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Polycube
A polycube is an orthogonal polyhedron formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.
Like polyominoes, polycubes can be enumerated in two ways, depending on whether chiral pairs of polycubes (those equivalent by mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 tetracubes are achiral and one is chiral, giving a count of 7 or 8 tetracubes respectively. Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, because one cannot turn a polycube over to reflect it as one can a polyomino given three dimensions. In particular, the Soma cube uses both forms of the chiral tetracube.
Polycubes are classified according to how many cubical cells they have:
Fixed polycubes (both reflections and rotations counted as distinct (sequence A001931 in the OEIS)), one-sided polycubes, and free polycubes have been enumerated up to n=22. Specific families of polycubes have also been investigated.
As with polyominoes, polycubes may be classified according to how many symmetries they have. Polycube symmetries (conjugacy classes of subgroups of the achiral octahedral group) were first enumerated by W. F. Lunnon in 1972. Most polycubes are asymmetric, but many have more complex symmetry groups, all the way up to the full symmetry group of the cube with 48 elements. There are 33 different symmetry types that a polycube can have (including asymmetry).
12 pentacubes are flat and correspond to the pentominoes. 5 of the remaining 17 have mirror symmetry, and the other 12 form 6 chiral pairs.
The bounding boxes of the pentacubes have sizes 5×1×1, 4×2×1, 3×3×1, 3×2×1, 3×2×2, and 2×2×2.
A polycube may have up to 24 orientations in the cubic lattice, or 48, if reflection is allowed. Of the pentacubes, 2 flats (5-1-1 and the cross) have mirror symmetry in all three axes; these have only three orientations. 10 have one mirror symmetry; these have 12 orientations. Each of the remaining 17 pentacubes has 24 orientations.
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Polycube
A polycube is an orthogonal polyhedron formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.
Like polyominoes, polycubes can be enumerated in two ways, depending on whether chiral pairs of polycubes (those equivalent by mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 tetracubes are achiral and one is chiral, giving a count of 7 or 8 tetracubes respectively. Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, because one cannot turn a polycube over to reflect it as one can a polyomino given three dimensions. In particular, the Soma cube uses both forms of the chiral tetracube.
Polycubes are classified according to how many cubical cells they have:
Fixed polycubes (both reflections and rotations counted as distinct (sequence A001931 in the OEIS)), one-sided polycubes, and free polycubes have been enumerated up to n=22. Specific families of polycubes have also been investigated.
As with polyominoes, polycubes may be classified according to how many symmetries they have. Polycube symmetries (conjugacy classes of subgroups of the achiral octahedral group) were first enumerated by W. F. Lunnon in 1972. Most polycubes are asymmetric, but many have more complex symmetry groups, all the way up to the full symmetry group of the cube with 48 elements. There are 33 different symmetry types that a polycube can have (including asymmetry).
12 pentacubes are flat and correspond to the pentominoes. 5 of the remaining 17 have mirror symmetry, and the other 12 form 6 chiral pairs.
The bounding boxes of the pentacubes have sizes 5×1×1, 4×2×1, 3×3×1, 3×2×1, 3×2×2, and 2×2×2.
A polycube may have up to 24 orientations in the cubic lattice, or 48, if reflection is allowed. Of the pentacubes, 2 flats (5-1-1 and the cross) have mirror symmetry in all three axes; these have only three orientations. 10 have one mirror symmetry; these have 12 orientations. Each of the remaining 17 pentacubes has 24 orientations.