Tessellations and Sweeping Nets: Advancing the Calculus of Geometric Logic

Tessellations and Sweeping Nets: Advancing the Calculus of
Geometric Logic
Parker Emmerson
Marlboro College
Abstract. This paper studies the advancement of geometric logic through tessellations and
sweeping nets, addressing the challenge of arranging reflecting points for efficient ray tracing
under limited time constraints. We introduce the concept of a sweeping subnet alongside a
causal barrier to encapsulate the geometrical limitations posed by time, thereby delineating
the boundary of influence for light propagation within a defined space. This work delves
into the underpinnings of tessellation dynamics, revealing how the spatial arrangement and
temporal evolution of tessellated patterns can be navigated and optimized through a novel
algorithmic framework. Through a combination of theoretical exploration and practical im-
plementation, including Python code for simulation and visualization, we provide a platform
for approximating optimal tessellations that adapt to the constraints dictated by the causal
barrier. This paper formalizes quasi-quanta notation in such a way that it may be imple-
mented.
Code is located at https://zenodo.org/records/10578751
1 Introduction
This paper focuses on the optimal arrangement of reflecting points for efficient ray tracing
given limited sweep time. We examine spatial configurations, employing our core concept
of a sweeping subnet and defining a causal barrier to capture constraints imposed by time.
We will also discuss the influence of these constructions on the design of an algorithm
for approximating optimal tessellations.
Code is provided for each of the graphs1, as the mathematics is demonstrated unequivocally
by their implementation. The reader can test out the reality of this system by visualizing
the graphs themselves using Python in a suitable environment like Google Colaboratory.
2 Fundamentals
2.1 Sweeping Subnet
A sweeping subnet refers to the set of reachable points on a surface from a light source
within a time constraint. To formalize:
∥ri − xi∥ ≤ 2∥n(Xi)∥ < 2ξ. (1)
This equation establishes the geometrical constraints required to capture the notion of
sweeping efficiency rigorously.
1 https://zenodo.org/records/10578751andgithub.com/sphereofrealization/PythonCode/blob/main/Tessellations.zip.
International Journal of Game Theory and Technology (IJGTT), Vol.10, No.3, September 2024
DOI : 10.5121/ijgtt.2024.10301
1

Fig. 1: Schematic representation of a sweeping subnet.
2.2 Causal Barrier
The causal barrier is the spatial limit reachable by rays within a defined temporal boundary:
rbarrier(t) = max
x,y∈C
G(x, y) · P(y, t). (2)
It delineates the boundary of influence for any point within our geometric configuration.
Causal Barrier Dynamics A pivotal factor in tracing rays within a limited timeframe is
the concept of a causal barrier. This barrier represents the set of points that are unreachable
by light within a given period, restricted by past events and influences. The causal barrier
is a manifest constraint shaped by the maximum sweep time τ and the network of reflecting
points across the surface Ω.
The causal barrier’s radius, rbarrier(t), quantifies the spatiotemporal limit of causal
influence for light propagating in the medium, and is expressed as:
rbarrier(t) = max
(x,y)∈C
G(x, y) · P(y, t), (3)
where G(x, y) characterizes a geometric factor dependent on the spatial coordinates of
the medium, and P(y, t) signifies the probability of light or causal influence reaching point
y at time t.
Notably, the causal barrier encompasses not just the physical impediments to light’s
movement but also integrates the historical dependencies influencing its progress. Factors
such as prior ray paths and reflecting surface orientations are embodied within this struc-
ture. For a discretized model:
2

