our research team that are not isolated from the national and
international abundance of such productions. We want this work to be an
educational tool for students, geographic researchers, developers and computer
scientists who wish to learn more about modeling in geography.
The mathematical aspect of certain developments should not alienate the literary
reader as the formulae and mathematic notations are not necessary for its general
comprehension and may be disregarded during a qualitative reading. These
developments are generally associated with text explaining them in today’s terms.
The formal aspect must therefore enable the reader to learn about this area and to
deal with these notions. They can seem repellent at times but we need to overcome
that perception if we wish to numerically test or program these methods or models.
Nevertheless, many of these formalisms deal with very simple concepts, and in this
work, we have made a constant effort to accompany these formalisms with a simple
explanation and to give meaning to the symbols and notations in the text.
Starting with the most general concepts of structure, organization and system, we
will firstly approach the fundamental notion of space. The richness of this concept is
shown through different formalizations that lead us to geometries, topologies and
metrics defined through space. Then, we approach the concepts of matter and object
to finally introduce time. This allows us to approach the notions of processes and
interaction that are fundamental in dynamic geographic modeling. After this section,
presenting the foundations of geographic space modeling we will work with the
computing tools of dynamic modeling, which are the geographic cellular automatons
(GCA) which enable us to have a general model of a GCA. Then, we generalize it to
construct a general system of geographic agents (SGA) model, based on a formal
ontology constructed on the Agent-Organization-Behavior triptych where the

General Introduction xiii
geographic object appears as a dual entity between the individual and the group.
This formal ontology is mathematically formalized as it lets us elaborate a
construction totally independent of all technological constraints and provides a
rigorous theoretical framework. Thus, we can think of geographic objects according
to a more realistic approach, even if it remains simplified. The mathematic
formalization enables us to think of continuum or infinity without being preoccupied
by the limitations of a computer in which everything must be explicit1
, enumerated
and finished. We need a theoretic and suppler framework to formalize this
construction.
Firstly, the set theory and the logic of predicates currently form an elementary
basis which is recognized for all mathematical formalizations. We have come a long
way from the beginning of the set theory of entities which was initiated by Cantor at
the end of the 19th century, a time when fundamental paradoxes shook its axiomatic
structure. The set theory has reached its maturity while being conscious of its limits,
for example, knowing how to distinguish between what is a set and what is not
(which we will call a “family” or a “collection”). There is no formal definition of the
notion of “set”. It is a primary definition of the theory. Nevertheless, the family of
all sets is not a set in itself, as a set must be clearly defined, either by the thorough
and non superfluous list of its elements, (it is therefore defined “in extension”) or by
a property characteristic of its elements, (it is then defined “in comprehension”).
Another essential rule exists so that the theory does not contradict itself. This has to
do with the relation of belonging: a set cannot belong to itself. However, the notion
of sub-sets gives birth to the relation of inclusion, which is a relation of order
defined on the set P(E) of the parts of a set E.
The inclusion relation is reflexive, as opposed to belonging, which is anti-
reflexive. Thus, in the set theory, a set contains itself but does not belong to itself.
With such precautions, Russel’s paradox no longer exists. In fact, this paradox
rested upon a particular set, formed by all the sets which do not contain themselves.
This paradox resulted from the fact that this set could either contain or not contain
itself. These improvements are linked with others in Zermelo-Fraenkel’s axiomatic.
The latter confers great weight to this theory. Even if it is not the only one at the
basis of a set theory, it is widely used today. It will eventually be accompanied by
other complementary axioms, such as the choice axiom, and the continuum
hypothesis.
1 Contrary to mathematics that are based on an implicit syntactic construction (a definition
once stated is assumed to be known afterwards) and on the implicit contents. For example, we
only know out of the real numbers those which can be formulated or made explicit, but there
is an infinity of numbers that will never be made explicit. Many mathematical objects are
implicitely definite by theorems of existence, but we either don’t know or cannot always
determine them effectively.

xiv Simulation of Complex Systems in GIS
A few other set theories have been formed, such as the theory of types
(Whitehead, Russell) and the theory of classes (von Neumann, Godel). In spite of
their differences, these theories now appear to be converging translations of the
same mathematical reality. Other tentatives of axiomatization were developed in
different directions and some of those were formalized. Such is the case for
mereology which is a more logical theory formalized by the logician Stanisław
Leśniewski (1886-1939). This theory does not form a more fecund advance for our
work than the “standard” set theory. For example, one of the main principles of the
complexity paradigm is that the whole is more than the sum of its parts. In the set
theory, like in mereology, this affirmation is false. The definition of a complex
system rests upon a richer concept than a simple set formed of elements (and of
parts).
We propose to formulate this enrichment, which is not contradictory by the use
of the set theory. This ontological construction is not limited to the single use of the
set theory. The whole structure of algebra, geometry, topology and analysis, whose
coherence and language rest upon the set theory, will be useful for us at many levels.
Nevertheless, we do not want to elaborate a mathematical theory formulated by a
series of theorems and demonstrations. We also do not want to elaborate on new
axioms. We will only use the mathematical language to define entities of our
ontology and to show its coherence. Thus, the level of mathematical knowledge used
remains elementary.
In order to define this ontological construction step in a geographic realm, we
will start from nothingness with the localizations constituted by what is left of the
world, keeping only localizations and coordinates. This nothingness is formed by the
space2
of geometry which is void of all matter and content. It allows for the
construction of geometric forms and permits them to be put in relation, through
topology, in order to construct more complex abstract objects. In the meantime, the
profound essence of objects only appears with the introduction of the concept of
matter and energy. How can matter be formalized in this geometric space? Does a
point, a line or a surface still exist when space becomes material? We will finally
examine how the acknowledgement of time permits us to construct facts and
behaviors. For example, it permits the birth, the development and the death of either
physical, living, social or imaginary beings. It also enables us to add depth of history
and incertitude of the future to the diversity of spatial reality. Thus, it seems that the
physical triptych of space-time-matter is the preliminary conceptual pedestal on
which our ontological construction Agent-Organization-Behavior (AOB) is based.
This confirms that the laws of physics do not only apply to life sciences of man and
2 This term is used in a voluntarily ambiguous manner to evoke geography’s disciplinary
area, but also to indicate that we are situated in a physical space, which is mathematically
formalized.

General Introduction xv
society. If each level of reality possesses its own laws, they keep their vertical
coherence, which is to say that each level cannot contradict laws acquired at lower
levels.
The concepts of agent and organization are at the heart of geographic object
construction. They define a geographic object dually, which consists of a more or
less abstract membrane, the external side of which is turned towards the exterior
world with which it acts. This realm is formed on the one hand by a diverse part of
agent-objects of the same level, more or less evolved but nevertheless of the same
general conception, and on the other hand by an englobing system into which all of
these objects are integrated. It also consists of an internal side which presents the
object as an organization turned towards the isolated depth of its interior for which
its parts are still agent-objects forming a system. These two sides express the
fundamental interaction which is the object’s essence that is active and evolutive
(some would say “inactive”). These qualities permit a co-construction (or even a co-
evolution) from both the collective exterior and interior universes. If the AOB
ontology was initially inspired by Jacques Ferber’s AGR work (Agent-Group-Role),
it defers from the duality of agent and organization which integrates this auto-
reference and its internal and external environments which derive from it.
Geography’s main interest with respect to this structure derives from the fact that it
expresses a systematic, multi-level vision. Furthermore, it permits us to identify the
exterior and interior limits of the model. Thus, we can often identify three levels of
modeling (but this number is not limited, i.e. macro, meso and micro).
The macro level is limited by the global system’s envelope (which corresponds
to the entire model) and contains the highest level of organization. The main level of
the system’s objects is constructed in this environment (the one that contains objects
we study) which are at the meso-level. These objects can themselves be structured
by terminal objects so they cannot be deteriorated by more elementary objects. This
is what we call the “particular” or “micro” level. If the problem persists, we can
always add more levels. This representation (see Figure I.1) is evident to an
individual observer (or an individual observed by the modeler), who can see at its
meso-level the grouping of the other individuals of this level, who can internally
“feel” the grouping of these micro-level internal components and who are also in
relation to this system in which it evolves (at the macro-level).
Moreover, these two formalization steps that we use, mathematically and
informatically, are not antagonistic but complementary and can be mutually
enriched. Our method will thus be presented more often as an object or as a concept
in the form of a description, as is customary in geography. Then we will
mathematically formalize it, and/or see how it can be translated in a conceptual,
structural or algorithmic computer science formalization.

xvi Simulation of Complex Systems in GIS
Figure I.1. Relations between the individual, collective, internal and external
Notations used
We constantly use two formalization methods: mathematic and algorithmic.
These two methods conform with slightly different conventions so it is therefore a
good idea to know the difference, depending on the context. The mathematic
language generally uses one symbol (typically a letter), sometimes accompanied by
an index to represent an entity as either a variable, an element, a set, a function, an
unknown, etc. When we associate two letters which represent numbers, this often
signifies that we multiply them. On the contrary, in computer science, as the number
of symbols in a program or in an algorithm can be large, we represent an entity by a
rather explicit name, by using many letters. The same formula or series of
calculations can be written in two manners, and the same symbols can have different
significations:
− In mathematical language, the symbols are written in italics to differentiate
them from common language. The multiplication operation is implied (or more
rarely indicated by a point). The expression of equality a=b indicates a mathematical
equality, which means that a and b are two ways of signifying the same quantity or
the same element of a set. We use specific symbols for operations (summation,
integration, fraction line, square root, etc.).
So,
min
y y jp
= +
the number is y equal to the y minimum added to the product of j by p numbers

General Introduction xvii
1
1
n
i
i
x x
n
=
= ∑
indicates that the number x bar (showing an average in statistics), is equal to the
opposite of n multiplied by the sums of xi for the index i varying from 1 to n, which
makes us divide the sum of xi by n:
1 2 ... ...
i n
x x x x
x
n
+ + + + +
=
− In algorithmic language, we use the “typewriter” font where we often use a
syntactic approach close to the Pascal language. The multiplication is then
represented by a star. The symbol of equality “=” does not have the same sense as it
does in mathematics. Here it is a logical operation that gives the “true” result if the
left and right members represent the same quantity or quality, otherwise giving a
“false” result. We must not confuse the symbol of equality with the symbol of
allocation, noted in the Pascal language “:=” or sometimes in the algorithmic
language, by an arrow “←”. The expressions “a := b+c” or “a ← b +c” mean that
we read the values contained in the memory files named b and c, that we use the b+c
addition and that we write (or store) the result in the “a” memory file. In
algorithmics, we do not use special symbols (Greek, etc.), we only use keyboard
symbols. The entities are often named by chains of many characters. The point
represents a separator between a complex entity (an object) and a component of this
entity (an attribute, property or method).
For example, the two following formulae could be written as:
y := DTM.yMin + j*DTM.PasY;
where “yMin” and “PasY” are fields (attributes) of the “DTM” object. We can also
have an algorithmic style of writing, like in the following example that calculates the
average of values contained in the X chart.
AveX :=0 ;
for i :=1 to n do
AveX := AveX + X[i]
end ;
AveX := AveX/n

PART 1
The Structure of the Geographic Space
Geometry and movement are the two inseparable problems in geographic theory.
Regardless of the movement, they leave their mark on the terrestrial surface. They
produce a geometry, then the geometry produces movements: circulations in states
are created by national frontiers, and in return they contribute to create these
frontiers.
William Bunge
Simulation of Complex Systems in GIS Patrice Langlois
© 2011 ISTE Ltd. Published 2011 by ISTE Ltd.

Part 1
Introduction
The concept of geographic space has been used by geographers and spatial
economists since the end of the 19th century, by such people as von Thunen, Weber,
Losch, Christaller and many others. It is mostly done through network studies,
taking into account locations, distances, and terrestrial surfaces. There are also
“functional distances” which are no longer expressed in kilometers, but in transport
cost, in travel time, in energy spent, etc. Surfaces are measured not only in hectares
or in square kilometers but also in population size, density and revenues. Thus,
geographic space appears as though it has been constituted by all of its “geographic
matter”, (natural or constructed, human or social). It is a space of diverse activities
that consumes energy and thus possesses an economic dimension. Then, there is a
sort of generalized or abstract roughness that expresses the degree of difficulty to
deal with the fundamentally heterogeneous space. For example, Jacques Levy
speaks of different pedestrianized “metrics” to express this notion. This has led to
different types of cartographic representations where geometric space is deformed in
order to better visualize this spatial roughness through anamorphosic methods
(Tobler, Charlton, Cauvin-Raymond, Langlois, etc.).
Also, the concept of space is not totally foreign to the concept of geographic
objects, which has been used for a long time in human geography. We will elaborate
on the precise and concrete definition of this notion of geographic object later on in
geomatics. In addition, the notion of objects is also a central concept in computer
science, where object-oriented languages have an important place, and are well
adapted to multi-agent modeling. In the context of geographic phenomena modeling,
the use of the term “object” may cause confusion. Nevertheless, we use it here not to
refer to oriented-object programming, but in a more general, systemic and auto-

4 Simulation of Complex Systems in GIS
referential “physics” sense. We will demonstrate how the object is the central
concept through which the first concepts of space, time and energy-matter were
structured. It is also the interface between the observation and modeling levels of
reality.
The object is not only defined as its inanimate material element but covers the
whole disciplinary field, as we believe that in the field of computer modeling the
same elementary principles of structuring and function are applicable, from a pebble
to a social group. The important differences between objects come from the
differences in the levels of complexity and not because they come from the essence
or from fundamental epistemic differences, in particular between inanimate and
living things. We must then be able to formalize and program them with the same
methods and modeling language, on the same platform of computer modeling.
If we were to reflect upon the concept of a modeling platform, we would need a
clear conceptual and mathematic formalization of the concepts of space, spatial
structure, objects and spatial systems. We could then elaborate on the notions of
dynamics, process and behavior, which gives these objects an “agent” status.

Chapter 1
Structure and System Concepts
1.1. The notion of structure
According to Raymond Boudon, “structure appears as indispensable in all human
sciences, judging by the increase of its employment, and it being difficult to
pinpoint”. Amongst the definitions contained in the Universalis Encyclopedia, there
are four concerning our subject:
– complex organization (administrative structure);
– the way in which things are organized to form a set (abstract or concrete);
– in philosophy the stable set of interdependent elements, such that each one is
dependent on its relation with others;
– in mathematics, a set composed of certain relations or laws of composition.
Let us observe at which point these definitions converge towards our subject.
The first reintroduces complexity; the second brings us back to the notion of an
organized set; the third, in its simplified version, refers to the structuralist theories
(Saussure, Merleau-Ponty, Piaget, Lévi-Strauss, etc.) but does not contradict the way
in which mathematics formalizes it through the fourth definition. Furthermore, it
corresponds to a contemporary trend consisting of defining an object, not by its
intrinsic properties, but by its connections with others. Its function is defined
because it consumes and produces on the outside and not by its content or its
internal mechanism of functioning. In particular, it is the systemic paradigm of the
“black box”.

