Notes

This document was produced on my Macs*, using htlatex to convert latex to html, and then hand editing the html file. (There's got to be a better way!) The graphics were done in mostly using Maple. Although I did some things with Pov-ray, which I find harder to use, but the graphics are spectacular. It's also open source (and in fact I prefered to compile the unix source since the Mac binaries seemed to contain a lot of obsolete stuff). Finally, the hyperbolic tiling in the penultimate section was created using David Joyce's applet.

Below are some samples of the code I used. I'm two lazy to include everything.

Maple Code

The first plot was given by

plot3d([r*cos(theta), r*sin(theta), sqrt(r)*cos(theta/2)], r=0..1, theta=0..4*Pi, color= sqrt(r)*sin(theta/2),grid=[20,50]);

Since the functions involved in the second set of plots are singular, I used the "view" option to truncate values. For example the plot of the real part of the ℘-function
was given by

P := (x,y) -> WeierstrassP(x + y*I,1,0);
plot3d(Re(P(x,y)), x= 0..8, y=0..8, view=-4..4, grid=[50,50]);

For example, the frames of the second animation of an elliptic curve were generated using the function "Gr" below. (Some tweaking had to be done to get the graphs to come out right, which explains some of the strange constants.)

P := (x,y) -> WeierstrassP(x + y*I,1,0);
PP := (x,y) -> WeierstrassPPrime(x + y*I,1,0);
IR := (theta, z) -> cos(theta)*Re(z) + sin(theta)*Im(z);
Gr1 := theta -> plot3d([Re(P(x,y)), Im(P(x,y)), 0.3*IR(theta,PP(x,y))] , x=0..3.74,y=0..3.74, color=[sin(2*Pi*x/3.74), 0.5, sin(2*Pi*y/3.74)], view=[-1..1,-1..1, -1..1], grid=[35,35]);
L1 := theta -> spacecurve([Re(P(x,1.854)), Im(P(x,1.854)), 0.3*IR(theta,PP(x,1.854))] , x=0..3.74, color=red, thickness=2);
L2 := theta -> spacecurve([Re(P(x,0)), Im(P(x,0)), 0.3*IR(theta, PP(x,0))] , x=1..2.7, color=red, thickness=2);
L3 := theta->spacecurve([Re(P(1.854,y)), Im(P(1.854,y)), 0.3*IR(theta,PP(1.854,y))] , y=0..3.74, color=yellow, thickness=2);
Gr := theta-> display({Gr1(theta) , L1(theta),L2(theta),L3(theta)});

The code for the plot of the the j-function is given below. Unfortunately, Maple doesn't what this function is, so it has to expressed in terms of theta functions (to simplify matters, I'm dropping some constant factors). Also I had to use a cutoff function "trun" so that Maple could handle the plot properly.

lambda := proc (tau)
local q;
q := exp(I*Pi*tau);
return JacobiTheta2(0, q)^4/JacobiTheta3(0, q)^4
end proc;
J := proc (tau)
local lam;
lam := lambda(tau);
return (1-lam+lam^2)^3/(lam^2*(1-lam)^2)
end proc;
trun := x -> 1000*arctan((1/1000)*x); plot3d(0, x = -1.5 .. 1.5, y = 0.001 .. 0.9, color = trun(Re(evalf(J(x+I*y)))), grid = [180, 180], style = patchnogrid);

Povray Code

The code for the genus two surface in the section on higher genus curves.

#include "colors.inc"
#include "functions.inc"
#declare T = function {f_torus(x,y,z,0.8,0.18)}
background{White}

camera {
location <0, 4, 4>
look_at 0
angle 36
}

light_source {<-100,200,100> colour rgb 1}

isosurface {
function { T(x-0.7,y,z)*T(x+0.7,y,z) - 0.02}
max_gradient 2
accuracy 0.001
contained_by{sphere{0,2}}
pigment {Green}
finish {phong 1}
}

The degenerate quartic in the same section:

#include "colors.inc"
camera {
location <0, .1, -65>
look_at 0
angle 30
}
background { color Gray50}
light_source { <300, 300, -1000> White }

#declare HT =difference {
torus {
4, 1
rotate -90*x
pigment { Blue }
}
box {<-5, -5, -1>, <5, 0, 1> }
}

#declare Cyl = cylinder {
<0,8,0>, <0,-8,0>, 1
pigment{Blue}
}

#declare Cyl2 = blob {
threshold .5
cylinder {
<0,8,0>, <0,-8,0>, 2, 1
pigment{Blue}
}
sphere {<1,0,0>, 3, 1 pigment {Blue}}
sphere {<1,-4,0>, 3, 1 pigment {Blue}}
sphere {<1,4,0>, 3, 1 pigment {Blue}}
}

#declare Cyl3 = blob {
threshold .5
cylinder {
<0,4,0>, <0,-4,0>, 2, 1
pigment{Green}
}
sphere {<-1,0,0>, 3, 1 pigment {Green}}
sphere {<-1,-4,0>, 3, 1 pigment {Green}}
sphere {<-1,4,0>, 3, 1 pigment {Green}}
}

union {
object {HT translate 8*y}
object {HT
rotate 180*x
translate -8*y
}
object{Cyl2 translate x*4 }
object{Cyl translate -x*4 }
object{Cyl3 translate x*9.2 }
}

*Purchasesd with funding from the NSF