Slicing Bagels: Plane Sections of Real and Complex Tori

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Slicing Bagels Plane Sections of Real and
Complex Tori
Bruce Cohen Lowell High School,
SFUSD bic_at_cgl.ucsf.edu http//www.cgl.ucsf.edu/hom
e/bic
David Sklar San Francisco State
University dsklar46_at_yahoo.com
Asilomar - December 2004
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Part I - Slicing a Real Circular Torus
Equations for the torus in R3 The Spiric
Sections of Perseus Ovals of Cassini and The
Lemniscate of Bernoulli Other Slices The
Villarceau Circles A Characterization of the
torus
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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)
Bibliography
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Elliptic curves and number theory
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
Within a year it was shown that Fermats 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 Fermats Last Theorem.
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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
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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
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Geometric Closure an Introduction to Projective
Geometry Part 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.
Two distinct lines intersect at one and only one
point.
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A Projective View of the Conics
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A Projective View of the Conics
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Graph of with x and y
complex Algebraic closure
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Graph of with x and y
complex Algebraic closure
Some comments on why the graph of the system
is a surface.
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Graph of with x and y
complex Algebraic closure
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Graph of with x and y
complex
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Graph of with x and y
complex
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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
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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)
Two distinct lines intersect at one and only one
point.
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Graph of with x and y
complex
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Graph of with x and y
complex
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A Generalization the Graph of
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A Generalization the Graph of
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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
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.