1.1.1. In mathematics and in physics
1.1.1.1. The mathematical structure of group and physics invariants
It is interesting to see how mathematics approaches this notion of structure.
There is a great diversity of meanings that are more or less general but each of them
is precisely defined, such as must be done in mathematics. The most general
definition is the following: a structure is a set composed of relations between its
elements. In the case of an algebraic structure, it is the operations (additions,
multiplications, etc.) that define the relations between the elements. Thus, the
number 5 is related to a couple of numbers (2, 3) by the addition operation. However
most of these operation properties are important (commutativity, associativity, etc.).
The example of the group’s structure1
is emblematic because it is both simple and
plays a fundamental role in mathematics and physics, translating a certain invariance
and symmetry properties in natural phenomena. We also speak of invariance in
Euclidian geometry by the group of trips (translations and rotations) that operate on
the points of space, or by the group of rotations that operate on the vectors (the
vectors already being invariant by translation). This means that a rigid object, such
as a box’s x width, y depth and z length, doesn’t change its dimension (its diagonal
length l being calculated by the Pythagorean theorem: 2 2 2 2
l x y z
= + + ). This is
translated by the invariance of length l when we operate its displacement of the box.
The group structure can be enriched if we add other operations such as a
multiplication or a scalar product, etc. We then see a swarm of algebraic structures
with flourishing images, such as modules, rings, bodies, algebra, vectorial space,
topologic space, Hibert space, etc. All of these structures play an essentially
intellectual role in mathematics and physics. If we can establish a bijection between
two sets of objects (often in very different domains), that can conserve their
respective algebraic structure (isomorphisms). We can apply all of these acquired
results from one domain to the other. Furthermore, each of these domains enlightens
the other one under a new representation and then improves the comprehension of
each of them.
1 A group is a set G which is defined by an internal operation called addition, denoted +,
which has the properties:
1) associativity: for all elements a, b, c of G, a + (b + c) = (a + b) + c;
2) the existence of a neutral element: noted e that checks for any element of G, e + a = a + e = a;
3) every element x of G has an inverse x’ (also called contrast and noted - x) as x + x' = x '+ x = e.
For example, the set Z of integers equipped with addition is a group (which has become
commutative, since for all integers a and b, the addition checks: a + b = b + a).

Structure and System Concepts 7
1.1.2. In computer science
In computer science the notion of structure is also fundamental but is presented
differently to the way it is presented in mathematics. In the programming languages
it appears at two levels: at control structures and at data structures.
The control structures enable us to organize the sequence of instructions of a
program, so that its execution can follow a particular and non-sequential order.
These structures organize the instructions in blocks and enable the operation of an
execution control from a block. In particular, the control structure, also known as the
conditional structure, enables us to execute or to skip an instruction sequence, so
that a condition is realized or not (if condition then action). The loop enables us to
repeat the execution of a continuous sequence in a block so long as a condition
remains valid (if condition then action). We have demonstrated that all algorithms
could be programmed with a language with only two control structures (on top of
the simple sequence of instructions). Programming languages permit us to share
codes in different parts, called sub-programs, that allow us to clarify and optimize
the size of a program by also creating components or modules that we can summon,
without limit. Furthermore, these control blocks, such as sub-programs, can fit into
each other recursively. Languages that have these characteristics are called
structured languages. The Algol language (algorithmic oriented language), defined
in 1958 by the Europeans, was the first programming language that was perfectly
structured and independent of machines. Today, all of these generalist programming
languages are structured.
However, the computer languages can create and manipulate data structures,
that are more or less complex. A data structure is generally described by a type that
defines the way to store different information. For example a graph is a structure
that enables us to store the same type of information, many times. We then get this
information through the intermediary of one or more indices according to the size of
the graph. Therefore, if T is a double-indexed graph from 1 to 31 and from January
to December and contains real numbers, the notation T[9, November] could
indicate the temperature corresponding to the date of November 9th. The address
type is an assembly of 5 information fields, written out in a certain order: the name
(chain of characters), followed by the number of the street (integer), the name of the
street (chain of characters), a postal code (a series of 5 numbers), the name of a city
(chain of characters) and the name of a country (chain of characters). In Delphi (or
Pascal) language, the modern drift of algol, a large number of data structures may be
created by the programmer. The Address type that we described above can be
expressed as an assembly of different fields of data (named record) in the following
manner:

Address=record
Name of the owner: string;
NumStreet: integer;
NameStreet: string
Postal code: 0..99999;
NameCity:string;
Country:string
End;
The data structure having been determined, we can use variables of this type that
allocate the memory place and enable us to store data inside. For example, if
myaddress is a variable of type Address, we can allocate a value of 7 to the
number of the street using the instruction.
Myadress,numStreet:=7
The structuring by objects, which is perfecting the classic record structure, is
also authorized by the Delphi language, which is an object-oriented language. It not
only enables us to define fields of data as a record, but also enables us to
“encapsulate” its own behavior methods in the structure. Furthermore, having
defined a class of object, we can define its sub-classes by inheritance. The sub-class
then inherits everything from the preceding class and we can add data and specific
methods. This programming style which is present in almost all current languages is
evidently very well adapted to agents programming.
When we speak of agent structure or other entities, unless otherwise mentioned,
it is in reference to the concept of data structure. Nevertheless, as for the diverse
steps of conception, we will generally not need to completely formulate a structure
like the one above, we will then explain these with a conceptual modeling language,
such as UML (Unified Modeling Language) [BLA 06].
1.1.3. In human, social and life sciences
At the end of the 1950s, the reference to the concept of structure was generally
used in the field of human sciences. Thus, structuralism, which has its roots in the
1960s, is not an easily identifiable school of thought. It is both pluridisciplinary and
transdisciplinary, or even interdisciplinary. Human science looks at structuralism to
find a general concept of structure, but we are in fact witnessing the topic-comment
of this concept that derives from heterogeneous rationalities through diverse
disciplines of human sciences. This is why it is more often conceived as a polymorph
concept. Through it a recomposition of knowledge is researched, to bring together
these sciences. At the beginning, there is a will to renew the forms and

representations in order to free up the connections and relationships between different
structures. Structuralism tries to free up a legitimacy of human sciences that could
move it closer to the so-called “hard” sciences in order to close the gap between
scientific and literary cultures. Thus, with the concept of structure, an illusion has
developed that there may be unity between different paradigms of human sciences.
If we look at the etymology of the word “structure”, we can see that it is
composed of the “structura” construct. The question is to study a construction of
knowledge and to bring sciences closer together, with the use of a similar concept.
However, as we have previously observed, each discipline has given a different form
to the concept of structure, but structuralism has, on its own, modified each
discipline by orienting their evolution, by renewing the representations and by
decompartmentalizing each discipline. Nonetheless, there is no homogeneity
between methodological and epistemological principles that would apply in the
same way to all life sciences. Thus, there is then no common definition for the
concept of structure in human and social sciences.
In life sciences, the terms “structure” and “organization” can be used in different
senses than the ones used here. For example Maturana and Varella [MAT 92]
provide the following definition: “The organization designs the relations that must
exist between the components of a system to be a member of a given class. The
structure designs the components and relations that constitute a particular unity and
materializes its organization”. The structure then has a sense of realistic material,
which is not the case in computer science where it has a more general sense and
often characterizes a class of objects. The materialization of this structure is then an
“instanced” object, where the structures’ defined attributes are then filled in by
effective values. The computer science structure then plays an organization role, as
Maturana defines it in biology.
Nor does human geography escape from structure. In [BAI 91], Rodolphe de
Konink defines it as “an arrangement of elements organized as a function of the
entity”, and Hubert Béguin defines it as “the set of proportions, relations and spatial
dispositions, of the elements and sub-sets of a system”, but if the term is employed
in many fields (urban, agrarian, landscape structure, etc.), then it is rare that it
appears in a formalized frame and it remains essentially descriptive. In the 1980s,
when Forrester’s ideas [FOR 80] were introduced in France, a true and formalized
modeling of geographic models [CHA 84], [UVE 84] appeared. In a more systemic
approach, B. Elissalde [ELI 04], defines the spatial structure in these terms, to which
we adhere: “If we consider geographic space as a set of interacting elements, the
spatial structure must be understood as the organization principle of the studied
geographic entity, which is materialized by a shape (axis, gradient, pole, etc).
Through this, the spatial structures belong to the theoretic field of systemics and
modeling”.

1.2. The systemic paradigm
Systemics also uses the structure notion, but goes even further by making it
dynamic. Modeling almost always implies the notion system. Isn’t modeling
building a system? It is important to note that we use the term “system” in different
contexts. The universe of reality is firstly perceived by our senses and by our
scientific observations, it will only ever be known through perspective filters. When
we try to understand or explain a part of this universe, we first try to identify a
“reality system”, which is informal at this stage, and unexplained, and only defined
by representation as a consequence of observation associated with a problem or a
scientific project. A “reality system” is foreign to us and exists independently of the
observations that we make and from the consciousness that derives from it. The
interpretation of the expression “reality system” is not unanimous.
In 1865, Claude Bernard stated that: “systems are not in nature but they are in
our minds”. Nevertheless, when we see an object or a group of objects in an
interrelation that produces effects which cannot be random, but that exhibit a certain
internal organization, unknown to us, we use the term “reality system”. Also, when
the knowledge is elaborated, we must also distinguish between those that are
theoretical (from our knowledge and representations) and those that are real. The
“reality system” is thus a term with one sense. To become scientific, this reality
system must surpass the level of elementary and individual observation and must be
structured and linked to a previous body of knowledge, giving it sense. It must also
give it a social existence which comes from the multiplicity and the independence of
the corroborating observations, permitting us to construct collectively recognized
knowledge. This existence can then be materialized by a formalization in a theoretic
or conceptual framework, expressed in a natural, algorithmic, graphic, mathematic
or other symbolic language. Here we speak of theoretical or conceptual systems.
1.2.1. The systemic triangle
The system is gradually materializing and developing through the scientific steps
of modeling, going back and forth towards observation and experimentation in order
to become an object belonging to the universe of knowledge. To do this, we must
simplify in many ways the inextricable complexity of reality. We simplify this
through problematics and hypotheses where thematic choices limit the space of
study in both depth and length. Then, by observation, we can retain only a small
portion of reality. We can only select some scattered images of elementary reality.
Then we bring in another way to simplify things, i.e. scientific formalization, which
tries to introduce observation by an economy of thought in an already known
theoretic framework (then leaves in order to evolve) through a concise and

simplifying formalism, thus permitting us to connect this reality to a pattern of
known semantic relations that will give it some meaning.
Furthermore, with the progress of technology, in particular that of computer
science, the researcher has the opportunity to prolong the purely intellectual
construction of the system, via a material construction (such as a laboratory
experiment, a model or a machine) or even via a purely informative virtual
construction by use of software. These constructions allow the researcher to obtain a
more precise representation, to manipulate the system almost like an existing object
and to attempt computer-related experiments to test hypotheses, etc.
The possibility of simulating the functioning of the system in a computer is very
flexible because it enables us to foresee many configurations, to vary the initial
conditions, even in an surrealist way, and to quickly see its consequences, which can
generally not be done in reality (meteorological simulations, social simulations,
etc.). This enables us to evolve the model in an incremental way, by confrontations
and successive validations through observation [GUE 04].
Universe of
reality
Reality
observed
Theoretic
formalization
Theoretic
system
Improvements
Computer science
modeling
Simulation observations
by confronting the
observations
Technological
system
Technological
universe
Figure 1.1. The three system universes

These three images of the same system (observed, theoretical and technological)
must not be confused. They give rise to two forms of modeling: theoretical and
conceptual for the first one and computational, energetic or mechanical for the
second. In geography, when we speak of a system, for example, the French
administrative system, we are referring to a reality, because even if it was created by
man, no-one knows it completely, even if for many centuries millions of people
have participated in the making of what now exists. It is made up of a collective,
social, historical, intellectual and technological “conscience”, by civil servants,
books, computers, networks and different infrastructures of society.
Technically, no one is supposed to ignore the law. Each of us must possess a
mental representation of this system. We can elaborate a partial, intellectual or
technological model that will often be incomplete or even formalized. While the
theoretical model has a primary objective of understanding the real system, the
computer science model can have many objectives: it can try to imitate the real
system (eventually by methods other than the theoretical model) in a utilitarian
approach, without necessarily wanting to better understand (meteorological,
climatic, hydrological previsions, nuclear tests, traffic regulation, etc.). It can
equally support and prolong the construction of the theoretical system, while
respecting its simplifications, in order to assist the theoretic step of comprehension.
This enables us to validate the behavior or certain properties of the theoretical
system, without necessarily adhering to reality.
Often, modeling finds an intermediate path between these two approaches,
knowing that unfortunately, the closer one model is to the description of reality, the
less explanatory it is and it becomes complicated; but the more explanatory a model
is, the more it moves away from the tangible reality, by deprivation.
1.2.2. The whole is greater than the sum of its parts
The systemic paradigm is partially based on the theory attributed to Durkheim:
“The whole is greater than the sum of its parts”. This expresses the fact that the set
theory is not sufficient to define the notion of system. It suggests the lack of the
fundamental notion of interaction between the different parts.
This proposition implies that by proceeding to a reductionist approach, by
reducing the problem to parts, the fullness of relationships and interdependencies
between the parts of the system is lost. This could justify the holistic approach.
However, it must be remembered that this reduction is not necessarily destructive, it
is actually the contrary.

By individualizing the successive levels of reduction of the system, this enables
us to clearly formalize the interactions between the components at the lower level,
thus permitting us to build the behavior of the system at a higher level. In a
methodological back and forth of descent and ascent between the levels, we can
consider both the simplifying reduction and the reconstruction of the interactions,
reinstating global cohesion. The emergence can then arise as a consequence of the
individual behaviors. Reduction to simplify is not sufficient, but it is not a reason not
to do it.
1.3. The notion of organization
The set theory enables us, to a large extent, to formalize the notion of a system,
at least as to its structure and its organization. In the systemic context, the notion of
structure appears as what remains invariant in the system. The structure is what
permits us, on the one hand, to define the “mold” of its constituent bodies and also
to serve as a local referential, as to what will be dynamic in the system.
This way of seeing the structure is the same in computer science, with the notion
of static structure2
of data, which defines the permanent way that information is
kept, then this information will evolve during the execution.
The notion of organization prolongs that of structure because an organization is a
structure made up of organs in interaction. It can thus possess a certain scalability.
The slight conceptual difference that we propose here between the structure and
the organization is principally what differentiates the notion of a cellular automaton
(CA) from the multi-agent systems (MAS). In fact, in a CA, the cells are connected
to each other by a permanent neighboring structure, while in an MAS, the agents are
connected to each other in a dynamic way. For example, if an agent is mobile, its
neighborhood will vary during the course of its displacement. Nevertheless, we will
often speak of spatial organization without it necessarily evolving significantly
during the study (we can always say that it has stationary dynamics).
What becomes invariable in an organization are its rules or laws that decide the
possible or applicable links between the elements. This way of organizing generally
has degrees of freedom in the construction and can become a huge combination of
possible forms.
2 As opposed to the dynamic structures of data that are defined to be able to vary in the course
of execution of the program to adapt to the data, such as a list, a tree, a graph, etc. (see for
example “HydroNetwork” explained in section 1.3.5.

