Icosahedral tiling of the sphere

- My original notes (also now published in Expositiones Mathematicae).
- The slides from the talk I gave about this for the IMSA conference in March, 2013.
- This GitHub repository where the solution is implemented in python.

Given the quintic equation: $$ y^5 + 5y + \gamma = 0 $$ set: $$ \nabla = \sqrt{\gamma^4 + 256}\\ Z = \frac{1}{2\cdot 1728}[2\cdot 1728 + 207\gamma^4 + \gamma^8 - \gamma^2 (81 + \gamma^4)\nabla]\\ z = \frac{{}_2F_1(\frac{31}{60}, \frac{11}{60}; \frac{6}{5}; Z^{-1})} {(1728Z)^{1/5}{}_2F_1(\frac{19}{60}, -\frac{1}{60}; \frac{4}{5}; Z^{-1})} $$ and: $$ f(z) = z(z^{10} + 11z^5 - 1)\\ H(z) = -(z^{20} + 1) + 228(z^{15} - z^5) - 494z^{10}\\ T(z) = (z^{30} + 1) + 522(z^{25} - z^5) - 10005(z^{20} + z^{10})\\ B(z) = -1 - z - 7(z^2 - z^3 + z^5 + z^6) + z^7 - z^8\\ D(z) = -1 + 2z + 5z^2 + 5z^4 - 2z^5 - z^6 $$ Then: $$ y = -\gamma\cdot\frac{f(z)}{H(z)/B(z)} - \frac{7\gamma^2 + 9\nabla}{2\gamma(\gamma^4 + 648)} \cdot\frac{D(z)T(z)}{f(z)^2H(z)/B(z)} $$ is a root!

Replacing \( z\) with \( e^{2\pi\nu i/5}z\) for \( \nu=1, 2, 3, 4\) provides all the other roots.

The above might look a little messy but it's really quite neat from the right point of view. Given this solution, a strong hint at the geometric connection is the five auxiliary polynomials \( f, H, T, B, D\). The roots of the first three \( f, H, T\) are, respectively, the locations of the projection of the vertices, face centres and edge midpoints of a regular icosahedron onto its circumsphere (once this circumsphere has been identified with the extended complex plane by stereographic projection). The roots of the last two polynomials, \( B, D\) are, respectively, the locations of the vertices and face centres of a regular cube inscribed in the icosahedron. Indeed the vertices of this cube are the vertices of a pair of dual tetrahedra inscribed in the icosahedron, one of which I show below:

Icosahedron with inscribed tetrahedron

The even explains the grouping \( H/B\) above: \( B\) is a factor of \( H\) because the vertices of a cube can be placed at the face centres of 8 of the 20 icosahedral faces.
Another important detail to notice above is that the non-radical functions employed are certain hypergeometric functions. Although they may look somewhat arbitrary at first sight, these are extremely special hypergeometric functions which belong to Schwarz's list of algebraic hypergeometric functions (those with finite monodromy). The quotient of the pair used in the above equation inverts the 60-fold branched covering of the complex projective line with \( A_5\) monodromy. The picture at the top of this post is a picture of that branched covering as well as being a picture of the icosahedral tiling of the sphere.

Also important in the above is the invariant \( Z\) since it has a neat geometric interpretation. The vector of roots of a quintic defines an \( S_5\)-orbit in complex projective 4-space. If the quintic has no degree 4 or 3 terms, this orbit lies on the quadric surface and in fact the \( S_5\)-action on the quadric is induced from a natural action on the lines in this quadric. If we adjoin a square root \( \nabla\) of the discriminant then the \( S_5\)-action becomes an \( A_5\)-action and each of the two families of lines in the ruled surface carries and \( A_5\)-action. These families are parameterised by complex projective lines and the quotient by \( A_5\) defines the icosahedral invariants. Evidently, it is this quotient that we invert using the hypergeometric functions mentioned above.

- As the Galois group of a general quintic (together with a distinguished square root of its discriminant)
- As the group of rotations of the icosahedron
- As the monodromy group of the hypergeometric differential equation with appropriate parameters (from Schwarz's list)

To give but one instance, the set of all rotations which twirl a regular icosahedron (twenty-sided regular solid) about its axes of symmetry, so that after any rotation of the set the volume of the solid occupies the same space as before, forms a group, and this group of rotations, when expressed abstractly, is the same group as that which appears, under permutations of the roots, when we attempt to solve the general equation of the fifth degree. [...] This beautiful unification was the work of Felix Klein (1849-1925) in his book on the icosahedron (1884).

Last year (2011) I could no longer stand to live in ignorance so I bought a copy of Klein's "*Lectures on the Icosahedron and the Solution of the Fifth Degree*" and started reading it and various other sources. I made a few notes for myself and as they took shape I thought it might be worth making them available online. Part of my motivation for doing so was this Mathoverflow question.

```
import mpmath, numpy as N
def nabla(cc):
return N.exp(N.log(cc**4 + 256 + 0j)/2)
def Z(cc):
return (2*1728 + cc**4*(207 + cc**4) - cc*cc*(81 + cc**4)*nabla(cc))/(2*1728)
def z(ZZ):
mpmath.mp.dps = 20
return N.exp(-N.log(1728*ZZ) / 5) * mpmath.hyp2f1(31.0/60, 11.0/60, 6.0/5, 1.0/ZZ) /\
mpmath.hyp2f1(19.0/60, -1.0/60, 4.0/5, 1.0/ZZ)
def HB(zz):
return (zz**4 - 3*zz**3 - zz**2 + 3*zz + 1) *\
(zz**8 + 4*zz**7 + 7*zz**6 + 2*zz**5 + 15*zz**4 - 2*zz**3 + 7*zz**2 - 4*zz + 1)
def D(zz):
return -1 + 2*zz + 5*zz**2 + 5*zz**4 - 2*zz**5 - zz**6
def f(zz):
u = zz**5
return zz*(-1 + u * (11 + u))
def T(zz):
u = zz**5
return 1 + u * (-522 + u * (-10005 + u * u * (-10005 + u * (522 + u))))
def y(cc):
ys = []
eps = N.exp(2*N.pi*1j / 5)
zz = z(Z(cc))
for n in range(5):
ys.append(-cc * f(zz) / HB(zz) -\
(7*cc*cc + 9*nabla(cc))/(2*cc**5 + 2*648*cc)*D(zz)*T(zz)/(HB(zz)*f(zz)**2))
zz *= eps
return [(yy, abs(yy**5 + 5*yy + cc)) for yy in ys]
for (z, err) in y(-2.8234): # Randomly chosen value for quintic (see xkcd #221).
print '%.6f + %.6fi (%.10f)' % (z.real, z.imag, err)
```