Apolycube is a solid figure formed by joining one or more equalcubes face to face. Polycubes are the three-dimensional analogues of the planarpolyominoes. TheSoma cube, theBedlam cube, theDiabolical cube, theSlothouber–Graatsma puzzle, and theConway puzzle are examples ofpacking problems based on polycubes.[1]
Likepolyominoes, polycubes can be enumerated in two ways, depending on whetherchiral pairs of polycubes (those equivalent bymirror 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.[2] 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, theSoma cube uses both forms of the chiral tetracube.
Polycubes are classified according to how many cubical cells they have:[3]
| n | Name ofn-polycube | Number of one-sidedn-polycubes (reflections counted as distinct) (sequenceA000162 in theOEIS) | Number of freen-polycubes (reflections counted together) (sequenceA038119 in theOEIS) |
|---|---|---|---|
| 1 | monocube | 1 | 1 |
| 2 | dicube | 1 | 1 |
| 3 | tricube | 2 | 2 |
| 4 | tetracube | 8 | 7 |
| 5 | pentacube | 29 | 23 |
| 6 | hexacube | 166 | 112 |
| 7 | heptacube | 1023 | 607 |
| 8 | octacube | 6922 | 3811 |
Fixed polycubes (both reflections and rotations counted as distinct (sequenceA001931 in theOEIS)), one-sided polycubes, and free polycubes have been enumerated up ton=22. More recently, specific families of polycubes have been investigated.[4][5]
As with polyominoes, polycubes may be classified according to how many symmetries they have. Polycube symmetries (conjugacy classes of subgroups of the achiraloctahedral 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).[2]
12 pentacubes are flat and correspond to thepentominoes. 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.[6]
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.
Thetesseract (four-dimensionalhypercube) has eight cubes as itsfacets, and just as the cube can beunfolded into ahexomino, the tesseract can be unfolded into an octacube. One unfolding, in particular, mimics the well-known unfolding of a cube into aLatin cross: it consists of four cubes stacked one on top of each other, with another four cubes attached to the exposed square faces of the second-from-top cube of the stack, to form a three-dimensionaldouble cross shape.Salvador Dalí used this shape in his 1954 paintingCrucifixion (Corpus Hypercubus)[7] and it is described inRobert A. Heinlein's 1940 short story "And He Built a Crooked House".[8] In honor of Dalí, this octacube has been called theDalí cross.[9][10] It cantile space.[9]
More generally (answering a question posed byMartin Gardner in 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.[9][11]
Although the cubes of a polycube are required to be connected square-to-square, the squares of its boundary are not required to be connected edge-to-edge.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. It is also not required that the boundary of a polycube form amanifold.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.
If a polycube has the additional property that its complement (the set of integer cubes that do not belong to the polycube) is connected by paths of cubes meeting square-to-square, then the boundary squares of the polycube are necessarily also connected by paths of squares meeting edge-to-edge.[12] That is, in this case the boundary forms apolyominoid.
Everyk-cube withk < 7 as well as the Dalí cross (withk = 8) can beunfolded to a polyomino that tiles the plane.It is anopen problem whether every polycube with a connected boundary can be unfolded to a polyomino, or whether this can always be done with the additional condition that the polyomino tiles the plane.[10]
The structure of a polycube can be visualized by means of a "dual graph" that has a vertex for each cube and an edge for each two cubes that share a square.[13] This is different from the similarly-named notions of adual polyhedron, and of thedual graph of a surface-embedded graph.
Dual graphs have also been used to define and study special subclasses of the polycubes, such as the ones whose dual graph is a tree.[14]
Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube (tesseract)..
