1603 plane sections of real and complex tori

Plane Sections of Real and
Complex Tori
Sonoma State - February 2006
or
Why the Graph of is a Torus( )2 2
1y x x= −
Based on a presentation by David Sklar and Bruce Cohen
at Asilomar in December 2004

Part I - Slicing a Real Circular Torus
The Spiric Sections of Perseus
Ovals of Cassini and The Lemniscate of Bernoulli
Equations for the torus in R3
Other Slices
The Villarceau Circles
A Characterization of the torus

The Spiric Sections of Perseus:
The sectioning planes are parallel to the axis of rotation

ϕ θ
a b
( cos )cosx a b θ ϕ= +
( cos )siny a b θ ϕ= +
sinz b θ=
( ) ( )( ) ( ) ( )
2 22 2 2 2 2 2 2 2 2 2 2 2 2
2 2 0x y z a b x y z a b z a b+ + − + + + − − + − =
Equations of a Circular Torus
Parametric equations:
Cartesian equations:
Note: we can get a cartesian equation for a spiric section by setting y equal to a
constant. In general the left hand side equation will be an irreducible fourth degree
polynomial, but for y = 0, it factors.
( ) ( )( ) ( ) ( )
2 22 2 2 2 2 2 2 2 2 2 2
2 2 0x z a b x z a b z a b+ − + + − − + − =
( ) ( )
2 22 2 2 2
0x a z b x a z b   + + − − + − =
   
x
y
z
( ), ,x y z

( ) ( )( ) ( ) ( )
2 22 2 2 2 2 2 2 2 2 2 2 2 2
2 2 0
with 1.2, 0.4, and 1.6
x y z a b x y z a b z a b
a b y
+ + − + + + − − + − =
= = ≤

Sections with planes rotating about the x-axis
Villarceau circles

More sections with planes rotating about the x-axis
Villarceau circles

1603 plane sections of real and complex tori

A Characterization of the Torus
A complete, sufficiently smooth surface with the property
that through each point on the surface there exist exactly four
distinct circles (that lie on the surface) is a circular torus.

Bibliography
Part II - Slicing a Complex Torus
and 2 2 2 2 2 2 2
( 1 )( 2 ) ( )y x x x x g= − − −L
Elliptic curves and number theory
2 2
( 1)y x x c= − +Some graphs of
Hints of toric sections
Two closures: Algebraic and Geometric
2 2
( 1)y x x= −Algebraic closure, C2
, R4
, and the graph of
Geometric closure, Projective spaces
P1
(R), P2
(R), P1
(C), and P2
(C)
2 2
( 1),y x x= −The graphs of 2 2 2
( ),y x x n= −

2
( )( )n n
y x x a x b= − +
Roughly, an elliptic curve over a field F is the graph of an equation of the
form where p(x) is a cubic polynomial with three distinct roots
and coefficients in F. The fields of most interest are the rational numbers,
finite fields, the real numbers, and the complex numbers.
2
( )y p x=
Within a year it was shown that Fermat’s last theorem would follow from a
widely believed conjecture in the arithmetic theory of elliptic curves.
In 1985, after mathematicians had been working on Fermat’s Last Theorem
for about 350 years, Gerhard Frey suggested that if we assumed Fermat’s
Last Theorem was false, the existence of an elliptic curve
where a, b and c are distinct integers such that with integer
exponent n > 2, might lead to a contradiction.
n n n
a b c+ =
Less than 10 years later Andrew Wiles proved a form of the Taniyama
conjecture sufficient to prove Fermat’s Last Theorem.

The strategy of placing a centuries old number theory problem in the
context of the arithmetic theory of elliptic curves has led to the complete or
partial solution of at least three major problems in the last thirty years.
The Congruent Number Problem – Tunnell 1983
The Gauss Class Number Problem – Goldfeld 1976, Gross & Zagier 1986
Fermat’s Last Theorem – Frey 1985, Ribet 1986, Wiles 1995, Taylor 1995
Although a significant discussion of the theory of elliptic curves and why
they are so nice is beyond the scope of this talk, I would like to try to show
you that, when looked at in the right way, the graph of an elliptic curve is a
beautiful and familiar geometric object. We’ll do this by studying the graph
of the equation 2 2
( 1).y x x= −

2 2
( 1)y x x= −
If we close up the geometry to include points at infinity and the algebra to include
the complex numbers, we can argue that the graph of is a torus.
2 2
( 1)y x x= − 2 2
( 1) 0.3y x x= − +
2 2
( 1) 1y x x= − +2 2
( 1) 0.385y x x= − +
Graphs of 2 2
( 1)y x x c= − + : Hints of Toric Sections
x
y
x
y
x
y
x
y