Thus, the notion of organization taken in this sense is totally adapted to the
distributed approach of the notion of a system, contrary to the compartmentalized
approach. A distributed system is one formed of many components of the same class
(or of a small number of classes), having between them a flexible organization,
where the rules always give them a certain data mapping freedom. The addition or
disappearance of a component is not destructive to the system, it is but a disruption
of its dynamics.
We distinguish this type of system from a compartmentalized system, which is
characterized by a small number of components of different classes, each being
essential to its structure. A system observed at a certain level can be
compartmentalized, much like the organs of a human being, and at another level, can
be distributed: this same human being as though composed of a multitude of cells.
1.3.1. Structure and organization
The organization has many possible forms. The system can then pass from one to
the other, depending on its evolution. This notion is comparable to dynamic data
structures in computer science. For example, a binary tree doesn’t have a defined
structure in advance. It’s a particular oriented graph that has basic entities that are
nodes (vertices of the graph) and the father-son connections (the arcs of the graph or
the branch which is a path in the tree) and “rules of construction”.
Rule 1: a tree starts with a node (called a root).
Rule 2: a father-node is connected to at maximal two other son-nodes.
Rule 3: a node that doesn’t have a son is called a sheet.
We often use this organized representation that distinguishes the connection to
the son-node on the left, and the connection to the son-node on the right from the
father-node (see formula tree, Figure 1.4). With such rules, an infinity of binary
trees can be built. We can also modify a binary tree into another binary tree, by
certain operations that respect organization rules. For example, we can switch two
branches attached to the same node, we then get another binary tree.
In computer science, we often use recursive data structures to represent an
organization in memory. These structures refer to themselves. It is the most elegant
way to create dynamic data structures, that is to say structures that are created at the
time of the execution of the program and whose elements that compose it can vary
from one instant to the other. This lets us represent realistic organizations that can be
very complex. We use them for example to model complex geographical models, for
instance zonal partitions (parcels, administrative cut-outs, level curves, etc.) or

networks (roads, hydrographics, etc.). Let us first examine a few simple
organizational examples.
1.3.2. Sequential organizations
The sequential organization is one of the most common and simple. It verifies
the relationship of total order3
. The elements are organized in an indexed suite. We
can use a data structure, also called a list, which is simply a table whose i’th box
contains a reference to the i’th object. If the number of elements remains the same
and is known in advance, we use a static table which is dimensioned in the program.
This is quite rare.
Otherwise, we use a dynamic table, dimensioned in the program and resized
during the course of the program in order to be adapted to the data. If we must
manipulate these insertions, removals, departures and entries often in the sequence
of objects, like for example to model a waiting line, a line of cars, etc., we then use a
chain (also known as a chained list), formed from a sequence of links that can easily
be detached and recombined with others:
Linktype=Class
data: link data...;
NextLink: Linktype
End;
Figure 1.2. Linked list
3 A relation of the order R defined on a set E, verifies the properties of reflexivity (1), of
antisymmetry (2) and of transitivity (3). The order is complete once all of the elements are
comparable (4):
( ) ( )
( ) ( )
(1) ,
(2) , and ;
(3) , , and
(4) , , or
x E xrx
x E y E xRy yRx x y
x E y E z E xRy yRz xRz
x E y E xRy yRx
∀ ∈
∀ ∈ ∀ ∈  =
∀ ∈ ∀ ∈ ∀ ∈ 
∀ ∈ ∀ ∈

A chain is then memorized by the address of the first link of the list. To go
through all the sequential links, it is sufficient to use NextLink. This gives the
following algorithm, which is far quicker than using a table T[i], where the
processor must make an index calculation to find the address of the stored object in
box i of the table:
Procedure TreatList (Chain:LinkType)
Begin
While Chain nil do begin
{Treat Chain . data}
Chain:=Chain.NextLink
End;
End;
To insert a new link in a list after a given C link, we can use the
InsertNewLinkAfter procedure:
procedureInsertNewLinkAfter(C,NewLink:TypeLink);
Begin
if (C<>nil) and (NewLink<>nil) then begin
NewLink.NextLink:= C.NextLinkt;
C.NextLink:= NewLink;
End
End;
To remove a link from the list, which is situated after a given C link we can use
the RemoveLinkAfter, which then removes the link taken out of the chain:
Function RemoveLinkAfter(C:TypeLink): TypeLink;
Begin
if (C<>nil) and (C.NextLinkt<>nil) then begin
Result:= C.NextLink;
C. NextLink:=
C. NextLink. NextLink;//skip link
Result.NextLink:=nil //we disconnect the series
of//the chain
End
Else
Result:= nil;
End;

1.3.3. Organization in classes and partitions
Another very simple type of organization is that of a set of objects partitioned in
many classes. This is associated with an equivalence relation4
. It is of great
importance due to its simple mathematical properties. Based upon the characteristics
(stored in its attributes) and the elements or objects of a set, we can use an
equivalence relation that induces a partition5
from the set into equivalent classes for
the relation. Thus, in a given class, all the elements are linked two at a time by this
relation, but two elements from two different classes cannot be linked. For example,
the relation “... has the same task as...” is an equivalence relation that partitions the
individuals in classes based on their jobs. A computer structure technique of such a
relation can consist of defining a list of classes. Each class is itself made of a list of
elements of its class. Such an organization is then often represented by a list of lists.
If the elements change classes, then these lists can be linked lists. This avoids having
to remove elements in a table during an addition or removal process.
It should be noted that a partition is not necessarily clarified in a data structure. It
can be defined by an attribute that lets us know in which class an object belongs. For
example, the set of communes in France is partitioned into departments. It suffices
to put a “DepartmentCode” in each of these commune objects, in order to solve
the problem. Everything depends on the process associated with the partition. For
example, if we often need to extract all the communes from the same department,
we should have an explicit structure to avoid having to look each time for the
concerned objects in the database.
1.3.4. Organizations in trees
An organization is rarely completely organized or purely sequential. For
example, an arithmetic expression combining multiplications and additions, with
parentheses, even if we write it in a sequential manner, possesses an organization
which is not sequential. In fact, what is important, is the order in which we can make
these calculations to obtain a correct result.
4 A relation of equivalence R defined on a set E, verifies the properties of reflexitivity (1), of
symmetry (2) and of transitivity (3):
( ) ( )
( ) ( )
(1) , ;
(2) , ;
(3) , , and .
x E xRx
x E y E xRy yRx
x E y E z E xRy yRz xRz
∀ ∈
∀ ∈ ∀ ∈ 
∀ ∈ ∀ ∈ ∀ ∈ 
5 A partition P of a set E is a set of parts from E, separated two by two (1) and that covers E
(2)
(1) , , ;
(2) , , .
A P B P A B
x E A P x A
∀ ∈ ∀ ∈ ∩ = ∅
∀ ∈ ∃ ∈ ∈

For example, if we want to calculate the numerical expression: 1-(2+4)*(3+8),
we cannot operate in the order we read it. We must multiply before adding and
subtracting and follow the order of the parentheses.
11
6
2 4 3 8
66
1
-65
−
×
+
+
X= 1- (2+4) × (3+8)
Figure 1.3. Example of a formula tree
The organization of calculations is expressed by a binary tree, where the sheets
contain the values of the expressions. Each operation combines two nodes to obtain
the intermediary result, stored in the node at the upper level, until we obtain the final
result of -65 in the root of the tree, as shown in Figure 1.3 above.
Figure 1.4. Structure of a binary tree in order to memorize an arithmetic expression

To create such an organization, we can use the following class of object, which
is recursive, as the fields “LeftValue” and “RightValue” are also objects of
the same type.
ClassCalculationTree = class
Result:real;
LeftValue: ClassCalculationTree;
Operation: (None, Add, Subtract, Multiply, Divide);
RightValue: CalculationTree;
Procedure Calculate;
End;
If the node is a sheet, the fields “Left value” and “Rightvalue” do not
point to anything (meaning they contain the nil value). The selected operation is then
“none” and the field “result” contains a numerical value of the expression, for
example 2. If the node is not terminal, then the two fields “LeftValue” and
“RightValue” point to two lower nodes. The expression calculation is done based
on the Calculate method from the root of the tree. This method determines if the
node is terminal, in which case there is nothing to do as the result field already
contains a value; or else, it calls us to Calculate methods of LeftValue and
RightValue nodes which enables us to calculate, step by step, the result of each
lower node. Finally, we will be able to perform our own operation based on these
results stored in the Leftvalue and Rightvalue nodes.
1.3.5. Network organization
Let us now look at a geographical example such as a hydrographic network, i.e. a
river, this organization is more complex. From the source to the river mouth, the
different tributaries are connected by successive confluences. The organization is
less strict than the preceding one as the confluences are not necessarily binary. Also,
certain sections can separate and reunite downstream. We are thus no longer
presented with a tree-shaped organization. Nevertheless, this organization has strict
rules which must be verified and each section has a single direction of flow. There is
an entering extremity (upstream) and an existing extremity (downstream) in each
section. A linking point connects at least three sections. When three sections or more
are reunited, there has to be at least one entering and exiting section at the linking
point. Finally, there does not need to be a loop in the network (as in Escher’s
fountain drawn in “trompe l’oeil” style): one drop that runs in the river cannot pass
the same section many times. The network must then verify the properties of a
partial order for the relation “upstream to...”.

Figure 1.5. Network organization
Such an organization must contain a procedure to validate these rules, by
verifying each of these properties. The most complex is the one that verifies the
absence of loops. For this we try and make a total order relation (by numbering all
the sections) in a way that one given numbered section is always running towards
higher numbered sections. If there is an impossibility of construction, it is that there
is a circulation issue in the network that must be corrected.
The data structure of such geographical objects can have very varied forms and
constitutions. To be economical, we can represent it by the HydroNetwork Class
where a section is connected downstream to different sub-networks and receives
upstream tributaries, that are each the same type of network. This gives the
following structure:
TypeSection = record
Axis : array of points3D;
Number:integer;
Mediumwitdth: real;
{ etc. }
End;
ClassHydroNetwork = class
Section: TypeSection;
ConflueTowards: array of ClassHydroNetwork;
Tributaries: array of ClassHydroNetwork;
Property Sectionlength:real read CalcLength;
Function CalculDebtOfExit(ApportExt:real):real;
End;
We can see that the HydroNetworkClass structure is defined in a recursive
manner as the ConflueToward and Tributary fields refer to the

HydroNetworkClass structure itself. It has a field of data, Section, which
defines the geometry of a portion of network situated between two confluences,
ConflueVers field that references a HydroNetworkClass object, and
Tributaries that contain the dynamic table of all the sub-networks that are the
Tributaries. Thus, the SectionLength property allows us to return after
calculation to the length of the section defined by the stored points in the Axis field.
Finally, the ExitFlowCalculation allows one to calculate the exit flow of the
section from an exterior source (rain, watershed, etc.) and these exit flows are
calculated (recursively) by each of these Tributaries.
We will see in topological structures (section 3.3.6) that it is often useful in a
network to add another type of entity to that of a section that enables us to represent
the junctions between different sections, that we will call nodes.
1.3.6. Hierarchical organizations
A hierarchy is a slightly more complex organization than a partition. In fact, if
we consider a system as constituted of “elementary bricks” connected to one
another, then the system firstly appears as a set E of elements e1, e2,...en. But then
these bricks can themselves be considered as sets formed of elements from a lower
level, and so on and so forth. We then come to the notion of a hierarchy formed
from a recursive series of decompositions of a group that we partition and/or where
each part obtained is itself partitioned. This is a current organization of elements of a
system, but it already engenders great complexity because the combination of these
hierarchies is immensely larger than the parts of a set (for a set of 10 elements, there
are 2.5 billion completely different hierarchies!). We formalize these objects by
introducing the notion of a mesh that generalizes the notion of hierarchies.
DEFINITION OF A FORMAL HIERARCHY.− Hierarchy H defined on set E is a family of
parts of E that verifies:
− the set E and all the parts from E with a single element (also called singletons)
belonging to H;
− ∀A∈H, ∀B∈H, or A⊂B, or B⊂A, or A∩B=∅.
Hierarchy can be qualified as “vertical”, as it often represents overlapping levels
of administrative partitions of a territory, such as regions, departments, cantons and
communes in France. We often associate with each level of a vertical organization a
“horizontal” structure formed by neighborhoods between the elements that play an
important role in geographic modeling.

We can also associate a hierarchy with a “balanced tree” type dynamic data
structure, that is to say that all the branches are of equal length. However, the nodes
that represent the “bricks” vary in degree. The common length of all the branches
indicates the depth of the hierarchy, that is that the number of successive layers on
which we made a partition.
Level 0 (high)
Level 1
Level 2 (low)
Figure 1.6. Stratified hierarchy
Another method that is often used to structure a hierarchy is called hierarchized
layers or stratified hierarchy, more in conformity with the organization in successive
layers. The objects of each level are organized in a list, which allow us to directly
access the objects of a given level. Then each object has an ascending connection
that points to the parent object, situated at the level immediately above it. Finally,
each object has a list of descending connections that points towards its children from
the level immediately below it. Such a structure can be defined as:
ClassObjectGeo = class
IdObject: integer; // general object identifier
Data : TData; // access to its data
Geom : TGeom; // access to its geometry
IdLayer: integer; // order# in its layer
IdLevel: integer; //order # in its hierarchy
Neighbors: list of ClassObjectGeo;// topological
links
Parent: ClassObjetGeo; // reference towards its
parent
// if none)
Children : list of ClassObjetGeo;// links towards
its children
//(nil if non)
{ … here, its behavior methods … }
End;
ClassLayer = list of ClassObjetGeo;
ClassHierarchyOfLayers = List of ClassLayer;

1.3.7. The use of graph theory for complex organizations
The order structures, of equivalence or hierarchy, are not the only ones that can
organize a system. Other types of organizations that do not have particular
mathematical properties, or on the contrary, that have very complex properties, are
also used to organize a system. We then use the graph theory as a theoretical
framework to manipulate these structures. In the chapter concerning topological
relations in particular, we will see efficient data structures, based on graph theories.
Different definitions of the graph notion are presented in the literature. We also
add our own definitions, as this notion is often used later on in this work.
1.3.7.1. Elementary graph (or 1-graph)
A G elementary graph is formed from a set S of vertices and from a set A of arcs,
where each arc is a couple (si, sf) of S vertices, called the initial vertex (si) and final
vertex t (sf). In this sense, a G graph = (S,A) is equivalent to a relation R, defined on
S, in which two vertices are connected by R, if and only if these two vertices are
forming an arc of A:
∀x∈S, ∀y∈S, xRy ⇔ (x, y)∈A
In this definition, a graph is naturally oriented, which means that we distinguish
the (x, y) arc from the (y, x) arc, except when x = y. In this case, an (x, x) type arc is
called a loop. If we are not interested in the orientation of the arcs, we speak of non-
oriented graphs.
1.3.7.2. Simple graph
A simple graph is a non-oriented elementary graph without a loop. In a non-
oriented graph, the arcs are called edges. The edges in a simple graph can be defined
by (x, y) pairs of vertices. We must remember that a pair is a sub-set of two
elements, which is naturally a non-oriented notion (the (x, y) and (y, x) pairs are
identical as they contain the same elements).
Furthermore, a pair cannot be in the (x, x) form because a set written in extension
cannot contain the same element twice. What excludes the presence of loops in this
type of graph would be described as singletons. Thus, a simple S vertices graph is
defined by a sub-set of pairs of S. We can generalize this notion by taking at random
a family of parts of S, the hyper-edges, instead of the edges. We then get to the
hypergraph notion, which we will not use.

