Polycube

Polycube symmetries (conjugacy classes of subgroups of the achiral octahedral group) were first enumerated by W. F. Lunnon in 1972.

Of the pentacubes, 2 flats (5-1-1 and the cross) have mirror symmetry in all three axes; these have only three orientations.

Salvador Dalí used this shape in his 1954 painting Crucifixion (Corpus Hypercubus)[7] and it is described in Robert A. Heinlein's 1940 short story "And He Built a Crooked House".

[9] More generally (answering a question posed by Martin Gardner in 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.

For instance, the 26-cube formed by making a 3×3×3 grid of cubes and then removing the center cube is a valid polycube, in which the boundary of the interior void is not connected to the exterior boundary.

For instance, one of the pentacubes has two cubes that meet edge-to-edge, so that the edge between them is the side of four boundary squares.

Every k-cube with k < 7 as well as the Dalí cross (with k = 8) can be unfolded to a polyomino that tiles the plane.

All 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
A puzzle involving arranging nine L tricubes into a 3×3×3 cube
A chiral pentacube
The Dalí cross
Unlike in three dimensions in which distances between vertices of a polycube with unit edges excludes √7 due to Legendre's three-square theorem , Lagrange's four-square theorem states that the analogue in four dimensions yields square roots of every natural number