Geometric Closure: an Introduction to Projective Geometry
Part I – Real Projective Geometry
One-Dimension - the Real Projective Line P1
(R)
The real (affine) line R is the
ordinary real number line
The real projective line P1
(R) is
the set { }∪ ∞R
0
It is topologically equivalent to the open
interval (-1, 1) by the map (1 )x x x+a
01− 1
and topologically equivalent to a punctured
circle by stereographic projection
0
It is topologically equivalent to a closed
interval with the endpoints identified
0 PP
0
and topologically equivalent to a circle
by stereographic projection
∞ ∞
∞ ∞
0
P∞

Two-Dimensions - the Real Projective Plane P2
(R)
The real (affine) plane R2
is
the ordinary x, y -plane
disk with antipodal points on the
boundary circle identified.
x
y
2 2 2 2
( , ) ,
1 1
x y
x y
x y x y+ + + +
a
It is topologically equivalent to
the open unit disk by the map
( )
x
y
x
y
The real projective plane P2
(R) is the
set . It is R2
together with a
“line at infinity”, . Every line in R2
intersects , parallel lines meet at the
same point on , and nonparallel
lines intersect at distinct points.
Every line in P2
(R) is a P1
(R).
2
L∞∪R
L∞
L∞
L∞
L∞
Two distinct lines
intersect at one and only
one point.

Two-Dimensions - the Real Projective Plane P2
(R)
The real (affine) plane R2
is
the ordinary x, y -plane
disk with antipodal points on the
boundary circle identified.
x
y
2 2 2 2
( , ) ,
1 1
x y
x y
x y x y+ + + +
a
It is topologically equivalent to
the open unit disk by the map
( )
x
y
x
y
The real projective plane P2
(R) is the
set . It is R2
together with a
“line at infinity”, . Every line in R2
intersects , parallel lines meet at the
same point on , and nonparallel
lines intersect at distinct points.
2
L∞∪R
L∞
L∞
L∞
L∞
Two distinct lines
intersect at one and only
one point.
Every line in P2
(R) is a P1
(R).

A Projective View of the Conics
x
y
Ellipse
x
y
x
y
Parabola
x
y
x
y
Hyperbola

A Projective View of the Conics
Ellipse Parabola
Hyperbola

2 2
( 1)y x x= − 2 2
( 1) 0.3y x x= − +
2 2
( 1) 1y x x= − +2 2
( 1) 0.385y x x= − +
Graphs of 2 2
( 1)y x x c= − + : Hints of Toric Sections
including point
topological view
at infinity
If we close up the algebra, by extending to the , and the geometry,complex numbers
2 2
by including points at infinity we can argue that the graph of ( 1) is a torus.y x x= −
x
y
x
y
x
y
x
y

Graph of with x and y complex
Algebraic closure
2 2
( 1)y x x= −
1 2 1 2Let andx x ix y y iy= + = +
2 2 2 2
1 2 1 2 1 2Then ( 1) becomes ( ) ( )[( ) 1]y x x y iy x ix x ix= − + = + + −
Equating real and imaginery parts we have
2 2 3 2 3 2
1 2 1 1 1 2 1 2 2 2 1 23 and 2 3y y x x x x y y x x x x− = − − = − − +
2 2 3 2 3 2
1 2 1 2 1 1 1 2 2 2 1 2( ) (2 ) ( 3 ) ( 3 )y y i y y x x x x i x x x x− + = − − + − − +
Expanding and collecting terms we have
It's not so easy to graph a 2-surface in 4-space, but we can look
at intersections of the graph with some convenient planes.
is a system of two equations in four real
2 2 3 2
1 2 1 1 1 23y y x x x x− = − −
unknowns whose graph is a 2-dimensional surface in real 4-dimensional space
3 2
1 2 2 2 1 22 3y y x x x x= − − +

Algebraic closure
2 2
( 1)y x x= −
1 2 1 2Letting and , then solving for and in terms of and ,x s x t y y s t= =
1 2 1 1 2 2we would essentially have , , ( , ) and ( , )x s x t y y s t y y s t= = = =
1 2for ( , ) and ( , ) which can be pieced together to get the whole graph.y s t y s t
The situation is a little more complicated in that the algebra leads to several solutions
2 2 3 2
1 2 1 1 1 23y y x x x x− = − −
3 2
1 2 2 2 1 22 3y y x x x x= − − +
Some comments on why the graph of the system
is a surface.
1 2 1 2These are parametric equations for a surface in , , , spacex x y y
1 2 1 2(a nice mapping of a 2-D , plane into 4-D , , , space.)s t x x y y

Algebraic closure
2 2
( 1)y x x= −
1x
1y
1x
2y
2 2
1 1 1( 1)y x x= −
2 1for 0, 0x y= =
2 2
2 1 1( 1)y x x= − −
2 2 3 2
1 2 1 1 1 23y y x x x x− = − −
3 2
1 2 2 2 1 22 3y y x x x x= − − +
2 2for 0, 0x y= =
1 1(the , - plane)x y
2 2
( 1) becomesy x x= −
1 2(the , - plane)x y
2 2
( 1) becomesy x x= −
2
1 1( )[( ) 1]x x= − − −