ki1...in := lim
x→pi
0
s(x), (4)
which signifies the link between the sites’ tessellation patches and the causal barrier.
Here, the function s(x) describes the geometric state at each point x along a path pi
0,
converging on the reflective elements’ configurations. The optimized tessellation, qi1...im ,
thus not only examines spatial coverage but also adapts to temporal dimension constraints
imposed by the causal barrier.
Figure 2 illustrates the causal barrier’s structure, highlighting its variation with differ-
ent discretization parameters and tessellation configurations. As the discretization gran-
ularity refines, the causal barrier grows dynamically, underscoring the coupling between
discretization strategies and the operational envelope of the tessellated reflective elements.
In essence, it becomes a data structure encoding the system’s global dynamics, encom-
passing discretized ray paths, reflective interactions, and time-sensitivity constraints. This
characterization permits an informed assessment of tessellation costs and provides insight
into the computational complexity of potential paving schemes across the surface Ω.
Fig. 2: Visualization of Causal Barrier with varying discretization parameters.
Further complicating the causal barrier’s role is its influence on evolving system con-
straints. It guides how computational processes unfold within the discretized model, calcu-
lating emerging constraints through time-evolving tessellations. The causal barrier, there-
fore, has a symbiotic relationship with the tessellation process: each contributing to and
shaped by the logical calculus of ray propagation.
Lemma 1. Let A =
S
i Ai and Wi, we define larger residue sets, Ri, where each sub-
set’s union within Wi is Ai, ensuring their existence due to the infinite cardinality of the
Cartesian product space.
In summary, the causal barrier serves as a critical component in the tessellation prob-
lem, reflecting the complexities of both spatial arrangement and temporal evolution. It
encapsulates the ultimate bounds of light’s propagation and provides a comprehensive
framework for calculating ray trajectory efficiencies across the discretized landscape of Ω.
3 Optimal Tessellation Framework
We now introduce the formalism for our tessellation strategy starting with the radius of
spheres Sr.
3

This section demonstrates the method for approximating surfacing singularities of sad-
dle maps using a sweeping net. The method involves constructing a densified sweeping
subnet for each individual vertex of the saddle map, and then combining each subnet to
create a complete approximation of the singularities. The authors also define two functions
f1 and f2, which are used to calculate the charge density for each subnet. The resulting
densified sweeping subnet closely approximates the surfacing saddle map near a circular
region.
{⟨∂θ × r∞⟩ ∩ ⟨∂x × θ∞⟩} →

(Ar ⊕ Br) ∩ S+
r . (5)
This is the implication of a calculus structure combining spatio-temporally to form a
oneness denoted 1E {⟨∂θ × r∞⟩ ∩ ⟨∂x × θ∞⟩} → {(Ar ⊕ Br) ∩ S+
r }.
Here S+
r is the right half of the unit circle, defined as
S+
r

(x, y) ∈ R2
| x2
+ y2
= r2
, x ≥ 0 , (6)
and Ar, Br are specified as follows
Ar

(x̃, ỹ)|x̃ ≥ 0, ỹ ≥ 0, x̃2
+ ỹ2
= 1, arcsin x̃ ≥ f1(arcsin (r−1
x̃)) , (7)
Br

(x̃, ỹ)|x̃ ≥ 0, ỹ ≥ 0, x̃2
+ ỹ2
= 1, arcsin ỹ ≥ f2(arcsin (r−1
ỹ)) , (8)
(9)
In the above, ⊕ indicates the direct sum of two sets and r+ = r. x is a curve where the
slope of tangent line is greater than the vertex in the line function (See Fig.(??)b upper
line). In the same way, ∂x is the vertex set of x (single point set). θ∞ is a direct sum of
line lmn := {(x, y) ∈ R2|x + ry = n} (n is constant) and the line with infinite slope.
We define f1, f2 : [0, π/2] → [0, π/2] as follows
∂ arcsin (sin θ)
∂θ
= ...
1
p
1 − sin2
θ
Z 1
0
d
dθ
sin θds

= ...
cos θ
p
1 − sin2
θ
When we take θ = π
2 , f1(0) = f2(0) = 0. It implies that f1 and f2 continuously connect
with straight line to positively going. The ω calculates as follows
ω S+
r
=
Z π
2
0

K−1
f′
i(s)∂s

× (x̃(s, l) − x̃(0, l)) , i = {1, 2}(10)
where K and charge density ∂s are constant and expressed as
x̃(s, l)x̃(0)
+ r sin sỸ (l), (11)
x̃(0, l)x̃(0)
+ rỸ (l), (12)
respectively. Here x̃(0) = (1, 1)t, and Ỹ (l) = (cos l, sin l)t normalize. Consequently, the
net (5) approximates the surfacing saddle map around the right circle, when r 0 is
sufficiently small(Since only around a right circle), approximately satisfying charge density
of sweeping generic singular saddle case around the right circle.
4