Figure 1.7. Graphic representation of two graphs
1.3.7.3. Graph (or multigraph or p-graph)
An elementary graph does not allow us to differentiate arcs with the same initial
and final vertex. Yet, this case appears in many situations. We generalize the
definition, to include this type of case.
A G graph is defined by a set S of vertices, a set A of arcs and a hitching function
f of arcs to the vertices, which is an application of A towards S×S, which to each a
arc, associates the couple formed of its initial and final vertex. Thus f(a) = (si, sf). If
there is a risk of confusion about the term “graph”, we can also use the terms
“multigraph” or “p-graph”.
We can group the arcs into classes of equivalence, a class being the set of arcs
with the same image by f, that is to say the set of the arcs hitched to the same pair of
(si, sf) vertices. The multiplicity of an a arc is the number of its class. If p is the
maximal multiplicity of the G arcs, we say also that G is a p-graph or that p is the
multiplicity of the G graph.
From this definition, an elementary graph is a graph of multiplicity 1 or a 1
graph. We can see that the different organizations defined above, lists, trees and
hierarchies, are particular graphs.
Note that the graph concept does not call upon geographic notions. A graph is a
purely relational concept that we must not confuse with a chart, even if, to reason or
to show examples, we often associate a chart with a graph.
Thus, set theory, like graph theory through the notion of a relation, gives the
essential formalization tools for the links between the components of a system.

We notice that the mathematical structures concern mostly “simple systems”, in
the sense that the elements, even if they are often of an infinite number, are at the
same time different (distinct), but identical in their properties relative to the
structure. They all have the same “behavior” with regard to operative rules. The
reality is obviously more complex as it is formed by diversified elements, as much
in content as in behavior. It must be extremely simplified so that we can use these
structures.
Nevertheless, using them allows us to correctly explain very general complex
phenomena that do not depend too much on these variations. For example, in the
study of gravitation, it is useless to know the composition, the form and the color of
the objects in action. All of these objects are only characterized by their position,
their speed and their mass, and we can use a very general mathematical theory to
formalize and explain this phenomenon, without keeping all the real complexity,
which is useless for explaining gravitation.
1.3.8. Complexity of an organization, from determinism to chaos
The organization concept appears as one of the characteristics of the complexity
notion. In fact, an organization defined by rules allows a system to progressively
construct itself, to transform itself in time with a view to adapting itself to the
changing environmental conditions, to answer goals which are themselves in
evolution. We come to this frequent property of complex systems which is
expressed by the fact that the only way to model, to foresee or to explain their form
and their state at a given time, is to reproduce the series of all steps of their
construction. This is the case in urban increase or even that of a tree (and of all
living things). This property often comes from the large number of degrees of
freedom in organization that brings with it an almost infinite combination of
possible forms, so well that at each step of the construction, a large number of
equally probable choices are admissible to construct the next step.
Nevertheless, we should note that this maximal complexity6
is not always the
result of a large freedom of the system. To illustrate this case, we can use the
example of the logistic suite, which is an interesting one due to its simplicity. We
will refer to [LAN 07] for more details. A term of the series is calculated from the
precedent by the following calculation:
xn+1 = xn + r.xn.(1 - xn) [1.1]
6 In the sense of Kolmogorov-Solomonoff-Chaitlin (KSC), that measures the complexity of a
series of numbers, by simplifying the size of the algorithm capable of producing it.

The series (x0, x1, …, xn, xn+1,…) represents the evolution of an organization in
which each term represents the state of an organization at any given time. This series
is studied here in a purely theoretical way apart from any physical modeling.
Nevertheless, it is often used to represent the evolution model of a physical size
fueled by a limited resource, such as, for example in the case of an epidemic.
In this case, xn, is a proportion of people infected in the population, on the
condition that r remains within reasonable limits. Thus the r value must not produce,
in one single iteration, a proportion of xn+1 infected population greater than 1, which
excludes chaos. In general, the construction of this organization has only one degree
of freedom, that corresponds to the choice of it is initial x0 value (after having fixed
the parameter of growth r).
Furthermore, this organization has only one deterministic rule that allows us to
calculate xn+1 from xn. This rule is a function that gives no degree of freedom for the
choice of xn+1, knowing xn. This example is remarkable, because according to the
value of r, either the complexity measure is very weak, or it is very high. In fact, for
positive r values that are small in front of 1, there is a simple way of “foreseeing”
the xn term. We can use the following formula that corresponds to the resolution of
the differential equation associated with this series. We then have:
0
.
0
1
1
with
1
n r n
x
x C
x
Ce−
−
≅
+
However, we wrote “foreseeing” and not “calculating” in order to express this
formula in that we give only an approximation of the xn value. This will be better
when the r stays close to 0. The only r value for which we know how to calculate xn
exactly, from n is the value r = 1. In this case we have:
( )2
0
1 1
n
n
x x
= − −
For r values that are greater than 1, and especially for values greater than 2.6, the
logistical series, though perfectly deterministic, becomes totally unpredictable. It is
the most simple example of deterministic chaos. The only way to calculate an xn
term, is to start from x0 and to reconstruct with the (l) rule, each term of the series
until xn. The construction algorithm is at its maximal size, which makes it an
organization of maximal complexity in the KSC sense, but above all a maximal
complexity in the sense of calculation time.

Figure 1.8. Comparison between series and logistical function
(
s
Figure 1.9. Complex behavior of the logistical series

Chapter 2
Space and Geometry
2.1. Different theories of space
Since Euclid, Pythagoras, Herodotus, Eratosthenes and other surveyors-
geographers, space has been mathematically modeled in the framework of Euclidian
geometry. In ancient times, people had perfectly witnessed boats disappear
progressively into the horizon. They had understood that the world was round.
Eratosthenes calculated the first circumference of the Earth. Nevertheless, the fact
that the Earth was round does not prevent the space in which it is placed, along with
the entire universe, from being Euclidian. This is what we had considered with
Copernicus, Keplet and Newton, way before the theory of relativity.
Nowadays, a certain number of space models are available. We cannot get them
confused. Each one corresponds to a scale level, but also to abstraction, to
properties, etc. Some of them are purely mathematical models, others are physical
models. We can therefore distinguish:
− geometry of an entire universe, modeled with general relativity, by a non-
Euclidian 4D space curved by the masses. This space is locally Euclidian (for
example at the terrestrial level, it is Euclidian, with great precision);
− terrestrial surface geometry is modeled much like a 2D curved surface put into
this Euclidian 3D space. In this framework, there are many geometric models of this
terrestrial surface that are more or less refined: the sphere, the ellipsoids and the
geoids;
− these curved surface models can be locally confused with a plane, for example
the scale of a city;

− for a lower scale (for example at a national level), this terrestrial surface (with
its relief) can be projected onto a plane (with elevated altitudes) by mathematical
transformations that minimize the deformations (for example Lambert’s conical
projection).
In fact, knowing that the Earth’s diameter measures1 40,000
6,370 km,
2
R
π
= ≅ we
can easily calculate by locally merging the surface of the sphere with its projection
on the tangent plane in this area, deviated to a point 10 km from the tangency point.
We made an error in the horizontal distance of less than 10 cm (which is halved with
a secant plane that splits the errors), perhaps a relative error less than a millionth.
With more sophisticated projections, we achieve a far greater precision.
The most recent theories, such as relativity, do not invalidate the more ancient
theories. However we now know the limits and the scales of validity of these
different theories. We know that the Earth’s surface can be modeled very well using
a plane surface, as long as we stay on a local scale (a piece of land or even a city).
We know that the Earth’s surface can be modeled very correctly itself using a piece
of spherical surface, which gives sufficient precision for the majority of usages
(urbanism, transportation, navigation, etc.). Also, for more precise or for more
global usages, we must use an ellipsoidal type, where we can model a 2D location
with a lower precision to the decameter. Here we are still reasoning with a Galileo-
Newtonian model where time is absolute and the space’s referential is linked to the
Earth (geodesic referential). For finer usages, such as the construction of big works
(dams, high-speed railways) where the verticality of the zero altitude is important,
we must refer to the geoid, that follows gravitational irregularities of the Earth very
closely, by a lightly bumpy surface, that can reach more than several dozen meters
of altitude gaps with the elipsoid. Finally, with GPS2
navigation or geodesic
measurements DGPS3
lower 3D precision is obtained to a decimeter across the
entire globe at the cost of relativist calculations, which take into account the
fundamental link between time and space. In fact, the principle of positioning a GPS
receptor, is based on the extremely precise synchronization of time between the
clocks launched into the system’s 24 satellites. Each satellite emits “top”
synchronized schedules. A GPS receptor picks up these signals with a very light jet
lag, due to light’s travel time, which allows us to calculate the distances between
1 We use this formula to calculate and to remember that Earth’s circumference historically
measures exactly 40,000 km, which is not the result of a physical measure, but results from
the first definition of the meter, in the framework of a spherical representation of Earth that
remains sufficient for the calculation we wish to do here.
2 GPS: Global Positioning System.
3 DGPS: differential GPS.

Space and Geometry 31
visible satellites and the receptor. Thus, by knowing the precise position of satellites,
the receptor knows its own position.
To avoid deformations, we should represent the Earth’s curved surface on a curved
support. However, the cartographer prefers slightly deforming reality and projecting
the world on a plane. We then obtain maps of the world and projection maps that
evidently present a number of practical advantages. We need a good understanding
of geometry to conserve some fundamental properties depending on the usage that
we want from the map. For example, for navigation the angles must be respected in
order to calculate the capes we need to follow, a map with this property is also
known as compliant. In this case the map’s local scale varies from one point to
another. If on the contrary we wish to preserve the scales, the projection is then
known as equivalent. We can also construct projections (known as aphylactics) that
compromise between these two properties. We can obviously not keep all the initial
geometric properties when we deform a surface. We will not describe the geometry
of projections here, as it would lead us to off-topic developments. The calculations
reveal geodetics and are now well integrated to the actual treatment tools of
geographic information that are known as GIS (geographic information systems).
Before going into more detail with the computer science data mapping of spatial
information, we must reflect upon the actual concept of space, by examining how
the different mathematical models that mathematicians or physicians have
developed since ancient times and that have enabled this conception to evolve. This
will allow us to discuss the geographic pertinence of the usage of various
mathematical models to represent space, make geometric calculations or structure
objects in space.
2.1.1. Euclidian models
First of all, we must remember what a Euclidian model is; when there is an abuse
of language, we often confuse two different mathematics: Euclidian space and affine
space. A Vn
Euclidian space is a vectorial space, which is therefore a group of
vectors. The n exponent indicates the dimension of space. The Euclidian qualifier
comes from the fact that this space (on top of the addition of vectors and the
multiplication of a vector by a real number) is equipped with a scalar product (that
associates a real number with two vectors) from which we construct the notion of
the length of an X vector, that we also call the X norm, written ||X||, that must be
positive or null. With the norm, we construct a space metric using d(X, Y) = ||YX||.
However, the physical space that surrounds us, the one that helps the geographer
to define his location, cannot be modeled by a Euclidian space as defined above, as
locations are not vectors but points. In fact, the adapted model corresponds to the

notion of linear En
space, as it is really a set of points. Nevertheless, every linear
space is associated to a vectorial Euclidian space, for example V3
is the 3D space,
because all V

vectors of V3
define a translation, that transforms each P point in
linear space in a unique P’ point, with '
PP V
=

 
. Thus, each E3
affine space point is
defined by its coordinates with relation to a landmark (O, X, Y, Z) where O is the
starting point and (X, Y, Z) a V3
is a vectorial space base. One must notice that we
can establish a bijection4
between the points and the vectors, because at each V3
point we can uniquely associate the V OP
=



vector. This might be the reason why
there is a widely spread confusion between Euclidian space and linear space. A
fundamental conceptual difference between these two types of spaces is
characterized by the fact that a vector is invariant by translation, as opposed to a
point.
The Euclidian model refers as much to Euclidian vectorial space as to linear
space of the points that are associated with it, because what unites these two models
is the chosen metric. A Euclidian model thus implies the method which the scalar
product is defined in order to give the metric of space.
2.1.2. Metric spaces
In fact, not all metrics are Euclidian. This signifies that the metric space notion is
a larger concept than the notion of Euclidian space. Formally, a metric space is a set
E in which we define a d application that associates with each pair of E elements, a
positive or null real number, written d(x, y) which is the distance between x and y.
This d application must verify the three following elementary properties:
( ) ( )
( )
( ) ( )
( ) ( ) ( )
1) , 0
2) , ,
3) , , ,
x y d x y
d x y d y x
d x z d x y d y z
= ⇔ =
=
≤ +
The distance will or will not be Euclidian depending on whether or not we can
associate it with a scalar product. The third property, also known as triangular
inequality, indicates what we call the distance between two A and B points that
verifies the minimality condition in a sense that the distance between A and B is
always the length of the shortest trajectory between these two points. This shortest
trajectory is called a geodesic and this definition is easily generalized with curved
spaces (such as the surface of a sphere). Here we must insist on the ambiguity of the
4 A bijection between two A and B sets is a one on one relation, which is to say that at each A
element, a B element is associate and vice-versa.

term “distance” that we employ in geography. Already in mathematics, “distance”
means both the function (that associates a positive or null real value with a pair of
points) and the result of this function (which is the number associated with two fixed
A and B points). However, in geography, this term is mostly used to signify the
length of a particular distance we cover in order to get from one place to another and
by sometimes indicating a particular mode of displacement. This is how J. Levy, for
example, speaks of “pedestrian metrics”. We then come to a considerable confusion
between a metric concept in the mathematical sense that does not depend on the
global structure of space, to a geographic metric concept, that would only be the
measure of a particular path associated with conditions or particular usages along
this path. In order to raise this terminological ambiguity, from now on we will use
“metric” for the mathematical concept and that of distance (better yet, of
geographical distance) for the “soft” version of the concept (length of a journey).
To mathematically model the geographic distance in relation to a network, we
must consider the space constituted of a set of lines (called sections) that are
continuous and measurable (in length), that only connect with each other at their
extremities (the junction point called nodes). We will name the group of nodes and
sections forming a geometric structure, network (whereas a graph does not include
geometry, it is only the relation expression). This network can always, as we will see
it, be associated to a simple graph, (which is to say a non-oriented graph, without
multiple edges and loops). This graph can or cannot be planar, as the geometric
crossing of two network sections does not necessarily signify that there is a junction
between these lines to form a node. This can represent a crossing, i.e. a bridge. We
construct this graph in a way by which the summits are network nodes and we
define an edge (i, j) between two distinct vertices when there is at least one section
between the two associated nodes to i and j.
We can define real metrics on the set of vertices of such a graph, where the d(i, j)
distance between i and j summits is the shortest length joining i and j. This length is
either the number of the path’s edges if we confine ourselves to the graph, either in a
more realistic way, the sum of the lengths of the edges, the length of the (i, j) edges
being the length of the shortest section that joins i to j. We can still make the model
more sophisticated by considering as space, not only the graph’s vertices but also the
group of all of the networks’ points forming the sections’ continuous lines. In this
reticular R space (which is 1D), we can associate to each pair of (A, B) of R points,
the length of the shortest path that connects these two points on the network. This
length then correctly models the geographic distance in the reticular R space. This
distance has truly metrical properties and presents a good degree of realism. Later
we will come back to geographic network structures.

