A box, cut into eight

What the piece has to be

Eight rotations of the cube permute the cells. The group acts freely: no rotation in it fixes a cell. That is what makes the eight pieces interchangeable. A piece has to meet these conditions.

1. exactly 8 cells
2. one cell from each of the 8 rotation orbits
3. face-connected
4. no sealed cavity — the complement stays connected to the outside
5. no solid 2 × 2 × 2 block
6. stricter reading: no 2 × 2 × 1 slab either. This one is optional. It is a filter, not a rule.
   Require it
7. never touches itself only at an edge or corner (well-composedness)
8. no tunnel through it (Euler characteristic = 1)

The eight orbits

Sixty-four cells, eight orbits, eight cells in each. A piece takes one cell from each orbit. The numbers are the orbits.

Where the name comes from

A Yemeni square is a square cut into four congruent polyominoes by a quarter turn about its centre. The name comes from a tiling pattern in Yemeni architecture. Wichmann and Rigby studied them. On a 4 × 4 square there are two: a T pinwheel and an L pinwheel. Both sit inside this cube, as a tetromino two layers thick.

In #1 the cube is two of those squares, stacked. Layers z = 0 and z = 1 match, and so do z = 2 and z = 3.

From here

• All 65, with the layers and the seams
• Twelve shapes each build two cubes
• Fourteen pairs that look the same from outside
• The pinwheel, with room around it
• Print a set
• What was computed, and what is open