Plane Sections of Real and Complex Tori

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

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 R3 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. Villarceau circles Sections with planes rotating about the x-axis

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

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. Some graphs of Algebraic closure, C2, R4, and the graph of Part II - Slicing a Complex Torus Elliptic curves and number theory Hints of toric sections Two closures: Algebraic and Geometric Geometric closure, Projective spaces P1(R), P2(R), P1(C), and P2(C) The graphs of Bibliography

12. 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. where a, b and c are distinct integers such that with integer exponent n > 2, might lead to a contradiction. Elliptic curves and number theory 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 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. Less than 10 years later Andrew Wiles proved a form of the Taniyama conjecture sufficient to prove Fermat’s Last Theorem.

13. 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 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. TheCongruent Number Problem – Tunnell 1983 TheGauss Class Number Problem – Goldfeld 1976, Gross & Zagier 1986 Fermat’s Last Theorem – Frey 1985, Ribet 1986, Wiles 1995, Taylor 1995

14. Graphs of : Hints of Toric Sections 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.

15. The real projective line P1(R) is the set It is topologically equivalent to the open interval (-1, 1) by the map 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 It is topologically equivalent to a closed interval with the endpoints identified and topologically equivalent to a punctured circle by stereographic projection and topologically equivalent to a circle by stereographic projection

16. 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). It is topologically equivalent to the open unit disk by the map ( ) Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry Two-Dimensions - the Real Projective Plane P2(R) The real (affine) plane R2is the ordinary x, y -plane It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified. Two distinct lines intersect at one and only one point.

17. 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. It is topologically equivalent to the open unit disk by the map ( ) Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry Two-Dimensions - the Real Projective Plane P2(R) The real (affine) plane R2is the ordinary x, y -plane Every line in P2(R) is a P1(R). It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified. Two distinct lines intersect at one and only one point.

18. Parabola Ellipse Hyperbola A Projective View of the Conics

19. Ellipse Parabola Hyperbola A Projective View of the Conics

21. Graphs of : Hints of Toric Sections

22. Graph of with x and y complex Algebraic closure

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

24. Graph of with x and y complex Algebraic closure

25. Graph of with x and y complex

26. Graph of with x and y complex

27. The complex projective line P1(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. Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry One-Dimension - the Complex Projective Line or Riemann Sphere P1(C) (Note: 1-D over the complex numbers, but, 2-D over the real numbers) 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

28. 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. The complex (affine) “plane” C2or 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) Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry Two-Dimensions - the Complex Projective Plane P2(C) (Note: 2-D over the complex numbers, but, 4-D over the real numbers) 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. Two distinct lines intersect at one and only one point.

29. Graph of with x and y complex

30. Graph of with x and y complex

31. A Generalization: the Graph of

32. A Generalization: the Graph of

34. Bibliography 1. E. Brieskorn & H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag, Basel, 1986 2. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York 1987 2. B. Cohen, Website; http://www.cgl.ucsf.edu/home/bic 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 5. K. Kendig, Elementary Algebraic Geometry, Springer-Verlag, New York 1977 6. Z. A. Melzak, Invitation to Geometry, John Wiley & Sons, New York, 1983 7. Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons, New York, 1973 8. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford 1997 9. J. Stillwell, Mathematics and Its History, Springer-Verlag, New York 1989 10. M. Villarceau, "Théorème sur le tore." Nouv. Ann. Math.7, 345-347, 1848.