The Pyraminx is a face/vertex-turning tetrahedron puzzle with three layers.
The turning axes of the Pyraminx are lines that pass through a vertex and the centre of the opposite face. These are axes of rotational symmetry for the tetrahedron, so turns move layers from isometry to isometry, and thus the tetrahedron shape is always retained: there is no jumbling.
According to Wikipedia (uncited), the Pyraminx can take 75,582,720 possible positions.
However the first thing anyone discovers when they pick up a Pyraminx is that the corner pieces rotate independently of the rest of the puzzle, so it is trivial to align them with their neighbouring centres. Since there are four of them, and each can take three positions, this alone inflates the number of possible positions by a factor of 81. Once these trivial corners are dealt with, there remain only 933,120 possible positions.
The second thing the prospective solvers of the Pyraminx realise is that the neighbours of the corner cubies also turn on only one axis. Have a look at the picture above: look at the orange triangle in the middle of the part of the orange face being turned. That orange triangle doesn’t turn around any other axis. Nor do any of the other corner neighbours. So again it is trivial to position these corner neighbours so they are all on the same size. That again reduces the number of possible positions by a factor of 81, leaving only 11,520 positions.
With the corners and centres trivally solved, this leaves only the edge cubies. There are only six of these, and each one rotates around only two axes. As you would expect from this, and with so few possible positions, solving the Pyraminx from here requires only a couple of simple algorithms and is not at all difficult.
According to Wikipedia (again uncited), God’s Number for the Pyraminx is 11. Interestingly, trivial orientations of the corners and centres would account for 8 of these. A cube with only the edges left to solve certainly cannot always be solved in just 3 moves, so it would seem that God’s Algorithm saves a lot of moves by deferring orienting each centre until the proper time.
The Pyraminx is the first of the Pyraminx series of puzzles — there is no two-layer Pyraminx, as this would be a trivial puzzle comprising four independent corners, for a total of 12 possible positions. After the Pyraminx comes the four-layer Master Pyraminx.
Jing’s Pyraminx is another 3-layer face/vertex-turning tetrahedron puzzle but with a slightly different cut. It is very similar in play but slightly harder to solve. The Pyramorphix series are edge-turning puzzles and therefore unrelated to the Pyraminx except that they are all tetrahedron puzzles.
The trivially oriented corners on the Pyraminx are just silly. The Tetraminx is a Pyraminx in which these corners have been removed, resulting in a puzzle in the shape of a truncated tetrahedron. It is otherwise identical to the Pyraminx.
The Vertex-turning Octahedron Puzzle, on the other hand, despite having the same internal mechanism as the face-turning cube puzzles, is extremely similar to the Pyraminx. Like the Pyraminx, it has corners and centres that are trivial to solve, and the algorithms for solving the edges are the same.