Graph of with x and y complex2 2
( 1)y x x= −
2 2 3
1 2 1 1
1 22 0
y y x x
y y
 − = −

=
The system of equations becomes
2 2 2
Recall, the graph of ( 1) in is equivalent to the graph of the systemy x x= − C
2 2 3 2
1 2 1 1 1 2
3 2
1 2 2 2 1 2
3
2 3
y y x x x x
y y x x x x
 − = − −

= − − +
Now lets look at the intersection of4
in .R
2the graph with the 3-space 0.x = 1x
2y
1y
2 1so 0 or 0y y= =
1 1 1 2and the intersection (a curve) lies in only the , - plane or the , - plane.x y x y
1x
1y
1x
2y
2for 0,y = 2 2
1 1 1( 1)y x x= − 1for 0,y = 2 2
2 1 1( 1)y x x= − −

( 1)y x x= −
1x
1y
2y
1x
1y
1x
2y
2
1 1 1 2
The intersection of the graph with the 3-space 0 is a curve whose branches
lie only the , - plane or the , - plane so we can put together this picture.
x
x y x y
=
2 2 4
2( 1) in intersecting the 3-space =0y x x x= − R
P∞ 1− 0 1+
2
Topological view in projective C
2
(roughly with points at infinity)C

Part II – Complex Projective Geometry
One-Dimension - the Complex Projective Line or Riemann Sphere P1
(C)
The complex (affine) line C is the
ordinary complex plane where (x, y)
corresponds to the number z = x + iy.
x
y
It is topologically a punctured sphere
by stereographic projection
The complex projective line P1
(C) is
the set the complex plane
with one number adjoined.
{ }∪ ∞C
∞
It is topologically a sphere by
stereographic projection with the
north pole corresponding to . It is
often called the Riemann Sphere.
∞
(Note: 1-D over the complex numbers, but, 2-D over the real numbers)

Part II – Complex Projective Geometry
Two-Dimensions - the Complex Projective Plane P2
(C)
The complex (affine) “plane” C2
or
better complex 2-space is a lot like
R4
. A line in C2
is the graph of an
equation of the form ,
where a, b and c are complex
constants and x and y are complex
variables. (Note: not every plane in
R4
corresponds to a complex line)
ax by c+ =
(Note: 2-D over the complex numbers, but, 4-D over the real numbers)
Complex projective 2-space P2
(C) is
the set . It is C2
together with
a complex “line at infinity”, . Every
line in R2
intersects , parallel lines
meet at the same point on , and
nonparallel lines intersect at
distinct points.
2
L∞∪C
L∞
L∞
L∞
L∞
Two distinct lines intersect at one and
only one point.
Every complex line in P2
(C) is a P1
(C),
a Riemann sphere, including the “line
at infinity”.
Basically P2
(C) is C2
closed up nicely
by a Riemann Sphere at infinity.

P∞ 1− 0 1+
P∞ 1− 0 1+
( 1)y x x= −
1x
1y
2y
2 2 4
2( 1) in intersecting the 3-space =0y x x x= − R
2intersecting the 3-space = > 0x ε 2
Topological view in projective C

2 2 2 2 2 2 2
( 1 )( 2 ) ( )y x x x x g= − − −L
A Generalization: the Graph of
2 2 2 2 2 2 2 2 2 2 2
The graph of ( 1 )( 2 )( 3 )( 4 )( 5 )y x x x x x x= − − − − −
2intersected with the 3-space 0x =
1x
1y
2y

2 2 2 2 2 2 2
( 1 )( 2 ) ( )y x x x x g= − − −L
2 2 2 2 2 2 2 2 2 2 2
( 1 )( 2 )( 3 )( 4 )( 5 )y x x x x x x= − − − − −
A Generalization: the Graph of

A depiction of the toric graphs
of the elliptic curves
2 2 2
( )y x x n= −
by A. T. Fomenko
This drawing is the frontispiece
of Neal Koblitz's book
Introduction to Elliptic Curves
and Modular Forms

Bibliography
8. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford 1997
1. E. Brieskorn & H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag,
Basel, 1986
5. K. Kendig, Elementary Algebraic Geometry, Springer-Verlag, New York 1977
7. Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons,
New York, 1973
9. J. Stillwell, Mathematics and Its History, Springer-Verlag, New York 1989
6. Z. A. Melzak, Invitation to Geometry, John Wiley & Sons, New York, 1983
3. D. Hilbert & H. Cohn-Vossen, Geometry and the Imagination, Chelsea
Publishing Company, New York, 1952
4. N. Koblitz, Introduction to Elliptic Curves and Modular Forms,
Springer-Verlag, New York 1984
10. M. Villarceau, "Théorème sur le tore." Nouv. Ann. Math. 7, 345-347, 1848.
2. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York
19872. B. Cohen, Website; http://www.cgl.ucsf.edu/home/bic

1603 plane sections of real and complex tori

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1603 plane sections of real and complex tori