Tetrahedron

The tetrahedron is the simplest possible polyhedron. The regular tetrahedron is one of the five Platonic solids, having four equilateral triangle faces, four vertices, and six edges.

The tetrahedron is its own dual. This is of particular interest to our puzzle taxonomy: it renders the distinction between face-turning and vertex-turning meaningless for tetrahedron puzzles. Notice, for example, how the turning axis of a Pyraminx passes through both a vertex and the centre of its opposite face. Thus any turn can be interpreted as turning about a vertex, or as turning a face about its centre.

Symmetry

Tetrahedra have a total of twelve rotational isometries. Notice how each Pyraminx below is shown in the same perspective, but each shows a different combination of coloured faces.

There are seven axes of rotation, falling into two conjugacy classes. The first conjugacy class contains the four axes that pass through a vertex and the centre of its opposite face. These are the familiar turning axes of a Pyraminx series of puzzles. They have 120° symmetry, and therefore allow us to cycle through three isometries each.

Master Pyramorphix, solved

The other conjugacy class contains the remaining three axes of symmetry: lines passing through the centres of opposite edges. These are the turning axes of the Pyramorphix series of puzzles. They have 180° symmetry, so each one allows us to cycle back and forth between two isometries.

We can be sure there are no other axes of rotation by noting that all our isometries are accounted for. Starting from any chosen isometry (that’s one), we  can reach two different isometries by rotating around any one of the four vertices (that’s eight, so we’re up to nine). Or we can reach one other isometry by rotating around any of the three edge centres (that’s twelve!)

Recall that for every pair of isometries there is an axis of rotation that lets us go directly from one to another. But we don’t need every axis of rotation to be able to reach every isometry if we don’t mind going indirectly; that is, using multiple turns. In fact, the first conjugacy class is a generating set for the isometries of the tetrahedron: every isometry can be reached from any other, using one or more face/vertex rotations only. The other conjugacy class most definitely is not a generating set. Given an isometry, one can reach only three others using edge rotations. This suggests that the Pyramorphix puzzles, when played using only non-jumbling moves, must contain far, far fewer positions than it might at first seem.