The following image shows the possible orientations of a dodecahedron, laid out symmetrically:

and the following diagram is a key to the layout of the pattern:

You may remember, when you were young, attempting to solve a sliding-block puzzle that looked like this:

---------------- |[ 1][ 2][ 3][ 4]| |[ 5][ 6][ 7][ 8]| |[ 9][10][11][12]| |[13][14][15] | ----------------

and you may have read the story about how some combinations were impossible: those that involved switching the positions of exactly two adjacent tiles.

The group formed by all possible permutations of 15 objects
is called S_{15}, or the symmetric group of order 15, and
the group of even permutations only, the ones reachable in this
puzzle, is called the alternating group of order 15, or A_{15}.

The group formed by the rotations of the dodecahedron (the operation
being first perform one rotation, then the other) is isomorphic to
the group of even permutations of five objects, or A_{5}.

The dodecahedron (and the icosahedron, and also the rhombic tricontahedron, the shape of a perfect garnet and the shape of the 30-sided dice offered for use in RPG games) has a symmetry with many interesting mathematical properties.

A book once explained the proof in Galois theory that quintic equations cannot be solved by radicals in terms of the dodecahedron.

Also, the binary Golay code, which is the only nontrivial perfect error-correcting code (with the optional addition of a parity bit) other than the Hamming codes (which also may have a parity bit optionally added) can be represented in terms of the symmetries of the dodecahedron:

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1

and more about the implications of this representation can be read at the following web site.

Note that this version, encoding 12 bits to 24, is a version of the form of the Golay code with the parity bit added, but, in this case, hidden.