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Plane Sections of Real and
Complex Tori
or
Why the Graph of y  x  x  1 is a Torus
2
2
Based on a presentation by David Sklar and Bruce Cohen
at Asilomar in December 2004
Sonoma State - February 2006
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
The Spiric Sections of Perseus:
The sectioning planes are parallel to the axis of rotation
More Spiric Sections
Equations of a Circular Torus
z
Parametric equations:
 x, y, z 
x  (a  b cos )cos 
y  (a  b cos )sin 
a

z  b sin
y
b

x
Cartesian equations:
 x2  y 2  z 2   2  a2  b2  x2  y 2  z 2   2  a2  b2  z 2   a2  b2   0
2
2
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.
x
2
z

2 2
 2a  b
2
2
 x
2
z
2
  2a
2
 b  z  a  b
2
2
2
 x  a 2  z 2  b2   x  a 2  z 2  b2   0




2 2
0
 x2  y 2  z 2   2  a2  b2  x2  y 2  z 2   2  a2  b2  z 2   a2  b2   0
2
2
with a  1.2, b  0.4, and y  1.6
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.
Part II - Slicing a Complex Torus
Elliptic curves and number theory
Some graphs of y 2  x( x 2  1)  c
Hints of toric sections
Two closures: Algebraic and Geometric
Algebraic closure, C2, R4, and the graph of y 2  x( x 2  1)
Geometric closure, Projective spaces
P1(R), P2(R), P1(C), and P2(C)
The graphs of y 2  x( x 2  1), y 2  x( x 2  n2 ),
and y 2  x( x 2  12 )( x 2  22 )
Bibliography
( x2  g 2 )
Elliptic curves and number theory
Roughly, an elliptic curve over a field F is the graph of an equation of the
form y 2  p( x ) 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.
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
y 2  x( x  a n )( x  bn )
where a, b and c are distinct integers such that a n  bn  c n with integer
exponent n > 2, might lead to a contradiction.
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.
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 y 2  x( x 2  1).
Graphs of y 2  x( x 2  1)  c : Hints of Toric Sections
y
y
x
y 2  x( x 2  1)
x
y 2  x( x 2  1)  0.3
y
y
x
y 2  x( x 2  1)  0.385
x
y 2  x( x 2  1)  1
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 y 2  x( x2  1) is a torus.
Geometric Closure: an Introduction to Projective Geometry
Part I – Real Projective Geometry
One-Dimension - the Real Projective Line P1(R)
The real projective line P1(R) is
the set R  
The real (affine) line R is the
ordinary real number line


0
0
It is topologically equivalent to the open
interval (-1, 1) by the map x x (1  x )
1
0
It is topologically equivalent to a closed
interval with the endpoints identified
P
1
0
P
and topologically equivalent to a punctured and topologically equivalent to a circle
circle by stereographic projection
by stereographic projection
P
0

0

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
y
x
It is topologically equivalent to
the open unit disk by the map
(x , y )
(1
x
,
y
x2  y2 1  x2  y2
)
The real projective plane P2(R) is the
set R 2  L. It is R2 together with a
“line at infinity”, L . Every line in R2
intersects L , parallel lines meet at the
same point on L , and nonparallel
lines intersect L at distinct points.
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.
y
y
x
x
Two distinct lines intersect
at one and only one point.
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
y
x
It is topologically equivalent to
the open unit disk by the map
(x , y )
(1
x
,
y
x2  y2 1  x2  y2
)
The real projective plane P2(R) is the
set R 2  L. It is R2 together with a
“line at infinity”, L . Every line in R2
intersects L , parallel lines meet at the
same point on L , and nonparallel
lines intersect L at distinct points.
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.
y
y
x
x
Two distinct lines intersect
at one and only one point.
A Projective View of the Conics
y
y
x
y
x
x
Parabola
Ellipse
y
y
x
Hyperbola
x
A Projective View of the Conics
Ellipse
Parabola
Hyperbola
Graphs of y 2  x( x 2  1)  c : Hints of Toric Sections
y
y
x
x
topological view
including point
at infinity
y 2  x( x 2  1)
y
y 2  x( x 2  1)  0.3
y
x
y 2  x( x 2  1)  0.385
x
y 2  x( x 2  1)  1
If we close up the algebra, by extending to the complex numbers, and the geometry,
by including points at infinity we can argue that the graph of y 2  x( x 2  1) is a torus.
Graph of y 2  x( x 2  1) with x and y complex
Algebraic closure
Let x  x1  ix2 and y  y1  iy2
Then y 2  x( x 2  1) becomes ( y1  iy2 )2  ( x1  ix2 )[( x1  ix2 ) 2  1]
Expanding and collecting terms we have
( y12  y2 2 )  i (2 y1 y2 )  ( x13  x1  3x1 x2 2 )  i (  x2 3  x2  3x12 x2 )
Equating real and imaginery parts we have
y12  y2 2  x13  x1  3x1 x2 2
and
2 y1 y2   x2 3  x2  3x12 x2
y12  y2 2  x13  x1  3x1 x2 2
2 y1 y2   x2 3  x2  3x12 x2
is a system of two equations in four real
unknowns whose graph is a 2-dimensional surface in real 4-dimensional space
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.
Graph of y 2  x( x 2  1) with x and y complex
Algebraic closure
y12  y2 2  x13  x1  3x1 x2 2
Some comments on why the graph of the system
2 y1 y2   x2 3  x2  3x12 x2
is a surface.
Letting x1  s and x2  t , then solving for y1 and y2 in terms of s and t ,
we would essentially have x1  s, x2  t , y1  y1 (s, t ) and y2  y2 (s, t )
These are parametric equations for a surface in x1 , x2 , y1 , y2 space
(a nice mapping of a 2-D s, t plane into 4-D x1, x2 , y1, y2 space.)
The situation is a little more complicated in that the algebra leads to several solutions
for y1 ( s, t ) and y2 ( s, t ) which can be pieced together to get the whole graph.
Graph of y 2  x( x 2  1) with x and y complex
Algebraic closure
y12  y2 2  x13  x1  3x1 x2 2
2 y1 y2   x2 3  x2  3x12 x2
for x2  0, y2  0
(the x1 , y1 - plane)
y 2  x( x 2  1) becomes
y12  x1 ( x12  1)
y1
for x2  0, y1  0
(the x1, y2 - plane)
y 2  x( x 2  1) becomes
y2 2   x1 ( x12  1)  (  x1 )[(  x1 )2  1]
y2
x1
x1
Graph of y 2  x( x 2  1) with x and y complex
Recall, the graph of y 2  x( x 2  1) in C2 is equivalent to the graph of the system
 y12  y2 2  x13  x1  3x1 x2 2
4
in
R
. Now lets look at the intersection of

