Mathematical fumblings: the Klein j-invariant and near-integers

[mmcirvin]

On Google+: Why e^(pi * sqrt(163)) is almost, but not quite, an integer, and how it relates to a pretty function called the Klein j-invariant. With some pictures, and bonus references to Ozma of Oz and an April Fool prank by Martin Gardner.

I am a piker compared to oonh, and there's some handwaving in the math because I don't understand it all, but I

Comments

Here's the script

[oonh]

#!/usr/bin/python

from mpmath import *
import pylab
import mpmath.libmp

def g(t,z):
return (fp.cos(pi*t/2.0)*z-fp.sin(pi*t/2.0))/(fp.sin(pi*t/2.0)*z+fp.cos(pi*t/2.0))

for a in range(0,1024):

fp.cplot(lambda z: fp.kleinj(tau=g(a/1024.0,z)), [-1,2], [0,1.5], points=100000, verbose=True, file='gen%04d.png' % a)