Plane Sections of Real and Complex Tori

Sonoma State - February 2006

orWhy the Graph of is a Torus 2 2 1y x x

Based on a presentation by David Sklar and Bruce Cohenat Asilomar in December 2004

Part I - Slicing a Real Circular Torus

The Spiric Sections of Perseus

Ovals of Cassini and The Lemniscate of BernoulliEquations 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

More Spiric Sections

a b

( cos )cosx a b ( cos )siny a b

sinz b

2 22 2 2 2 2 2 2 2 2 2 2 2 22 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 22 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 22 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

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

Elliptic curves and number theory2 2( 1)y x x c Some graphs of

Hints of toric sections

Two closures: Algebraic and Geometric2 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

Elliptic curves and number theory

2 ( )( )n ny 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 na b c

Less than 10 years later Andrew Wiles proved a form of the Taniyama conjecture sufficient to prove Fermat’s Last Theorem.

Elliptic curves and number theory

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 GeometryPart I – Real Projective Geometry

One-Dimension - the Real Projective Line P1(R)

The real (affine) line R is theordinary 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

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

Geometric Closure: an Introduction to Projective GeometryPart I – Real Projective Geometry

Two-Dimensions - the Real Projective Plane P2(R)

The real (affine) plane R2 is the ordinary x, y -plane

It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified.

x

y

2 2 2 2 ( , ) ,

1 1x yx yx y x y

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 LR

L

L

L

L

Two distinct lines intersect at one and only one point.

Geometric Closure: an Introduction to Projective GeometryPart I – Real Projective Geometry

Two-Dimensions - the Real Projective Plane P2(R)

The real (affine) plane R2 is the ordinary x, y -plane

It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified.

x

y

2 2 2 2 ( , ) ,

1 1x yx yx y x y

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 LR

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 numbers2 2by 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 complexAlgebraic closure

2 2( 1)y x x

1 2 1 2Let and x x ix y y iy

2 2 2 21 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 21 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 21 2 2 2 1 22 3y y x x x x

Graph of with x and y complexAlgebraic 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 tThe situation is a little more complicated in that the algebra leads to several solutions

2 2 3 21 2 1 1 1 23y y x x x x

3 21 2 2 2 1 22 3y y x x x x

Some comments on why the graph of the systemis 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

Graph of with x and y complexAlgebraic closure

2 2( 1)y x x

1x

1y

1x

2y

2 21 1 1( 1)y x x

2 1for 0, 0x y

2 22 1 1( 1)y x x

2 2 3 21 2 1 1 1 23y y x x x x

3 21 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 31 2 1 1

1 22 0y y x x

y y

The system of equations becomes

2 2 2Recall, the graph of ( 1) in is equivalent to the graph of the systemy x x C2 2 3 2

1 2 1 1 1 23 2

1 2 2 2 1 2

32 3y y x x x x

y y x x x x

Now lets look at the intersection of4in .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

2y2for 0,y 2 2

1 1 1( 1)y x x 1for 0,y 2 22 1 1( 1)y x x

Graph of with x and y complex2 2( 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.

xx y x y

2 2 42 ( 1) in intersecting the 3-space =0 y x x x R

P 1 0 1

2 Topological view in projective C2 (roughly with points at infinity)C

Geometric Closure: an Introduction to Projective GeometryPart 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)

Geometric Closure: an Introduction to Projective GeometryPart 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 LC

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.

Graph of with x and y complex2 2( 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.

xx y x y

2 2 42 ( 1) in intersecting the 3-space =0 y x x x R

P 1 0 1

2 Topological view in projective C2 (roughly with points at infinity)C

P 1 0 1

P 1 0 1

Graph of with x and y complex2 2( 1)y x x

1x

1y

2y

2 2 42 ( 1) in intersecting the 3-space =0 y x x x R

2 intersecting the 3-space = > 0 x 2 Topological view in projective C

2 2 2 2 2 2 2( 1 )( 2 ) ( )y x x x x g

A Generalization: the Graph of

2 2 2 2 2 2 2 2 2 2 2The graph of ( 1 )( 2 )( 3 )( 4 )( 5 )y x x x x x x

2intersected with the 3-space 0x

1x1y

2y

2 2 2 2 2 2 2( 1 )( 2 ) ( )y x x x x g

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