4 Example Tessellations from the Method
The provided Python code illustrates an algorithm to visualize a tessellated surface
using hexagons whose properties are influenced by a decision graph with logical nodes and
geometry-modulating functions.
Mathematical Framework The tessellation process involves two critical functions, f1
and f2, which appear to be scalar fields that map the polar coordinate to a calculated
value that influences tessellation:
f1(θ) = arcsin (sin(θ)) +
π
2

1 −
π
2θ

(13)
f2(θ) = arcsin (cos(θ)) +
π
2

1 −
π
2θ

(14)
These functions are continuous for all 0 and are utilized in the tessellation to modulate
the properties of individual hexagons in the pattern. The choice of arcsine function suggests
a periodic influence within the tessellation, potentially attending to the natural constraints
of the surface.
The mesh initialization relies on an evenly spaced grid determined by hex centers in
Cartesian coordinates, which are then mapped to polar coordinates within the visualization
function. The hexagonal tessellation operates in this 2D domain described by the variable
domain.
In the scope of topology, each point (x, y) is mapped to the tessellation efficiency using
the functions f1 and f2 after converting to polar coordinates, . The radial aspect evokes a
natural coordinate system, possibly intended to align with radially symmetric properties
of the surface or light source distribution.
Logic Modulation and Visualization:
A directed graph G serves as a decision model, perhaps encoding logic or data pertaining
to each hexagon’s fitness regarding an overarching tessellation strategy. The nodes within
G may represent choices or properties deemed significant in the tessellation:
5

k =
where k is the logical output that could symbolize decision-making processes, such as
the reflectivity or permissibility of a hexagonal tile within the tessellation.
Visualization merges combinatorial logic and geometric heuristics to derive the tes-
sellated surface. The output hexagon’s color (color value) combines geometric modulating
variables (from f1, f2) and the logic-driven decisionvalue, illustratingviaacolormaptheareasinfluencedbylogi
Interpretation and Usage The intersection of computational geometry and logic in this
visualization has a broad applicability in fields like robotics, spatial analysis, and computa-
tional optics. In these fields, tessellations often underpin mesh generation for simulations,
photorealistic rendering, and path planning.
It is worth noting that the validity of the tessellation approach hinges on the semantic
linkage between polar coordinates (θ, r) and f1, f2 functionality. Moreover, the logic vec-
tor decisions influenced by G’s logical nodes imply a flexibility that allows for optimized
tessellations tailored to varied application-specific conditions.
Commentary on Code Structure and Style The code elegantly combines functional
abstraction with procedural execution. The modeling of the problem in terms of polar co-
ordinates implies an understanding of spatial symmetry, and the use of a decision graph
suggests an appreciation for the mathematical rigor of computational logic. Computation-
ally, converting Cartesian coordinates to polar within a loop is suboptimal, and caching
these conversions could improve performance. Additionally, the logic model G is not fully
exploited within the provided code but serves as a scaffold for refining the decision-making
process for each element of the tessellation.
Incorporating dynamic logic that modulates the visual representation of computational
tessellations presents an innovative approach, blending discrete mathematics with contin-
uous geometries to produce a visualization rich in information and adaptable to diverse
scenarios in computational design and analysis.
Examples:
6

5 Extrapolating into 3D
6 Logic Vectors as Directed Graphs and Geometric Logic
We want to essentially iterate the tessellation over a space that evolves through the logic
vector directed graphs. The edges of the hexagon units of the tessellation are geometric logic
vectors, and they manipulate the orientation and direction of the tessellation depending on
the logical deductions and inferences based on the geometric interpretation of the vectorial
logic assessments of other activities in the space.
The provided code above represents various computational concepts from different do-
mains, ranging from symbolic logic operations to graph theory and mathematical transfor-
mations.
The first two code snippets demonstrate the applications of Sympy, a Python library for
symbolic mathematics. They contain functions and symbols which allow for manipulation
and representation of symbolic expressions. The Fourier series approximations mentioned
are indicative of attempting to express a function as a series of sines and cosines, capturing
the frequency domain representation of spatial patterns.
6.1 Network Graph Logic Modulation
We see the instantiation of a decision graph via NetworkX, a library suitable for the
creation, manipulation, and study of the dynamics of complex networks. The directed graph
G simulates logical connections between hypothetical state representations ψ1, ψ2, and
their combined effect on some resultant state X. The visualization function visualizegraph
generates a representation of these logical relationships. Tessellation Visualization
7