2.1.3. Normed spaces
We have seen that metrics, which are defined by very general properties, are not
necessarily Euclidian, which is to say connected to a scalar product. They can be
constructed, for example, from a norm or from a semi-norm. Let us keep in mind
that a norm, defined in vectorial space, lets us measure the length of vectors.
Formally, a norm is an application where each x vector of a vectorial V space,
associates a real number, p(x) verifying the following 4 elementary properties, for all
λ real numbers and all x vectors of V:
– positivity: ( ) 0
p x ≥
– homogeneity: ( ) ( )
p x p x
λ λ
=
– triangular inequality: ( ) ( ) ( )
p x y p x p y
+ ≤ +
– separation: if p(x)=0 then x = 0
A vectorial V space, equipped with a norm, is known as a normed space. When
we manipulated a particular norm, always the same one, we generally write it as x .
If the fourth property is not verified, we only have what we call a semi-norm.
In the vectorial Rn
space, real coordinated n vectors, a classic example of norms
is constituted of the family (infinite) of the following norms that depend on the real
p parameter, defined as:
1
1
n p
p
i
p
i
x x
=
 
 
=
 
 

The vectorial space, equipped of such a norm, is called space p
l . All norms
allow us to immediately construct a metric by:
( , )
p p
d x y y x
= −
whence, with x and y which are two coordinate vectors ( )
1 2
, ,..., n
x x x and
( )
1 2
, ,..., n
y y y :

1
1
( , )
n p
p
p i i
i
d x y y x
=
⎛ ⎞
⎜ ⎟
= −
⎜ ⎟
⎝ ⎠
∑
This metric is only Euclidian for p=2 where we find the:
( ) ( )2
2
1
,
n
i i
i
d x y y x
=
= −
∑
usual Euclidian metric. It is the only metric that is able to correctly measure distance
in straight lines between two points of our physical space on Earth, like the
measurement of an object (with n=3).
On top of the p=2 value, some other p values, particularly like p=1 and p = ∞ ,
provide examples of non-Euclidian p
l space, and yet that possess a certain utility in
geographic modeling.
2.1.3.1. Manhattan metric
For p=1 the norm of a vector is simply the sum of absolute values of its
coordinates. This gives the l1
space, whose noted d1 metric is spelled:
( )
1
1
,
n
i i
i
d x y y x
=
= −
∑
In geography, it is sometimes used under the name of “Manhattan distance”.
2.1.3.2. Max metric
When we extend p towards infinity, we can show that the distance can almost be
expressed as:
( ) { }
1,
, i i
i n
d x y y x
Max
∞
=
= −

We also do not believe that a single Euclidian metric exists, corresponding to
geometry, that would be traced in real space. In fact, we can mathematically define
as many Euclidian metrics as we want. For example, ( ) ( )2
1
,
n
i i i
i
d x y y x
Λ λ
=
= −

( )
1 2
, ,..., ,
n
Λ λ λ λ
= where the series of positive λi coefficients weighs each
dimension of space, is an infinite family of metrics that are all Euclidian.
2.1.4. Pseudo-Euclidian spaces
On the other hand, if we give λi values that could be negative, we risk getting a
negative sum under the square root. We are then brought to a situation in theory of
the restricted relativity, where we have a ( )
1, 1, 1, 1
Λ = − − − weighting. The first
dimension being time, the others are of space. We are then forced to generalize the
notion of a complex numbered distance. Thus there is a scalar product, but the
distance between the two points can be imaginary. Such a space will be called
pseudo-Euclidian.
2.1.4.1. Restricted relativity or how to fuse the concepts of time and space
In the theory of relativity (restricted), we put ourselves in space (as Minkowski
said it) in 4D. The Poincaré group, is defined by the fact that it leaves the s distance
to the square from two points (spatio-temporal) invariable, which is translated as:
( )
2 2 2 2 2
s t x y z
= − + + [2.1]
where the three space coordinates x, y and z are expressed in meters and the t
coordinate, all while being connected to time, also represents a distance expressed in
meters. In fact, it expresses light-time5
which is the distance travelled by light during
the interval of a given time. We chose the light-meter as a unit of time which is the
time used by light to travel a meter. With these units, the speed of light c is equal
to 1. The four dimensions of the space of relativity are homogenous and are all
expressed in the same unit, the meter. It is in fact fundamental so that a space is
physically consistent, that the different dimensions of this space be of the same
nature and expressed with the same unit. If this was not the case, what would signify
a rotation, for example, in this space?
5 If we keep the second as a unit of time, we must replace t² by c²t² in formula [2.1].

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THE SCHOOL OF VENICE.
The conquests, commerce and possessions of Venice in the Levant,
and thence its uninterrupted intercourse with the Greeks, give
probability to the conjecture, that Venetian art drew its origin from
the same source, and that the first institution of a company, or, as it
is there called, a School (Schola) of Painters, may be dated up to the
Greek artists who took refuge at Venice from the fury of the
Iconoclasts at Constantinople. The choice of its Patron, which was
not St. Luke, but Sta
. Sophia, the patroness of the first temple at
that time, and prototype of St. Mark's, distinguishes it from the rest
of the Italian Schools. Anchona, the vulgar name of a picture in the
technic language, the statutes,[134] and documents of those times,
is evidently a depravation of the Greek Eikon. The school itself is of
considerable antiquity; its archives contain regulations and laws
made in 1290, which refer to anterior ones; and though not yet
separated from the mass of artisans, its members began to enjoy
privileges of their own.
In various cities of the Venetian State we meet with vestiges of art
anterior in date[135] to the relics of painting and mosaic in the
metropolis, which prove that it survived the general wreck of society
here, as in other parts of Italy. Of the oldest Venetian monuments,
Zanetti has given a detailed account, with shrewd critical conjectures
on their chronology; though all attempts to discriminate the nearly
imperceptible progress of art in a mass of works equally marked by
dull servility, must prove little better than nugatory; for it does not
appear that Theophilus of Byzantium, who publicly taught the art at
Venice about 1200, or his Scholar Gelasio[136], had availed
themselves of the improvements made in form, twenty years before,
by Joachim the Abbot, in a picture of Christ. Nor can the notice of

Vasari, who informs us that Andrea Tafi repaired to Venice to profit
by the instructions of Apollonios in mosaic, prove more than that,
from the rivalship of Greek mechanics, that branch of art was
handled with greater dexterity there than at Florence, to which place
he was, on his return, accompanied by Apollonios. The same torpor
of mind continued to characterise the succeeding artists till the first
years of the fourteenth century, and the appearance of Giotto, who,
on his return from Avignon 1316, by his labours at Padua, Verona,
and elsewhere in the state, threw the first effectual seeds of art, and
gave the first impulse to Venetian energy and emulation[137] by
superior example.
He was succeeded by Giusto, surnamed of Padova, from residence
and city rights, but else a Florentine and of the Menabuoi. To
Padovano, Vasari ascribes the vast work of the church of St. John
the Baptist; incidents of whose life were expressed on the altar-
piece. The walls Giusto spread with gospel history and mysteries of
the Apocalypse, and on the Cupola a glory filled with a consistory of
saints in various attire: simple ideas, but executed with incredible
felicity and diligence. The names 'Joannes Antonius de Padova,'
formerly placed over one of the doors, as an ancient MS. pretends,
related probably to some companions of Giusto, fellow pupils of
Giotto, and show the unmixed prevalence of his style, to which
Florence itself had not adhered with more scrupulous submission,
beyond the middle of the century, and the less bigoted imitation of
Guarsiento, a Padovan of great name at that period, and the leader
of Ridolfi's history. He received commissions of importance from the
Venetian senate, and the remains of his labours in fresco and on
panel at Bassano and at the Eremitani of Padova, confirm the
judgment of Zanetti, that he had invention, spirit, and taste, and
without those remnants of Greek barbarity which that critic pretends
to discover in his style.
Of a style still less dependant on the principles of Giotto, are the
relicks of those artists whom Lanzi is willing to consider as the
precursors of the legitimate Venetian schools, and whose origin he

dates in the professors of miniature and missal-painting, many
contemporary, many anterior to Giotto. The most conspicuous is
Niccolo Semitecolo, undoubtedly a Venetian, if the inscription on a
picture on panel in the Capitular Library at Padova be genuine, viz.,
Nicoleto Semitecolo da Venezia, 1367. It represents a Pietà, with
some stories of S. Sebastian, in no contemptible style: the nudities
are well painted, the proportions, though somewhat too long, are
not inelegant, and what adds most to its value as a monument of
national style, it bears no resemblance to that of Giotto, which,
though it be inferior in design, it equals in colour. Indeed the silence
of Baldinucci, who annexes no Venetian branch to his Tuscan
pedigree of Art, gives probability to the presumption, that a native
school existed in the Adriatic long before Cimabue.
A fuller display of this native style, and its gradual approaches to the
epoch of Giorgione and Tizian, were reserved for the fifteenth
century: an island prepared what was to receive its finish at Venice.
Andrea da Murano, who flourished about 1400, though still dry,
formal, and vulgar, designs with considerable correctness, even the
extremities, and what is more, makes his figures stand and act.
There is still of him at Murano in S. Pier Martire, a picture, on the
usual gold ground of the times, representing, among others, a Saint
Sebastian, with a Torso, whose beauty made Zanetti suspect that it
had been copied from some antique statue. It was he who formed to
art the family of the Vivarini, his fellow-citizens, who in uninterrupted
succession maintained the school of Murano for nearly a century,
and filled Venice with their performances.
Of Luigi, the reputed founder of the family, no authentic notices
remain. The only picture ascribed to him, in S. Giovanni and Paolo,
has, with the inscription of his name and the date 1414, been
retouched.[138] Nor does much more evidence attend the names of
Giovanni and Antonio de' Vivarini, the first of which belonged
probably to a German, the partner of Antonio,[139] who is not heard
of after 1447, whilst Antonio, singly or in society with his brother

Bartolommeo Vivarini, left works inscribed with his name as far as
1451.
Bartolommeo, probably considerably younger than Antonio, was
trained to art in the principles before mentioned, till he made himself
master of the new-discovered method of oil-painting, and towards
the time of the two Bellini became an artist of considerable note. His
first picture in oil bears the date of 1473; his last, at S. Giovanni in
Bragora, on the authority of Boschini, that of 1498; it represents
Christ risen from the grave, and is a picture comparable to the best
productions of its time. He sometimes added A Linnel Vivarino to his
name and date, allusive to his surname.
With him flourished Luigi, the last of the Vivarini, but the first in art.
His relics still exist at Venice, Belluno, Trevigi, with their dates; the
principal of these is in the school of St. Girolamo at Venice, where, in
competition with Giovanni Bellini, whom he equals, and with Vittore
Carpaccia, whom he surpasses, he represented the Saint caressing a
Lion, and some monks who fly in terror at the sight. Composition,
expression, colour, for felicity, energy, and mellowness, if not above
every work of the times, surpass all else produced by the family of
the Vivarini.
At the beginning of the century, Gentile da Fabriano, styled Magister
Magistrorum, and mentioned in the Roman School, painted, in the
public palace at Venice, a naval battle, now vanished, but then so
highly valued that it procured him an annual provision, and the
privilege of the Patrician dress. He raised disciples in the state:
Jacopo Nerito, of Padova, subscribes himself a disciple of Gentile, in
a picture at S. Michele of that place, and from the style of another in
S. Bernardino, at Bassano, Lanzi surmises that Nasocchio di Bassano
was his pupil or imitator. But what gives him most importance, is the
origin of the great Venetian School under his auspices, and that
Jacopo Bellini, the father of Gentile and Giovanni, owned him for his
master. Jacopo is indeed more known by the dignity of his son's than
his own works, at present either destroyed, in ruins, or unknown.
What he painted in the church of St. Giovanni at Venice, and, about

