The symmetry group of a polyhedron is normally seen as s.th. composed of bulk rotations (sometimes together with reflections as well). The tetrahedron gets A(4) ≅ PSL(2,3) , the cube S(4) ≅ PGL(2,3) (well, actually perhaps this is not allowed to say, because the Schur cover of S(4) is not unique, maybe not GL(2,3)?!), and the icosahedron A(5) ≅ PSL(2,4) ≅ PSL(2,5) . Strangely enough one also gets those groups as the lattice groups (those are the result of an elaborate construction applying to all our graphs, aren't they?).

The icosahedral group H(3) is the set of all rigid motions preserving an icosahedron. It is isomorphic to the direct product of the alternating group A(5) and the two-element group C(2). The subgroup isomorphic to A(5) acts as a set of rotations, and is denoted by H(3,+) (so reflections are left out now). If the icosahedron is situated so that the center of the icosahedron and the origin coincide, then the subgroup H(3,+) occurs as a subgroup of the orthogonal group SO(3), the rotations in 3-space fixing the origin.

The automorphism group (or symmetry group) of a lattice is the set of distance-preserving transformations (or isometries) of the space that fix the origin and take the lattice to itself (the hexagonal lattice gets a dihedral group of order 12), and furthermore adjoining the translations in lattice vectors gives the affine automorphisms (infinite group).

The (positive definite) quadratic form by which one can compare the vectors' length even in case they are not parallel / antiparallel is our goal now. One quadratic form for the hexagonal lattice is ξ 1 2 + ξ 1 ξ 2 + ξ 2 2 and is (bi-)invariant under the symmetry group. But such forms may hide underlying symmetry, so we prefer taking our graphs as geometric objects (coordinate-free approach).

As you can see there are connections to lattices, geometric algebra and invariants.

gap>GG:=SL(IsMatrixGroup,2,5);;
gap>m:=InvariantBilinearForm(GG).matrix;

[[ 0*Z(5), Z(5)^0 ], [ Z(5)^2, 0*Z(5) ]]

Besides there are polyhedral equations for the invariants retrieved using tools such as stereographic projection (it's of course quite ugly to involve extrinsic data from ambient space IMHO), Hessian and Jacobian forms (see [Toth00]). For example a tetrahedron gets Φ^3 - 12√3 i Ω^2 - Ψ^3 = 0 , the cube gets 108 Ω^4 + ((Φ^3 + Ψ^3)/2)^2 - (ΦΨ)^3 = 0 , and the icosahedron gets 1728 I^5 - J^2 - H^3 = 0 .

The latter is then used to solve the general polynomial of 5. degree involving Tschirnhausen transformation and hypergeometric functions. In general there are connections to modular functions. So obviously the question is if our generalized polyheda can be a starting point to attack higher polynomials. By the way those things get involved as well in case of iterative solutions (such as Newton's method), and the procedure then is only guaranteed to succeed if the Galois group is appropriate (see [Doy89]).

Btw. we encountered the formula mentioned above as well in the context of Belyi functions.

Actually what we find sort of most intriguing is that apparently the resulting (embedded) curves from the pieces movings in the section called “Chess Variant” look aesthetical (differentiable) in a continous sense, and one could probably discuss them in standard manner as in calculus, differential and projective geometry (integrate to get the path length, how many intersections are there, compute some normalform...).

In fact here is some algebraic curves homework for you: plot Klein's quartic x^3y + y^3z + z^3x = 0 using Singular™, remember the L2(7) group, and then play with the GAP™ curves package, compute the Riemann-Roch space of Klein's quartic over GF(7) for some divisors. Finally generalize all this for our framework:-)

Yet another approach: The Poincaré homology 3-sphere arises as well as the link of the Brieskorn singularity x^2 + y^3 + z^5 . It's the boundary of the 4-manifold obtained by plumbing on the E 8 graph (whatever that means).

More information: G ⊂ SL(2, ) a finite subgroup then the quotient variety X := ^2 / G is called a Kleinian singularity. As a hypersurface or analytic function ^3 with one relation the defining equations for our three objects (tetrahedron, cube, icosahedron) are x^2 + y^3 + z^4 , x^2 + y^3 + yz^3 and x^2 + y^3 + z^5 respectively. This belongs to complex analysis, algebraic geometry, differential topology and singularity theory.

If G is a finite subgroup of SL(2,C) or SO(3), there exists a complex reflection group 'G' containing G with ['G':G] = 2 (so both the polyhedral groups and their (binary) covers can be constructed this way).

Perhaps out group rings are graded cocommutative or something?! Anyway, while they are not semisimple, still they can be decomposed into some ideals, only this time the order is important.