Fig. 3: The base state
Fig. 4: Mild Evolution of Coloring
8

Fig. 5: Increased Color Change Indicates a kind of implied gradation movement
Fig. 6: Increased Color Change Indicates a kind of implied gradation movement
9

The visualization functions within the last two snippets aim to generate a tessellated
pattern modified by the logic vector—the tessellation here is achieved via hexagonal and
triangular units, wherein their properties are modulated by some logic vector, a collec-
tion of mathematical functions representing logical states. While the program is simple,
it illustrates the prinicple that we can evolve the pattern using different logic functions
illustrating a geometric interpretation of reasoning.
7 Quasi-Quanta and Evolving Chaos
Integrating Geometric and Logic Structures The collective goal of these programs is to
iterate tessellations through a dynamically evolving space, influenced by logic vector di-
rected graphs. Each tessellation element’s edges represent geometric logic vectors—that
is, the edges themselves have logical properties dictating the flow and structure of the
tessellation. This behavior is akin to using an evolving map, where decisions at each geo-
metric point affect the global arrangement of the tessellation, carrying implications for both
structural integrity and optimized design, possibly within a machine learning or artificial
intelligence context.
To illustrate this, we form an application of quasi-quanta symbolic transformations to
visualize the evolving chaotic states of the system. We see that the colorations represent
varying depths of, runnels.
The form of the quasi quanta expression is taken from, Combinations of Quasi Quanta
Expressions, (Emmerson, 2023), from the package, Quasi Quanta Language Package.
It serves as an example form that large language models can apply to actual programs to
visualize these kinds of pseudo-programs using Python.
f ◦ g =
[
x∈S1∪S2
x = Ω
∆iÅ ∼
♡H ⊕ ·
⋆
⋆H∆Å
i⊕ ∼ ·♡
Thus, it serves as a more concrete example of how these linguistic expressions and
linguistic synchronicity forms can be applied and continue to evolve.
10

(a) t = .2 (b) t = .4
(c) t = .6 (d) t = .8
(e) t = 1.2 (f) t = 1.7
In practical terms, this could reflect a system where the tessellation adapts based
on sensed or inferred information—a robot adjusting its path planning based on dynamic
environmental variables, or a rendering system adjusting the level of detail based on viewer
focus and processing power availability. Theoretically, the quasi quanta expressions then
represent chaotic runnels through which language flows into and through the oneness of
the living one.
11

(a) t = 2.2 (b) Image 8
Fig. 8: The pattern in the system resonance attains a higher degree of chaotic synchronicity
as time increases. This paper is published as a package with the Jupyter Python notebooks,
and I urge the reader to play with these programs as they see fit.
8 Logic Vectors, Directed Graphs, Tessellation Associations
This type of system requires careful orchestration of the logic and geometry interplay.
Defining a clear and coherent representation for the interaction between logical conditions
and spatial tessellation is key. It ultimately represents a complex adaptive system where
localized decisions and conditions propagate their influence through the network, affecting
the larger whole and leading to emergent spatial behaviors and patterns. This integration
has a multitude of applications, from computational physics simulations to optimizing
rendering engines in computer graphics and ray tracing.
However, for simplicity sake, I just want to draw the analogy between the potential for
logic statements to actively seek out proof environments by navigating geometric hexago-
nal tiling platform topologically. Essentially, this works by simply noting the relationship
between the interactive compass, the symbolic directed graph (representative of a neural
network), and the logic vector adaptation presented by the large language model. The large
language model essentially interprets the logic vector (logic space geometry) as a mode of
changing the functions of the tessellating pattern.
12

For instance,
Fig. 9: Example of an Arbitrary configuration of the Diredted Graph-Compass relationship.
Fig. 10: Example of an Arbitrary configuration of the Diredted Graph-Compass relation-
ship.
13