1456, at the Santo of Padova, the chapel of the family Gattamelata,
are works that exist in history only. One single picture, subscribed by
his name, Lanzi mentions to have seen in a private collection,
resembling the style of Squarcione, whom he seems to have
followed in his maturer years.
A name then still more conspicuous, though now nearly obliterated,
is that of Jacopo, or as he is styled Jacobello, or as he wrote himself,
Jacometto del Fiore, whose father Francesco del Fiore, a leader of
art in his day, was honoured with a monument and an epitaph in
Latin verse at S. Giovanni and Paolo: of him it is doubtful whether
any traces remain, but of the son, who greatly surpassed him,
several performances still exist, from 1401 to 1436. Vasari has
wantonly taxed him with having suspended all his figures, in the
Greek manner, on the points of their feet: the truth is, that he was
equalled by few of his contemporaries, for few like him dared to
represent figures as large as life, and fewer understood to give them
beauty, dignity, and that air of agility and ease, which his forms
possess; nor would the lions in his picture of Justice at the
Magistrato del Proprio, have shared the first praise, had not the
principal figures, in subservience to the time, been loaded with tinsel
ornament and golden glitter.
Two scholars of his are mentioned: Donato, superior to him in style,
and Carlo Crivelli, of obscure fame, but deserving attention for the
colour, union, grace, and expression, of the small histories in which
he delighted.
The ardour of the capital for the art was emulated by every town of
the state; all had their painters, but all did not submit to the
principles of Venice and Murano. At Verona the obscure names of
Aldighieri and Stefano Dazevio, were succeeded[140] by the vaunted
one of Vittore Pisanello, of S. Vito: though accounts grossly vary on
the date in which he flourished, and the school from which he
sprang, that his education was Florentine is not improbable, but
whoever his master, fame has ranked him with Masaccio as an

improver of style. His works at Rome and Venice, in decay at the
time of Vasari, are now no more; and fragments only remain of what
he did at Verona. S. Eustachio caressing a Dog, and S. Giorgio
sheathing his Sword and mounting his horse, figures extolled to the
skies by Vasari, are, with the places which they occupied, destroyed:
works which seem to have contained elements of truth and dignity in
expression with novelty of invention, and of contrast, style, and
foreshortening in design: a loss so much the more to be lamented,
as the remains of his less considerable works at S. Firmo and
Perugia, far from sanctioning the opinion which tradition has taught
us to entertain of Pisano, are finished indeed with the minuteness of
miniature, but are crude in colour, and drawn in lank and emaciated
proportions. It appears from his works, that he understood the
formation, had studied the expression, and attempted the most
picturesque attitudes of animals. His name is well known to
antiquaries, and to the curious in coins, as a medallist, and he has
been celebrated as such by many eminent pens of his own and the
subsequent century.[141]
From the crowd[142] of obscure contemporary artists, which the
neighbouring Vicenza produced, the name of Marcello, or as Ridolfi
calls him Gio. Battista Figolino, deserves to be distinguished: a man
of original manner, whose companion, in variety of character,
intelligence of keeping, landscape, perspective, ornament, and
exquisite finish, will not easily be discovered at Venice, or elsewhere
in the State, at that period; and were it certain that he was anterior
to the two Bellini, sufficiently eminent to claim the honours of an
epoch in the history of Art: in proof of which Vicenza may still
produce his Epiphany in the church of St
. Bartolommeo.
But the man who had the most extensive influence on Art, if not as
the first artist, as the first and most frequented teacher, was
Francesco Squarcione,[143] of Padova; in whose numerous school
perhaps originated that eclectic principle which characterised part of
the Adriatic and all the Lombard schools. Opulent and curious, he
not only designed what ancient art offered in Italy, but passed over

to Greece, visited many an isle of the Archipelago in quest of
monuments, and on his return to Padova formed, from what he had
collected, by copy or by purchase, of statues, basso-relievos, torsos,
fragments, and cinerary urns, the most ample museum of the time,
and a school in which he counted upwards of 150 students, and
among them Andrea Mantegna, Marco Zoppo, Girolamo Schiavone,
Jacopo Bellini.
Of Squarcione, more useful by precept than by example, little
remains, and of that little, perhaps, not all his own. From the variety
of manner observable in what is attributed to him, it may be
suspected that he too often divided his commissions among his
scholars; such as some stories of St
. Francis, in a cloister of his
church, and the miniatures of the Antifonario in the temple della
Misericordia, attributed by the vulgar to Mantegna. Only one
indisputably genuine, though retouched work of his, is mentioned by
Lanzi; which, in various compartments, represents different saints,
subscribed 'Francesco Squarcione,' and conspicuous for felicity of
colour, expression, and perspective.
These outlines of the infancy of Venetian art show it little different
from that of the other schools hitherto described; slowly emerging
from barbarity, and still too much busied with the elements to think
of elegance and ornament. Even then, indeed, canvass instead of
panels was used by the Venetian painters; but their general vehicle
was, a tempera, prepared water-colour: a method approaching the
breadth of fresco, and friendly to the preservation of tints, which
even now retain their virgin purity; but unfriendly to union and
mellowness. It was reserved for the real epoch of oil-painting to
develope the Venetian character, display its varieties, and to
establish its peculiar prerogative.
Tiziano, the son of Gregorio Vecelli, was born at Piave, the principal
of Cadore on the Alpine verge of Friuli, 1477.[144] His education is
said to have been learned, and Giov. Battista Egnazio is named as
his master in Latin and Greek;[145] but his proficiency may be

doubted, for if it be true that his irresistible bent to the art obliged
the father to send him in his tenth year to the school of Giov. Bellini
at Venice, he could be little more than an infant when he learnt the
rudiments under Sebastiano Zuccati.[146]
At such an age, and under these masters, he acquired a power of
copying the visible detail of the objects before him with that
correctness of eye and fidelity of touch which distinguishes his
imitation at every period of his art. Thus when, more adult, in
emulation of Albert Durer, he painted at Ferrara[147] Christ to whom
a Pharisee shows the tribute money, he out-stript in subtlety of
touch even that hero of minuteness: the hair of the heads and hands
may be counted, the pores of the skin discriminated, and the
surrounding objects seen reflected in the pupils of the eyes; yet the
effect of the whole is not impaired by this extreme finish: it
increases it at a distance, which effaces the fac-similisms of Albert,
and assists the beauties of imitation with which that work abounds
to a degree seldom attained, and never excelled by the master
himself, who has left it indeed as a single monument, for it has no
companion, to attest his power of combining the extremes of finish
and effect.
GIACOMO ROBUSTI, SURNAMED IL TINTORETTO.
1512-1594.
It might almost be said that vice is the virtue of the Venetian
school, because it rests its prerogative on despatch in execution, and
therefore is proud of Tintoretto, who had no other merit.[148] Such,
in speaking of the great genius before us, is the equally rash,
ignorant, unphilosophic verdict of a man exclusively dubbed The
Philosophic Painter.

G. Robusti of Venice was the son of a dyer, who left him that
byname as an heir-loom.[149] He entered the school of Tiziano when
yet a boy; but he, soon discovering in the daring spirit of his nursling
the symptoms of a genius which threatened future rivalship to his
own powers, with that suspicious meanness which marks his
character as an artist, after a short interval, ordered his head pupil,
Girolamo Dante, to dismiss the boy; but as envy generally defeats its
own designs, the uncourteous dismissal, instead of dispiriting,
roused the energies of the heroic stripling, who, after some
meditation on his future course, and comparing his master's
superiority in colour with his defects in form, resolved to surpass him
by an union of both: the method best suited to accomplish this he
fancied to find in an intense study of Michael Angelo's style, and
boldly announced his plan by writing on the door of his study, THE
DESIGN OF M. ANGELO, AND THE COLOUR OF TIZIAN.
But neither form nor colour alone could satisfy his eye; the
uninterrupted habit of nocturnal study discovered to him what
Venice had not yet seen, not even in Giorgione, if we may form an
opinion from what remains of him—the powers of that ideal
chiaroscuro which gave motion to action, raised the charms of light,
and balanced or invigorated effect by dark and lucid masses
opposed to each other.
The first essays of this complicated system, in single figures, are
probably the frescoes of the palace Gussoni;[150] and in numerous
composition, the Last Judgement, and its counterpart, the Adoration
of the Golden Calf, in the church of Sta
. Maria dell' Orfo.
It is evident that the spirit of Michael Angelo domineered over the
fancy of Tintoretto in the arrangement of the Last Judgement,
though not over its design; but grant some indulgence to that, and
the storm in which the whole fluctuates, the awful division of light
and darkness into enormous masses, the living motion of the agents,
notwithstanding their frequent aberrations from their centre of
gravity,[151] and the harmony that rules the whirlwind of that

tremendous moment, must for ever place it among the most
astonishing productions of art. Its sublimity as a whole triumphs
even over the hypercriticisms of Vasari, who thus describes it:
—Tintoretto has painted the Last Judgement with an extravagant
invention, which, indeed, has something awful and terrible,
inasmuch as he has united in groups a multitudinous assemblage of
figures of each sex and every age, interspersed with distant views of
the blessed and condemned souls. You see likewise the boat of
Charon, but in a manner as novel and uncommon as highly
interesting. Had this fantastic conception been executed with a
correct and regular design, had the painter estimated its individual
parts with the attention which he bestowed on the whole, so
expressive of the confusion and the tumult of that day, it would be
the most admirable of pictures. Hence he who casts his eye only on
the whole, remains astonished, whilst to him who examines the
parts it appears to have been painted in jest.
In the Adoration of the Golden Calf, the counterpart in size of the
Last Judgement, Tintoretto has given full reins to his invention; and
here, as in the former, though their scanty width does not very
amicably correspond with their height, which is fifty feet, he has
filled the whole so dexterously that the dimension appears to be the
result of the composition. Here too, as in the Transfiguration of
Raffaelle, some short-sighted sophist may pretend to discover two
separate subjects and a double action; for Moses receives the tables
of the decalogue in the upper part, whilst the idolatrous ceremony
occupies the lower; but the unity of the subject may be proved by
the same argument which defended and justified the choice of
Sanzio. Both actions are not only the offspring of the same moment,
but so essentially relate to each other that, by omitting either,
neither could with sufficient evidence have told the story. Who can
pretend to assert, that the artist who has found the secret of
representing together two inseparable moments of an event divided
only by place, has impaired the unity of the subject?

Nowhere, however, does the genius of Tintoretto flash more
irresistibly than in the Schools of S. Marco and S. Rocco, where the
greater part of the former and almost the whole of the latter are his
work, and exhibit in numerous specimens, and on the largest scale,
every excellence and every fault that exalts or debases his pencil:
equal sublimity and extravagance of conception; purity of style and
ruthless manner; bravura of hand with mental dereliction; celestial
or palpitating hues tacked to clayey, raw, or frigid masses; a
despotism of chiaroscuro which sometimes exalts, sometimes
eclipses, often absorbs subject and actors. Such is the catalogue of
beauties and defects which characterize the Slave delivered by St.
Marc; the Body of the Saint landed; the Visitation of the Virgin; the
Massacre of the Innocents; Christ tempted in the Desert; the
Miraculous Feeding of the Crowd; the Resurrection of the Saviour;
and though last, first, that prodigy which in itself sums up the whole
of Tintoretto, and by its anomaly equals or surpasses the most
legitimate offsprings of art, the Crucifixion.[152]
It is singular that the most finished and best preserved work of
Tintoretto should be one which he had least time allowed him to
terminate—the Apotheosis of S. Rocco in the principal ceiling-piece
of the Schola, conceived, executed, and presented, instead of the
sketch which he had been commissioned with the rest of the
concurrent artists to produce for the examination of the fraternity: a
work which equally strikes by loftiness of conception, a style of
design as correct as bold, and a suavity of colour which entrances
the eye. Though constructed on the principles of that sotto in su,
then ruling the platfonds and cupolas of upper Italy, unknown to or
rejected by M. Angelo, its figures recede more gradually, yet with
more evidence, than the groups of Correggio, whose ostentatious
foreshortenings generally sacrifice the actor to his posture.
That Tintoretto acquired, during his stay with or after his dismissal
from the study of Tiziano's principles, the power of representing the
surface and the texture of bodily substance with a truth bordering
on illusion, is proved with more irresistible because more copious

evidence, in the picture of the Angelic Salutation; though it cannot
be denied that the admiration due to the magic touch of the
paraphernalia is extorted at the expense of the essential parts:
Gabriel and Maria are little more than foils of her husband's tools; for
their display, the artist's caprice has turned the solemn approach of
the awful messenger into boisterous irruption, the silent recess of
the mysterious mother into a public dismantled shed, and herself
into a vulgar female. Nowhere would the superiority of refined over
vulgar art, of taste and judgment over unbridled fancy, have
appeared more irresistibly than in the sopraporta by Tiziano on the
same subject and in the same place, had that exquisite master been
inspired more by the sanctity of the subject than the lures of courtly
or the ostentatious bigotry of monastic devotion. If Maria was to be
rescued from the brutal hand that had travestied her to the mate of
a common labourer, it was not to be transformed to a young abbess,
elegantly devout, submitting to canonization, amongst her delicate
lambs; if the angel was not to rush through a shattered casement on
a timid female with a whirlwind's blast, the waving grace and calm
dignity of his gesture and attitude, ought to have been above the
assistance of theatrical ornament; nor should Palladio have been
consulted to construct classic avenues for the humble abode of pious
meditation. It must however be owned that we become reconciled to
this mass of factitious embellishments by a tone which seems to
have been inspired by Piety itself; the message whispers in a
celestial atmosphere,
Θειη ἀμφεχυτ' ὀμφη—
and so forcibly appears its magic effect to have influenced Tintoretto
himself, ever ready to rush from one extreme to another, that he
imitated it in the Annunciata of the Arimani Palace:[153] not without
success, but far below the mannerless unambitious purity of tone
that pervades the effusion of his master, and of which he himself
gave a blazing proof in the Resurrection of the Saviour,—a work in
which sublimity of conception, beauty and dignity of form, velocity
and propriety of motion, irresistible flash, mellowness and freshness

of colour, tones inspired by the subject, and magic chiaroscuro, less
for mastery strive, than relieve each other and entrance the
absorbed eye.

