**Pyritohedral symmetry**This animation by

**Tom Ruen** shows a cube morphing into a regular dodecahedron, then into a rhombic dodecahedron, and then back again. Even better, the morphing is done in such a way as to preserve as much symmetry as possible.

As I mentioned in a recent post, the

**rhombic dodecahedron** has 12 identical rhombus-shaped faces, and 24 rotational symmetries. Another way to say this is that there are 24 essentially different ways to pick up a rhombic dodecahedron from a table and then replace it in a way that looks identical. This is because there are 12 choices for the face that ends up touching the table, and 2 choices for which way round to place this face. (The latter is because a rhombus has rotational symmetry of order 2.)

These 24 rotational symmetries form what mathematicians call a

**group**; this particular group is called the

**symmetric group** S4, of order 24. The simplest way to think of this group is to think of it as the 24 ways to permute four objects; the 24 in this case comes from 4! (4 factorial), which is equal to 4x3x2x1.

In order to understand how the group S4 relates to the rhombic dodecahedron, it helps to identify four objects that are being permuted by the rotations. Although the rhombic dodecahedron has 14 vertices (corners), these vertices are of two distinct types. Six of the vertices occur where four rhombuses meet at their acute-angle corners, and the other eight vertices occur where three rhombuses meet at their obtuse-angle corners. The subset of eight vertices falls into four opposite pairs, and it is these four opposite pairs that are being permuted to give the correspondence with the symmetric group S4.

If you watch the animation, you will see that the set of six vertices corresponds to the six faces of the cube, and the set of eight vertices corresponds to the corners of the cube. The group of rotational symmetries of a cube is also given by S4, and in this case, the four objects being permuted are the four pairs of opposite corners.

Notice that the numbers 6 and 8 are factors of 24; this is not a coincidence, but relates to the orbit-stabilizer theorem, which is a result in the branch of mathematics known as

**group theory**. A group theorist would say that S4 acts

**transitively** on the 12 faces of a rhombic dodecahedron, which means that if one chooses two of the twelve faces and calls them A and B, then it is possible to find a rotation of the polyhedron that moves face A to face B. However, this is possible only because 12 is a factor of 24. Since there are 14 vertices and 14 is not a factor of 24, it follows from group theory that there must be more than one type of vertex. A mathematician would say that there are two

**orbits** of vertices under the action of the group of rotations: one of size 6, and one of size 8.

The cube in the animation is depicted with each of its six faces subdivided into two identical rectangles. Half the rotations of the cube respect this subdivision, and the other half do not. The 12 rotations of the cube that do respect the subdivision form a group of order 12 called the

**alternating group** A4; in this case, the number 12 is given by the formula 4!/2. There is also an alternating group A5, with 60 elements (because 60 = 5!/2 = 5x4x3x2x1/2). This is the group of rotational symmetries of the regular dodecahedron appearing at the midpoint of the animation.

Something that the groups A4, A5 and S4 have in common is that they all contain the group A4 as a substructure. All the polyhedra in the intermediate stages of the animation have A4 as part of their rotational symmetry group. In additional to this rotational symmetry, all of the intermediate shapes also have mirror symmetry. These intermediate shapes are dodecahedra with irregular pentagonal faces called

**pyritohedra**, and the A4-based symmetry that they exhibit is called

**pyritohedral symmetry**. The name comes from the fact that the mineral

**iron pyrite** (fool's gold) comes in two crystalline forms: cubical and pyritohedral.

**Relevant links**Rhombic dodecahedron:

http://en.wikipedia.org/wiki/Rhombic_dodecahedronSymmetric group:

http://en.wikipedia.org/wiki/Symmetric_groupAlternating group:

http://en.wikipedia.org/wiki/Alternating_groupIron pyrite:

http://en.wikipedia.org/wiki/PyriteThe animation also appears on Wikipedia.

This post is a sequel to a recent post by me about rhombic dodecahedra:

https://plus.google.com/101584889282878921052/posts/5mLoukdZAA5#mathematics #scienceeveryday