1. Slicing Bagels: Plane Sections of Real and Complex Tori Asilomar - December 2004 Bruce Cohen Lowell High School, SFUSD bic@cgl.ucsf.edu http://www.cgl.ucsf.edu/home/bic David Sklar San Francisco State University dsklar46@yahoo.com

2. Part I - Slicing a Real Circular Torus Equations for the torus in R 3 The Spiric Sections of Perseus Ovals of Cassini and The Lemniscate of Bernoulli Other Slices The Villarceau Circles A Characterization of the torus

3. 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

4. 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 Fermats last theorem would follow from a widely believed conjecture in the arithmetic theory of elliptic curves. In 1985, after mathematicians had been working on Fermats Last Theorem for about 350 years, Gerhard Frey suggested that if we assumed Fermats 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 Fermats Last Theorem.

5. 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 Fermats 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. Well do this by studying the graph of the equation

6. Graphs of: Hints of Toric Sections If we close up the algebra to include the complex numbers and the geometry to include points at infinity, we can argue that the graph of is a torus.

7. 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

8. 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.

9. A Projective View of the Conics Ellipse Parabola Hyperbola

10. A Projective View of the Conics Ellipse Parabola Hyperbola

11. Graphs of: Hints of Toric Sections

12. Graph of with x and y complex Algebraic closure

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

14. Graph of with x and y complex Algebraic closure

15. Graph of with x and y complex

17. 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)

18. 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. Every 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. Two distinct lines intersect at one and only one point.

19. Graph of with x and y complex

21. A Generalization: the Graph of

24. 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