FOOTNOTES
[134] Thus in an order of the Justiziarii we read: Mcccxxii.
Indicion Sexta die primo de Octub. Ordenado e fermado fo per
Misier Piero Veniero per Miser Marco da Mugla Justixieri Vieri, lo
terzo compagno vacante. Ordenado fo che da mo in avanti alguna
persona si venedega come forestiera non osa vender in Venexia
alcuna Anchona impenta, salvo li empentori, sotto pena, c. Salvo
da la sensa, che alora sia licito a zaschun de vinder anchone infin
chel durerà la festa, c. And a picture in the church of S.
Donato at Murano, has the following inscription: Corendo Mcccx.
indicion viii. in tempo de lo nobele homo Miser Donato Memo
honorando Podestà facta fo questa Anchona de Miser S. Donato.
[135] In the church at Cassello di Sesto, which has an abbey
founded in 762, there are pictures of the ninth century.
[136] Gelasio di Nicolo della Masuada di S. Giorgio, was of
Ferrara, and flourished about 1242. Vid. Historia almi Ferrariensis
Gymnasii, Ferraria, 1735.
[137] At that time he painted in the palace of Cari della Scala at
Verona, and at Padoua a chapel in the church 'del Sarto;' he
repeated his visit in the latter years of his life to both places. Of
what he did at Verona no traces remain, but at Padoua the
compartments of Gospel histories round the Oratorio of the
Nunziata all' Arena, by the freshness of the fresco and that
blended grace and grandeur peculiar to Giotto, still surprise.
[138] Fiorillo has confounded this questionable name with the real
one of Luigi, who painted about 1490.—See Fiorillo Geschichte, ii.
p. 11.
[139] In S. Giorgio Maggiore is a St. Stephen and Sebastian, with
the inscription:
1445.
Johannes de Alemania
et Antonius de Muriano.
P.
from which, another picture at Padova, inscribed Antonio de
Muran e Zohan Alamanus pinxit, and some traces of foreign style

where his name occurs, Lanzi suspects that the inscription in S.
Pantaleone, Zuane, e Antonio da Muran, pense 1444, on which
the existence of Giovanni is founded, means no other than the
German partner of Antonio.
[140] In no instance seems Vasari to have given a more decisive
proof of his attachment to the Florentine school, than by building
the fame of Pisano on having been the pupil of Andrea del
Castagno, and having been allowed to terminate the works which
he had left unfinished behind him about 1480; an anachronism
the more absurd as the Commendator del Pozzo was possessed of
a picture by Pisano, inscribed 'Opera di Vittor Pisanello de San V.
Veronese, mccccvi.' a period at which probably Castagno was not
born. The truth is, that Vasari, whose rage for dispatch and
credulity kept pace with each other, composed the first part of
Pisano's life nearly without materials, and the second from
hearsay.
[141] What Vasari says of the dog of S. Eustachio and the horse
of St. Giorgio, though on the authority of Frà Marco de' Medici,
warrants the assertion; and still more the foreshortened horse on
the reverse of a medal struck in 1419, in honour and with the
head of John Palæologus. The horse, like that of M. Antoninus,
has an attitude of parallel motion. The medal has been published
by Ducange in the appendix to his Latin Glossary, by Padre
Banduri, Gori and Maffei.
[142] See their lists in Descrizione delle Architetture, Pitture e
Sculture di Vicenza con alcune osservazioni, c. Vicenza, 1779,
8vo. p. I. II.
[143] Ridolfi, i. 68. Vasari, who treats his art with contempt, calls
him Jacopo; and Orlandi, afraid of choosing between them, used
both, and made two different artists.
[144] Vasari dates his birth 1480.
[145] Liruti, Notizie de' Letterati del Friuli, t. ii. p. 285.
[146] Sebastiano Zuccati of Trevigo, flourished about 1490. He
had two sons, Valerio and Francesco, celebrated for mosaic about
and beyond the middle of the sixteenth century. Flaminio Zuccati,
the son of Valerio, who inherited his father's talent and fame,
flourished about 1585. See Zanetti.

[147] See Ridolfi. The original went to Dresden; but Italy abounds
in copies of it. Lanzi mentions one which he saw at S. Saverio in
Rimini, with Tiziano's name written on the fillet of the Pharisee, a
performance of great beauty, and by many considered less a copy
than a duplicate. The most celebrated copy, that of Flaminio
Torre, is preserved at Dresden with the original.
[148] Si può quasi dire, che il vizio sia la virtù della Scuola
Veneziana, poichè fa pompa della sollecitudine nel dipingere; e
perciò fa stima di Tintoretto, che non avea altro merito. Mengs,
Opere, t. i. p. 175. ed. Parm.
[149] It has supplanted, was probably perpetuated in allusion to
his rapidity of execution, and remains familiar to ears that never
heard of Robusti.
[150] See Varie Pitture a fresco de' principali Maestri Veneziani,
c. Venez. fol. 1760. Tab. 8, 9, p. viii. No one who has seen the
original figures of the Aurora and Creposcolo in S. Lorenzo, can
mistake their imitation, or rather transcripts, in these.
[151] The frequent want of equilibration found in Tintoretto's
figures, even where no violence of action can palliate or account
for it, has not without probability been ascribed to his method of
studying foreshortening from models loosely suspended and
playing in the air; to which he at last became so used that he
sometimes employed it even for figures resting on firm ground,
and fondly sacrificed solidity and firmness to the affected graces
of undulation.
[152] It would be mere waste of time to recapitulate what has
been said on the efficient beauties of this astonishing work in the
lectures on colour and chiaroscuro, and in the article of Tintoretto,
in the last edition of Pilkington's Dictionary. It has been engraved
on a large scale by Agostino Carracci, if that can be called
engraving which contents itself with the mere enumeration of the
parts, totally neglecting the medium of that tremendous twilight
which hovers over the whole and transposes us to Golgotha. If
what Ridolfi says be true, that Tintoretto embraced the engraver
when he presented the drawing to him, he must have had still
more deplorable moments of dereliction as a man than as an
Artist, or the drawing of Agostino, must have differed totally from
the print.

[153] It is engraved by Pietro Monaco, as that of Tiziano, by Le
Fevre, but in a manner which makes us lament the lot of those
who have no means to see the original.
THE SCHOOL OF MANTOUA.
Mantoua,[154] the birth-place of Virgil, a name dear to poetry, by the
adoption of Andrea Mantegna and Giulio Pippi claims a distinguished
place in the history of Art, for restoring and disseminating style
among the schools of Lombardy.
Mantoua, desolated by Attila, conquered by Alboin, wrested from the
Longobards by the Exarch of Ravenna, was taken and fortified by
Charles the Great: from Bonifazio of Canossa it descended to
Mathilda; after her demise, 1115, became a republic tyrannized by
Bonacorsi, till the people conferred the supremacy on Lodovico
Gonzaga, under whose successors it rose from a marquisate, 1433,
to a dukedom, 1531, and finished as an appendage to the spoils of
Austria.
Revolutions so uninterrupted, aggravated by accidental devastations
of floods and fire, may account for the want of earlier monuments of
art in Mantoua and its districts, than the remains from the epoch of
Mathilda.[155] A want perhaps more to be regretted by the antiquary
than the historian of art, whose real epoch begins with the
patronage of Lodovico Gonzaga and the appearance of Andrea
Mantegna.[156]
This native of Padova[157] was the adopted son and pupil of
Squarcione, in whose school he acquired that taste for the antique

which marks his works at every period of his practice; if sometimes
mitigated, never supplanted by the blandishments of colour and the
precepts of Giovanni Bellino, whose daughter he had married.
Perhaps no question has been discussed with greater anxiety, and
dismissed from investigation with less success, than that of
Correggio's origin, circumstances, methods of study, and death.
The date of his birth is uncertain, some place it in 1475, others in
1490; were we to follow a MS. gloss in the Library at Gottingen,
mentioned by Fiorillo, which says he died at the age of forty in 1512,
he must have been born in 1472; but the true date is, no doubt, that
of the inscription set him at Correggio, viz. that he died in 1534,
aged forty. The honour of his birth-place is allowed to Correggio,
though not without dispute.[158] His father's name was Pellegrino
Allegri, according to Orlandi, countenanced by Mengs. He was
instructed in the elements of literature, philosophy, and
mathematics; however doubtful this, there can be no doubt
entertained on the very early period in which he must have applied
to Painting. The brevity of his life, and the surprising number of his
works, evince that he could not devote much time to literature, and,
of mathematics, probably contented himself with what related to
perspective and architecture. On the authority of Vedriani and of
Scannelli, Mengs and his follower Ratti make Correggio in Modena
the pupil of Franc. Bianchi Ferrari, and in Mantoua of Andr.
Mantegna, without vouchers of sufficient authenticity for either: the
passage quoted by Vedriani from the chronicle of Lancillotto, an
historian contemporary with Correggio, is an interpolation; and
Mantegna, who died in 1505, could not have been the master of a
boy who at that time was scarcely in his twelfth year.
Some supposed pictures of Correggio at Mantoua, in the manner of
Mantegna, may have given rise to this opinion. An imitation of that
style is visible in some whose originality has never been disputed:
such as in the St. Cecilia of the Palace Borghese, and a piece in his
first manner of the Gallery at Dresden.

Father Maurizio Zapata, a friar of Casino, in a MS. quoted by
Tiraboschi, affirms that the two uncles of Parmegianino, Michele and
Pier Stario Mazzuoli, were the masters of Correggio,—a supposition
without foundation; it is more probable, though not certain, that he
gained the first elements from Lorenzo Allegri his uncle, and not, as
the vulgar opinion states, his grandfather.
Equal doubts prevail on his skill and power of execution in
architecture and plastic: the common opinion is, that for this he was
beholden to Antonio Begarelli. Scannelli, Resta, and Vedriani,
pretend that Correggio, terrified by the enormous mass and variety
of figures to be seen foreshortened from below in the cupola of the
Domo at Parma, had the whole modelled by Begarelli, and thus
escaped from the difficulty, correct, and with applause. They likewise
tell in Parma, that by occasion of some solemn funeral, many of
those models were found on the cornices of the cupola, and
considered as the works of Begarelli: hence they pretend that
Correggio was his regular pupil, and as such finished those three
statues which a tradition as vague as silly has placed to his account
in Begarelli's celebrated composition of the Deposition from the
Cross in the church of St. Margareta.
That either Correggio himself or Begarelli made models for the
cupola admits no doubt, the necessity of such a process is evident
from the nature and the perfection of the work; but there is surely
none to conclude from it to that of a formal apprenticeship in
sculpture. He who had arrived at the power of painting the cupola at
Parma, may without rashness be supposed to have possessed that
of making for his own use small models of clay, without the
instructions of a master, especially in an age when painting,
sculpture, and architecture frequently met in the same artist; and, as
we have elsewhere[159] observed, when sketching in clay was a
practice familiar to those of Lombardy.
Correggio's pretended journey to Rome is another point in dispute:
two writers of his century, Ortensio Landi and Vasari, reject it. The

first says[160] Correggio died young without having been able to visit
Rome; the second affirms that Antonio had a genius which wanted
nothing but acquaintance with Rome to perform miracles. Padre
Resta, a great collector of Correggio's works, was the first who
opposed their authority.[161] He pretends, in some writing of his
own, to have adduced twelve proofs of Correggio's having twice
visited Rome, viz. in 1520 and 1530. But the allegations of a crafty
monk, a dealer in drawings and pictures, cannot weigh against
authorities like those of Vasari and Landi. His conjectures rest partly
on some supposed drawings of Correggio's in his possession, from
the Loggie of the Vatican, and partly on an imaginary journey, in
which, he tells us, Correggio traversed Italy incognito, and made
everywhere copies, which all had the good luck to fall into his own
reverend hands. These lures, held out to ensnare the ignorant and
wealthy, he palliated by a pretended plan of raising a monument to
the memory of the immortal artist at Correggio, the expenses of
which were to be defrayed by the produce of his stock in hand. He
had even face enough to solicit from that town an attestation that
their citizen had travelled as a journeyman painter.
Mengs, and of course Batti, embrace the same opinion. Mengs draws
his conclusion from the difference between Correggio's first and
second style, which he considers less as the imperceptible progress
of art than as the immediate effect of the works of Raphael and
Michel Agnolo. Mengs was probably seduced to believe in this
visionary journey on the authority of Winkelmann, who pretended to
have discovered, in the museum of Cardinal Albani, some designs
after the antique by Mantegna, Correggio's reputed master. Bracci, in
opposition, assert that Allegri was beholden to none but himself for
his acquirements, and appeals to a letter of Annibale Carracci, who
says that Correggio found in himself those materials for which the
rest were obliged to extraneous help. The words of Carracci,
however, with all due homage to the genius of Correggio and the
originality of his style, appear to refer rather to invention and the
poetic, than to the executive part of his works.

If there be any solidity in the observation of Mengs on Correggio's
first manner, as a mixture of Pietro Perugino's and Lionardo's style,
and of course not very different from Raphael's, how comes it that in
the works of his second and best manner all resemblance to either,
and consequently to Raphael, disappears? The simplicity of
Raphael's forms is little beholden to that contrast and those
foreshortenings which are the element of Correggio's style. Raphael
sacrificed all to the subject and expression; Correggio, in an artificial
medium, sacrifices all to the air of things and harmony. Raphael
speaks to our heart; Correggio insinuates himself into our affections
by charming our senses. The essence of Raphael's beauty is dignity
of mind; petulant naïveté that of Correggio's. Raphael's grace is
founded on propriety; Correggio's on convenience and the harmony
of the whole. The light of Raphael is simple daylight; that of
Correggio artificial splendour. In short, the history of artists scarcely
furnishes characteristics more opposite than what discriminate these
two. And though it may appear a paradox to superficial observation,
were it necessary to find an object of imitation for Allegri's second
and best style, the artificial medium, the breadth of manner and
mellowness of transition, with the enormous forms and
foreshortenings of Michel Angelo, though adopted by so different a
mind, from as different motives, for an end still more different, will
be found to be much more congenial with his principles of seeing
and executing, than the style of any preceding or coetaneous period.
The authenticity of Correggio's celebrated Anch' io son Pittore, is
less affected by the improbability of his journey to Rome, than by its
own legendary weakness: though not at Modena or Parma, for there
were no pictures of Raphael in either place during Antonio's life, he
might have seen the St. Cecilia at Bologna; and if the story be true,
perhaps no large picture of that master that we are acquainted with
could furnish him with equal matter of exultation. He was less made
to sympathize with the celestial trance of the heroine, the intense
meditation of the Apostles, and the sainted grace of the Magdalen,
than to be disgusted by a parallelism of the whole which borders on
primitive apposition, by the total neglect of what is called

picturesque, the absence of chiaroscuro, the unharmonious colour,
and dry severity of execution.
The next point is to fix the dates of Correggio's works; the certain,
the probable, the conjectural.
The theatre of Correggio's first essays in art is supposed to have
been his native place and the palace of its princes; but that palace
perished with whatever it might contain. From a document in the
parochial archive of Correggio, of 1514, it appears that in the same
year he painted an altar-piece for one hundred zechini, a
considerable price for a young man of twenty. This picture was in the
church of the Minorites, where it remained till 1638, when a copy
was unawares put into the place of the original. The citizens
alarmed, in vain made representations to Annibale Molza, their
governor; it even appears from a letter of his to the Court of
Modena, in whose name he governed, that, many years before, two
other pieces of Antonio had been removed from the same chapel by
order of Don Siro, the last prince of the House of Correggio; those
represented a St. John and a St. Bartholomew; the subject of the
altar-piece was the Madonna with the child, Joseph and St. Francis.
The fraternity of the Hospital della Misericordia possessed likewise
an altar-piece of Antonio. The centre piece represented the Deity of
the Father; the two wings, St. John and Bartholomew. According to
a contract which still remains in the archives, it was estimated by a
painter of Novellara, Jacopo Borboni, at three hundred ducats,
bought for Don Siro in 1613, and a copy put in its place. The
originals of all these pictures are lost.
The picture with the Madonna and child on a throne, St. John the
Baptist, the Sts. Catharina, Francis, and Antony, inscribed Antonius
de Allegris P. now in the gallery of Dresden, was, as Tiraboschi
correctly supposes, an altar-piece in the church of St. Nicolas of the
Minorites, at Carpi: a copy of it by Aretusi, is at Mantoua. To this
period, and perhaps even an earlier one, belongs the St. Cecilia of
the Borghese palace. The general style of this picture is dry and

