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

2. 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 R 3 Other Slices The Villarceau Circles A Characterization of the torus

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

4. More Spiric Sections

5. 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.

7. Sections with planes rotating about the x-axis Villarceau circles

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

10. 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.

11. Bibliography Part II - Slicing a Complex Torus Elliptic curves and number theory Some graphs of Hints of toric sections Two closures: Algebraic and Geometric Algebraic closure, C 2, R 4, and the graph of Geometric closure, Projective spaces P 1 (R), P 2 (R), P 1 (C), and P 2 (C) The graphs of

12. Elliptic curves and number theory 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. 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. Less than 10 years later Andrew Wiles proved a form of the Taniyama conjecture sufficient to prove Fermat’s Last Theorem.

13. 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

14. 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. Graphs of: Hints of Toric Sections

15. Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry One-Dimension - the Real Projective Line P 1 (R) The real (affine) line R is the ordinary real number line The real projective line P 1 (R) is the set It is topologically equivalent to the open interval (-1, 1) by the map and topologically equivalent to a punctured circle by stereographic projection It is topologically equivalent to a closed interval with the endpoints identified and topologically equivalent to a circle by stereographic projection

16. Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry Two-Dimensions - the Real Projective Plane P 2 (R) The real (affine) plane R 2 is the ordinary x, y -plane It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified. It is topologically equivalent to the open unit disk by the map ( ) The real projective plane P 2 (R) is the set. It is R 2 together with a “line at infinity”,. Every line in R 2 intersects, parallel lines meet at the same point on, and nonparallel lines intersect at distinct points. Every line in P 2 (R) is a P 1 (R). Two distinct lines intersect at one and only one point.

17. Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry Two-Dimensions - the Real Projective Plane P 2 (R) The real (affine) plane R 2 is the ordinary x, y -plane It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified. It is topologically equivalent to the open unit disk by the map ( ) The real projective plane P 2 (R) is the set. It is R 2 together with a “line at infinity”,. Every line in R 2 intersects, parallel lines meet at the same point on, and nonparallel lines intersect at distinct points. Two distinct lines intersect at one and only one point. Every line in P 2 (R) is a P 1 (R).

18. A Projective View of the Conics Ellipse Parabola Hyperbola

19. A Projective View of the Conics Ellipse Parabola Hyperbola

21. Graphs of: Hints of Toric Sections

22. Graph of with x and y complex Algebraic closure

23. Graph of with x and y complex Algebraic closure Some comments on why the graph of the system is a surface.

24. Graph of with x and y complex Algebraic closure

25. Graph of with x and y complex

27. Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry One-Dimension - the Complex Projective Line or Riemann Sphere P 1 (C) The complex (affine) line C is the ordinary complex plane where (x, y) corresponds to the number z = x + iy. It is topologically a punctured sphere by stereographic projection The complex projective line P 1 (C) is the set the complex plane with one number adjoined. 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)

28. Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry Two-Dimensions - the Complex Projective Plane P 2 (C) The complex (affine) “plane” C 2 or better complex 2-space is a lot like R 4. A line in C 2 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 R 4 corresponds to a complex line) (Note: 2-D over the complex numbers, but, 4-D over the real numbers) Complex projective 2-space P 2 (C) is the set. It is C 2 together with a complex “line at infinity”,. Every line in R 2 intersects, parallel lines meet at the same point on, and nonparallel lines intersect at distinct points. Two distinct lines intersect at one and only one point. Every complex line in P 2 (C) is a P 1 (C), a Riemann sphere, including the “line at infinity”. Basically P 2 (C) is C 2 closed up nicely by a Riemann Sphere at infinity.

29. Graph of with x and y complex

31. A Generalization: the Graph of

34. 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 1987 2. B. Cohen, Website; http://www.cgl.ucsf.edu/home/bic