In the above code, we see how the time compass concept can be integrated with the
connections of a directed graph. Noticing this, we also notice that the time compass can be
adapted to manipulate the tessellation coloration program based on the interpretation of a
set of logic vectors to form runnels. This provides us a platform to connect chaos theory of
an evolving dynamic system, logic-vector space, which shares the same supra-manifold as
space-time mathematically (Supramanifolds of Logic, Emmerson 2023 (Limbertwig)), and
the language of quasi-quanta synchronistic synergy. This illustrates that there are real-
number programmatic interpretations of the entire system and that it isn’t just nonsense-
dingbat statements when interpreted through a large language model into functional code.
This kind of synthesis brings together computational logic, graph theory, and interactive
visualization in a way that allows for rich user interaction.
Here, we associate the tessellation coloration as an ebb and flow transformation over
time associated with the time-compass angular velocity conception, as
Fig. 11: Slider (Angular Velocity adjusts the wave form within the circular band.)
Fig. 12: Time Compass
14

Fig. 13: In this example, the Time Compass adjusts the ebb and flow of the tidal coloring.
Not only this, though we can demonstrably prove that logic-vectors are a spatial lan-
guage that can be implemented in hard code using large language models to adapt the
pseudo code of generative mathematical language patterns (symbolic of quasi-quanta topo-
logical infinity meanings):
Fig. 14: Code for this program is attached in the python jupyter file included within the
.zip package of this paper.
Look:
15

Fig. 15: It stands to reason that logical associations that form a geometrically solid structure
will form deeper runnels in the chaos theory itself as quasi-quanta syngergistically combine
to form oneness indications from infinity meanings.
In this example, we use the logic vector:

∀x
δ , ∃x
ϵ , ∀x
τ , ¬P(y)
∆ , R(x) =⇒ S(x)
∆ , ∃x∈E,∀y∈U,P(x) ⇐⇒ F(y)
∆

In this program, users can input a logical expression at each step to influence the
visualization of the tessellation. The draw logic graph edge function has been added to
draw the logic graph on the hexagon edges. The visualize tessellation function now includes
the logic graph drawing inside the loop over each hexagon center to embed the logic graph
visualization within each hex tile. The while True loop at the end of the program handles
user input for interactivity.
9 Virtual Nerves
The tessellation, formed by a synergy that harnesses quasi-quanta significations to a one-
ness expression that maps the resulting conceptual energy number to the real number
programming language, this pseudo-coding method has yielded results. Thus, another po-
tential application of the chaos-expandin, runnel-forming, tessellation connection to logic
vectors is virtual nerves as described by Buchanan.
In Quantization and torsion on sheaves I, Buchanan states:
Let Pε be an extended p̂-complex, and Γl [Pε] the group of toric connections stratifying
every p̂ into a space E.
Fig. 2
Pε regular with 01, 10 poles
P◦
= 0 ∂(10)
1
θ = P◦
Π1 (P⋆
)

The theory has potential in the sense that we can form a kind of nerve scaffold that
connects via the time compass to the quasi-quanta linguistic synergy directing toward a
symbolic analogical oneness expression (isn’t it ironic that the directing toward a oneness
16

in the energy number notation is so similar to the directing of the oneness in the nerve
center).
Because this program can be tied to the, time compass, concept, and thus in turn,
phenomenological velocity, we can formulate a theory of consciousness as the in-tandem
existential interaction of the actualizing of the symbolic experiential plateau synergizing a
spontaneous big-bang cosmograph with the structural nature of the geometric-logic game,
but of course, we are left with just another lens on metaphors of consciousness. However,
it seems a promising mathematical metaphor.
Fig. 16: Slider (Angular Velocity adjusts the wave form within the circular band.)
10 Potential Applications of the Mathematical Linguistic Insight
Cellular Automata: These are mathematical models in which a grid of cells evolves through
discrete time steps according to a set of rules based on the states of neighboring cells. This
could be represented with a 3D tessellation where each layer represents a moment in time.
Graph Structures and Networks: Directed graphs and networks can visualize complex
relationships. These can be precisely defined with nodes representing logic states and edges
representing logical operations or transformations.
Fractals: Fractal structures emerge from simple rules applied recursively and can repre-
sent self-similar logic at different scales, much like how genetic information can be packed
densely within a genome.
Crystal Lattices: These structures can represent three-dimensional tessellations with
repeating logic units, much like how chemical structures crystallize according to molecular
interactions.
Quantum Logic Gates: These are used in quantum computing and could be represented
by a complex 3D tessellation where each cell represents a qubit in a superposition of states.
17

Fig. 17: Time Compass
Fig. 18: In this example, the Time Compass adjusts the ebb and flow of the tidal coloring.
18