hard, and the draperies in Mantegna's taste; but the light which
proceeds from a glory of angels, and imperceptibly expands itself
over the whole, is a characteristic too decisive to leave any doubt of
its originality.
In the gallery of Count Brühl was the Wedding (sposalizio) of St.
Catharine, with the following inscription on the back:—Laus Deo:
per Donna Metilde d'Este Antonio Lieto da Correggio fece il presente
quadro per sua divozione, anno 1517. This inscription appears,
however, suspicious, as at that time there was no princess of that
name at the court of Ferrara. At the purchase of the principal
pictures in the Modenese gallery by Augustus III. this was presented
by the Duke to Count Brühl; from him it went to the Imperial Gallery
at Petersburg. A similar one was in the collection of Capo di Monte at
Naples, and Mengs considers both as originals. Copies of merit by
Gabbiani and Volterrano are in England and Toscana. It is singular
that an artist, than whom none had more scholars and copyists, and
whose short life was occupied by the most important works, should
be supposed to have painted so many duplicates, and that a set of
men, as impudent as ignorant, should meet with dupes as credulous
as wealthy, eager to purchase their trash at enormous prices, in the
face of the few legitimate originals.
In 1519, Antonio went to Parma, and soon after his arrival is said to
have painted a room in the Nunnery of St. Paul. The authenticity of
this work, placed within the clausure of the convent and
consequently inaccessible, has been recently disputed, and the
author of a certain dialogue even attempts to prove the whole a
fable. To ascertain the fact, a special licence to visit the place was
obtained for some painters and architects of note, and on their
declaring the paintings one of Correggio's best works, Don
Ferdinando de Bourbon, with some of the courtiers and Padre
Iveneo Affo, followed to inspect it. What he tells us of monastic
constitution in those times accounts for the admission of so profane
an ornament in such a place; for in the beginning of the sixteenth
century, clausure was yet unknown to nunneries; abbesses were

elected for life, their power over the revenue of the convent was
uncontrolled, their style of life magnificent, and their political
influence not inconsiderable. Such was the situation of nunneries
when Donna Giovanna da Piacenza, descended from an eminent
family at Parma, the new-elected abbess of St. Paul's, ordered two
saloons of her elegant apartments to be decorated with paintings;
one by Correggio, and another, as it is conjectured, either by
Alessandro Araldi of Parma, or Cristoforo Casella, called Temperello.
Padre Affo proves that Correggio must have painted his apartment
before 1520, immediately after his arrival at Parma, and four or five
years before the introduction of the clausure. Of a work so singular
and questionable, it will not appear superfluous to repeat some of
the most striking outlines from his account:—The chimney-piece
represents Diana returned from the chase, to whom an infant Amor
offers the head of a new-slain stag; the ceiling is vaulted, raised in
arches over sixteen lunettes; four on each side of the walls; the
paintings are raised about an ell from the floor, and form a series of
mythologic and allegoric figures, which breathe the simplicity, the
suavity, and the decorum of Art's golden age. Of these the three
Graces naked, in three different attitudes, offer a charming study of
female beauty, and a striking contrast with the Parcæ placed
opposite; the most singular subject is a naked female figure,
suspended by a cord from the sky, with her hands tied over her head
—her body extended by two golden anvils fastened with chains to
her feet, floating in the attitude of which the Homeric Jupiter
reminds his Juno.[162] The high-arched roof embowers the whole
with luxuriant verdure and fruit, and is divided into sixteen large
ovals, overhung with festoons of tendrils, vine-leaves, and grapes,
between which appear groups of infant Amorini, above the size of
children, gamboling in various picturesque though not immodest
attitudes.
Neither the pretended inaccessibility of place, nor the veil thrown by
monastic austerity over the profaneness of the subject, can
sufficiently account for the silence of tradition, and the obscurity in
which this work was suffered to linger for nearly three centuries.

Supposing it, on the authorities adduced, to be the legitimate
produce of Correggio, and considering its affinity to the ornamental
parts of the Loggie in the Vatican, it affords a stronger argument of
Allegri's having seen Rome, studied the antique, and imitated
Raphael, than any of those that have been adduced by Mengs, who
(with his commentator D'Azara,) appears to have been totally
uninformed of it, notwithstanding his familiarity at Parma with every
work of Correggio, his perseverance of inquiry and eager pursuit of
whatever related to his idol, the influence he enjoyed at Court, and
unlimited access to every place that might be supposed to contain or
hide some work of art.
Soon after his arrival at Parma, Antonio probably received the
commission of the celebrated cupola of S. Giovanni, which he
completed in 1524, as appears from an acquittance for the last
payment subscribed 'Antonio Lieto,' still existing at Parma.
In the cupola he represented the Ascension of the Saviour, with the
Apostles, the Madonna, c. and the Coronation of the Virgin on the
tribune of the principal altar, whose enlargement in 1584 occasioned,
with the destruction of the choir, that of the painting: a few
fragments escaped; an exact copy had, however, been provided
before, by Annibale Carracci, from which it was repainted on the
same place by Aretusi. The same church preserved two pictures in
oil of Correggio, the martyrdom of St. Placidus and Flavia, and Christ
taken from the cross on the lap of his mother; both are now (1802)
in the collection of the Louvre.
The success of the cupola of S. Giovanni encouraged the inspectors
of the Domo to commit the decoration of theirs to the same master.
Of their contract with him, the original still remains in the archive of
their chapter; it was concluded in 1522, and amounted to about one
thousand zecchini, no inconsiderable sum for those times, and alone
sufficient to do away the silly tradition of the artist's mendicity. The
decorations of the chapel, next to the cupola, were distributed
among three of the best Parmesan painters at that time,
Parmegianino, Franc. Maria Rondani, and Michael Angelo Anselmi.

From all the papers hitherto found, it appears, however, that
Correggio did not actually begin to paint the cupola before 1526: it
represents the Ascension of the Virgin, and without recurring to an
individual verdict, has received the sanction of ages, as the most
sublime in its kind, of all that were produced before and after it; a
work without a rival, though now dimmed with smoke, and in decay
by time. These were the two first cupolas painted entire, all former
ones being painted in compartments. Nothing occurs to make us
surmise that Correggio had partners of his labour in these two
works; for Lattanzio Gambara of Brescia, mentioned by Rossi as his
assistant in the Domo, was born eight years after Correggio's death.
During the progress of these two great works, Correggio produced
others of inferior size but equal excellence; the principal of which are
the two votive pictures of St. Jerome, and La Notte. That of St.
Jerome represents the Saint offering his Translation to the Infant
Christ, who is seated in his Mother's lap, with St. Magdalen reclining
on and kissing his feet, and flanked by Angels. The commission for
this picture is said to have been given in 1523, by Donna Briseide
Colla, the widow of Orazio or Ottaviano Bergonzi of Parma, who in
1528 gave it as a votive offering to the church of S. Antonio del
Fuoco. The price agreed on, was 400 lire; 40,000 ducats were
offered for it afterwards by the King of Portugal; and the then Abbot
of the convent was on the point of concluding the bargain, when the
citizens of Parma, to prevent the loss, applied to the Infante Don
Philippo. He ordered it in 1749 to be transposed from S. Antonio to
the Domo; there it remained till 1756, when, on the application of a
French painter, expelled by the Canons for his attempt to trace it,
the Prince had it transferred under an escort of twenty-four
grenadiers to Colorno; and from thence to the newly instituted
academy, where it remained till 1797, and now, (1802,) with other
transported works of Art, glitters among the spoils of the Louvre.
The second picture known by the name of La Notte, represents the
birthnight of the Saviour, and was the commission of Alberto
Pratonieri, as appears from a writing dated in 1522, though it was

not finished till 1527 according to Mengs, or 1530 as Fiorillo
surmised, when it was dedicated in the Chapel Pratonieri of S.
Prospero at Reggio: from whence, 1640, it was carried to the gallery
of Modena, by order, of Duke Francesco I. and from thence at length
to that of Dresden.
A chapel in the church del S. Sepolcro at Parma, possessed formerly
the altar-piece known by the name of La Madonna della Scodella,
because the Virgin, represented on her flight to Egypt, holds a
wooden bowl in her hand: a figure, whom Mengs fancies the Genius
of the Fountain, pours water into it; and in the back ground an
angel, whose action and expression he considers as too graceful for
the business, ties up the ass. This picture, he tells us, was, thirteen
years before the date in which he wrote, nearly swept out of the
panel by the barbarous wash of a Spanish journeyman painter who
had obtained permission to copy it. It is now in the Louvre, and how
much of its present florid colour is legitimate, must be left to the
decision of the committee de la Restoration.
If the most sublime degree of expression be entitled to the right of
originality, Mengs must be followed in his decision on the Ecce
Homo, formerly in the Palace Colonna, without much anxiety
whether it be the same that belonged to the family of Prati at
Parma, or that which Agostino Carracci engraved.
The Madonna seated beneath a palm-tree, bending in somnolently
pensive contemplation over the Infant on her lap, watched by an
Angel above her, and attended by a Leveret, known by the name of
La Zingarella or the Egyptian, from the sash round her head,
formerly in the gallery of Parma, and now at Naples in that of Capo
di Monte, has suffered so much from a modern hand, that little of
the master remains but the conception. Nearly a duplicate of it was
presented by Cardinal Alessandro Albani to king Augustus of Poland;
but Mengs hesitates to pronounce it an original.
In the period of these, about 1530, we may probably place the two
celebrated pictures of Leda and Danae, than which no modern works

of art have suffered more from accident and wanton or bigoted
barbarity, or been tossed about by more contradictory tradition.
If the subject that takes its name from Leda be, as Mengs says,
rather an allegory than a fable, it alludes to what would aggravate
even the story of that mistress of Jupiter. The central figure
represents a female seated on the verge of a rivulet with a swan
between her thighs, who attempts to insinuate his bill into her lips;
but at her side, and deeper in the water, is a tender girl, who with
an air of innocence playfully struggles to defend herself from the
attacks of another swimming swan; farther on, a girl more grown up
to woman, gazes, whilst a female servant dresses her, with an air of
satiate pleasure after a swan on the wing, that seems just to have
left her; at some distance appears half a figure of an aged woman,
draped, and with looks of regret. On the other side of the principal
group, the graceful form of a full-grown Amor strikes the lyre, and
two Amorini contrive to wind some horn instruments. The scene of
all this is a charming grove on the brink of a pellucid lake.
The second picture represents the daughter of Acrisius, but with
poetic spirit. The virgin gracefully reclines on her bed; a full-grown
Cupid, perhaps a Hymen, lifts with one hand the border of the sheet
on her lap that receives the celestial shower, whilst his other
presents the mystic drops to her enchanted glance: two Amorini at
the foot of the bed try on a touchstone, that, one of the golden
drops, this, the point of an arrow, and he, says Mengs, has a vigour
of character much superior to the other, plainly to express, that Love
proceeds from the arrow, and its ruin from gold; he likewise finds
that the head and head-dress of Danae are imitated from those of
the Medicean Venus.
Vasari, and after him Mengs with others, tell that in 1530, Federigo
Gonzaga, then created Duke of Mantoua, intended to present
Charles the Fifth at the ceremonial of his coronation with two
pictures worthy of him, and in the choice of artists gave the
preference to Correggio. From this, a correct inference is drawn
against that pretended obscurity in which Correggio is said to have

lingered; for at that time Giulio Romano lived at the Court of
Mantoua, and Tizian was in the service of the Emperor. Vasari is
silent on the date of the pictures, but he affirms that, at their sight,
Giulio Romano declared he had never seen a style of colour
approaching theirs. So far all seems correct; but that they were
actually presented to Charles, sent to Prague, and after the sacking
of that city by Gustavus Adolphus, carried to Stockholm, is unproved
or erroneous. If it is not likely that the Emperor, instead of sending
them to Madrid, the darling depository of his other works of art,
should have sent them into a kind of exile to Prague, it is an error to
pretend they were removed from thence by Gustavus Adolphus, who
was slain at Lutzen sixteen years before the Swedes sacked that city,
1648. The truth is, that these pictures were not given to the
Emperor, but placed in his own gallery by the Duke, where they
remained till 1630, when the Imperial General Colalto stormed
Mantoua, sacked it, deprived it of its cabinet of treasures, of the
celebrated vase since possessed by the House of Brunswick, and
transmitted its beautiful collection of pictures to Prague, from
whence by the event of war we have mentioned, they became the
property of Queen Christina, at whose abdication, when the whole
was packing up for Rome, the two pictures in question were
discovered in the royal stables, where they had served as window-
blinds, mutilated and despised. Whether so unaccountable a neglect
be imputable to the Queen's want of taste, as Tessin asserts, or to
accident, or, what is most unlikely, to her modesty, cannot now be
decided. They were repaired, and at her demise left to Cardinal
Azzolini, of whose heirs they were purchased by Don Livio
Odescalchi, and by him left to the Duke of Bracciano, were sold to
the Regent of France, whose son, from a whim of bigotry, had the
picture of the Leda cut to pieces in his own presence, in which state
Charles Coypel requested and obtained it for his private study. At his
death it was vamped up, repieced, disposed of by auction, and, at a
high price, sold to the King of Prussia. What became of the Danae is
matter of dispute.[163]

The picture of Io embraced by Jupiter, inbosomed in clouds, by a
silent water in which a stag quenches his thirst, was their
companion: a work to which the most lavish fame has done no
justice, and beyond which no fancy ever soared. The Io shared a still
more barbarous fate. Not content with mangling her like the Leda,
the bigot prince burnt her head; and, were it not for the beautiful
duplicate which fortune preserved in the Gallery of Vienna,[164] we
should be reduced to guess at Correggio in the fragments at Sans
Souci, and the prints of Surregue and Bartolozzi. The Imperial
Gallery possesses, likewise, the Rape of Ganymede, by Correggio, of
the same size with the Io; a Mountain Scene; a full-grown Cupid,
seen from behind, with his head turned to the spectator, shaping a
bow, accompanied by a laughing and a weeping infant, in struggling
attitudes, which was likewise sold by the heirs of Don Livio
Odescalchi, has equally exercised opinion. Vasari, Tassoni, Du Bois,
de St. Gelais, c. ascribe it to Parmegiano; Mengs and Fiorillo, who
judge from the duplicate at Vienna, with greater probability give it to
Correggio. The contrast of the attitudes is produced more by naïveté
than affectation, the lines have more simplicity than the style of
Mazzuola admitted of, and the colour more breadth. The conception
of the whole, whether the infants be the symbols of successful and
unsuccessful love, or denote the dangers of love, or be simply
children, though not beyond the fancy of Parmegiano, has more the
air of a Correggiesque conceit. Numberless copies were made after
it, some by Parmegiano himself, whose handling may be recognized
in the picture at Paris.
We are now arrived at those works of Correggio's which cannot be
fixed to a certain period. Such are probably, in the Gallery of
Dresden, those known under the names of S. Giorgio and S.
Sebastiano, of both of which Mengs gives a circumstantial account.
He is, however, mistaken when he imagines the last to have been
voted by the City of Modena after a plague: the commission of it
was given by the fraternity of St. Sebastian.

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