3
2
y2
 2 y1 y2   x2  x2  3x1 x2
the graph with the 3-space x2  0.
The system of equations becomes
y1
x1
 y12  y2 2  x13  x1

2 y1 y2  0

so y2  0 or y1  0
and the intersection (a curve) lies in only the x1, y1 - plane or the x1, y2 - plane.
for y2  0,
y12  x1 ( x12  1)
for y1  0, y2 2   x1 ( x12  1)
y2
y1
x1
x1
Graph of y 2  x( x 2  1) with x and y complex
The intersection of the graph with the 3-space x2  0 is a curve whose branches
lie only the x1 ,y1 - plane or the x1,y2 - plane so we can put together this picture.
y1
y2
x1
x1
y 2  x ( x 2  1) in R 4 intersecting the 3-space x2 =0
y2
Topological view in projective C2
(roughly C2 with points at infinity)
y1
x1
P
1
0
1
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.
The complex projective line P1(C) is
the set C   the complex plane
with one number  adjoined.
y
x
It is topologically a punctured sphere
by stereographic projection
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
Two-Dimensions - the Complex Projective Plane P2(C)
(Note: 2-D over the complex numbers, but, 4-D over the real numbers)
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 ax  by  c,
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)
Complex projective 2-space P2(C) is
the set C2  L . It is C2 together with
a complex “line at infinity”, L . Every
line in R2 intersects L , parallel lines
meet at the same point on L , and
nonparallel lines intersect L at
distinct points.
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.
Graph of y 2  x( x 2  1) with x and y complex
The intersection of the graph with the 3-space x2  0 is a curve whose branches
lie only the x1 ,y1 - plane or the x1,y2 - plane so we can put together this picture.
y1
y2
x1
x1
y 2  x ( x 2  1) in R 4 intersecting the 3-space x2 =0
y2
Topological view in projective C2
(roughly C2 with points at infinity)
y1
x1
P
1
0
1
Graph of y 2  x( x 2  1) with x and y complex
y2
y1
P
x1
1
0
y 2  x ( x 2  1) in R 4 intersecting the 3-space x2 =0
intersecting the 3-space x2 =  > 0
P
1
0
Topological view in projective C2
1
1
A Generalization: the Graph of
y 2  x( x 2  12 )( x 2  22 )
( x2  g 2 )
y2
y1
x1
The graph of y 2  x( x 2  12 )( x 2  22 )( x 2  32 )( x 2  42 )( x 2  52 )
intersected with the 3-space x2  0
A Generalization: the Graph of
y 2  x( x 2  12 )( x 2  22 )
( x2  g 2 )
y 2  x( x 2  12 )( x 2  22 )( x 2  32 )( x 2  42 )( x 2  52 )
A depiction of the toric graphs
of the elliptic curves
y 2  x( x 2  n 2 )
by A. T. Fomenko
This drawing is the frontispiece
of Neal Koblitz's book
Introduction to Elliptic Curves
and Modular Forms
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.