Logic Solitons: In physics, a soliton is a self-reinforcing solitary wave that maintains its
shape while traveling at a constant speed. Similarly, we could envision a visual represen-
tation where logic states propagate through a medium without dispersal.
Viral Capsids: The geometrical shapes of viral capsids can inspire new forms based on
tessellation logic. They often exhibit icosahedral symmetry, which could be translated into
structured logic patterns in 3D space.
To visualize these forms, we would define rulesets that determine the logical interactions
and transformations between units, much like how physical laws govern the structure of
natural entities. By rendering these rulesets visually, we create a visual representation of
how complex logical structures might emerge from simple principles.
This type of modeling is highly abstract and crosses into areas of generative art, visual
mathematics, and computational biology. Each extrapolation represents an opportunity to
explore how patterns and logic can interplay to create rich and complex visual structures.
Each form can also embody different logical operations or interactions, allowing for a visual
analogy to computational processes or natural phenomena.
11 Conclusion
In this paper, we have explored geometrical constructs that define the optimal tessellation
for ray tracing within spatio-temporal constraints.
Our algorithmic approach serves as a substantial advancement in discretizing and han-
dling complex geometric configurations, permitting efficient computational simulations rel-
evant to a range of applications in computer graphics, optics, and robotic path planning.
The exploration of tessellation efficiencies using polar coordinates, logic vectors, and
directed graphs unveils a deeply interconnected framework between topology, computa-
tional geometry, and logic. This intricate blend not only advances our understanding of
mathematical landscapes but also opens avenues for innovative applications across robotics,
spatial analysis, and computational optics. By translating logical operations into visual rep-
resentations through tessellation, we create a richly informative method for interpreting
and manipulating complex datasets and processes.
The code structure and style discussed herein adeptly marry the abstract with the
procedural, unveiling an elegance in the computational modeling of spatial symmetries and
logic operations. Despite the computational challenges, such as the suboptimal conversion
of Cartesian to polar coordinates within iterative loops, the scaffold provided by the logic
model G is a testament to the adaptability and potential of this approach to optimize
tessellations tailored to specific applications.
Extending these concepts into three-dimensional spaces and beyond, we embark on a
journey through potential forms that mimic the foundational structures of nature, from
the double helix to crystal lattices and quantum logic gates. Each of these extrapolations
offers a unique perspective on how logical structures emerge from simple rules, invoking a
deeper appreciation for the inherent logic that shapes our world.
Potential applications, as discussed, span from visually representing quantum comput-
ing phenomena to The exploration of tessellation efficiencies using polar coordinates, logic
vectors, and directed graphs unveils a deeply interconnected framework between topology,
computational geometry, and logic. This intricate blend not only advances our understand-
ing of mathematical landscapes but also opens avenues for innovative applications across
robotics, spatial analysis, and computational optics. By translating logical operations into
visual representations through tessellation, we create a richly informative method for in-
terpreting and manipulating complex datasets and processes.
19

The code structure and style discussed herein adeptly marry the abstract with the
procedural, unveiling an elegance in the computational modeling of spatial symmetries and
logic operations. Despite the computational challenges, such as the suboptimal conversion
of Cartesian to polar coordinates within iterative loops, the scaffold provided by the logic
model G is a testament to the adaptability and potential of this approach to optimize
tessellations tailored to specific applications.
Extending these concepts into three-dimensional spaces and beyond, we embark on a
journey through potential forms that mimic the foundational structures of nature, from
the double helix to crystal lattices and quantum logic gates. Each of these extrapolations
offers a unique perspective on how logical structures emerge from simple rules, invoking a
deeper appreciation for the inherent logic that shapes our world.
Potential applications, as discussed, span from visually representing quantum comput-
ing phenomena to modeling the evolution of viral capsids. The symbolic and generative
capacities of these mathematical and computational models unfold a rich tapestry of logical
and geometric relationships that can be harnessed for varied and complex problem-solving
scenarios, from AI-driven analytics to the optimization of photorealistic rendering.
In conclusion, this paper has demonstrated the profound utility and versatility of com-
bining computational geometry with logic through tessellation. As we move forward, the
continued development and refinement of these methodologies promise to unlock further
innovations in the fields of computational design, analysis, and beyond. The path ahead
is rich with possibilities, inviting further exploration into the dynamic interplay between
geometry, logic, and the computational arts.
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