Polycube

Apolycubeis a solid figure formed by joining one or more equalcubesface to face. Polycubes are the three-dimensional analogues of the planarpolyominoes.TheSoma cube,theBedlam cube,theDiabolical cube,theSlothouber–Graatsma puzzle,and theConway puzzleare examples ofpacking problemsbased on polycubes.^[1]

Enumerating polycubes

Likepolyominoes,polycubes can be enumerated in two ways, depending on whetherchiralpairs 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 cubeuses 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) (sequenceA000162in theOEIS)	Number of freen-polycubes (reflections counted together) (sequenceA038119in 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 (sequenceA001931in theOEIS)) and one-sided polycubes have been enumerated up ton=22. Free polycubes have been enumerated up ton=16.^[4]More recently, specific families of polycubes have been investigated.^[5]^[6]

Symmetries of polycubes

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. Numerous other symmetries are possible; for example, there are seven possible forms of 8-fold symmetry.^[2]

Properties of pentacubes

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.^[7]

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.

Octacube and hypercube unfoldings

Thetesseract(four-dimensionalhypercube) has eight cubes as itsfacets,and just as the cube can beunfoldedinto 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 crossshape.Salvador Dalíused this shape in his 1954 paintingCrucifixion (Corpus Hypercubus)^[8]and it is described inRobert A. Heinlein's 1940 short story "And He Built a Crooked House".^[9]In honor of Dalí, this octacube has been called theDalí cross.^[10]^[11]It cantile space.^[10]

More generally (answering a question posed byMartin Gardnerin 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.^[10]^[12]

Boundary connectivity

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.^[13]That is, in this case the boundary forms apolyominoid.

Unsolved problem in mathematics:

Can every polycube with a connected boundary beunfoldedto a polyomino? If so, can every such polycube be unfolded to a polyomino that tiles the plane?

(more unsolved problems in mathematics)

Every $k$ -cube with $k < 7$ as well as the Dalí cross (with $k = 8$ ) can beunfoldedto a polyomino that tiles the plane. It is anopen problemwhether 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.^[11]

Dual graph

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.^[14]This is different from the similarly-named notions of adual polyhedron,and of thedual graphof 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.^[15]

References

^Weisstein, Eric W. "Polycube." From MathWorld
^^a ^bLunnon, W. F. (1972), "Symmetry of Cubical and General Polyominoes", in Read, Ronald C. (ed.),Graph Theory and Computing,New York: Academic Press, pp. 101–108,ISBN 978-1-48325-512-5
^Polycubes, at The Poly Pages
^Kevin Gong's enumeration of polycubes
^"Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, FrancePDF
^"Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013PDF
^Aarts, Ronald M."Pentacube".From MathWorld.
^Kemp, Martin (1 January 1998), "Dali's dimensions",Nature,391(27): 27,Bibcode:1998Natur.391...27K,doi:10.1038/34063
^Fowler, David (2010), "Mathematics in Science Fiction: Mathematics as Science Fiction",World Literature Today,84(3): 48–52,doi:10.1353/wlt.2010.0188,JSTOR 27871086,S2CID 115769478,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)..
^^a ^b ^cDiaz, Giovanna;O'Rourke, Joseph(2015),Hypercube unfoldings that tile $\mathbb {R} ^{3}$ and $\mathbb {R} ^{2}$ ,arXiv:1512.02086,Bibcode:2015arXiv151202086D.
^^a ^bLangerman, Stefan;Winslow, Andrew (2016),"Polycube unfoldings satisfying Conway's criterion"(PDF),19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016).
^Turney, Peter (1984), "Unfolding the tesseract",Journal of Recreational Mathematics,17(1): 1–16,MR 0765344.
^Bagchi, Amitabha; Bhargava, Ankur; Chaudhary, Amitabh;Eppstein, David;Scheideler, Christian (2006), "The effect of faults on network expansion",Theory of Computing Systems,39(6): 903–928,arXiv:cs/0404029,doi:10.1007/s00224-006-1349-0,MR 2279081,S2CID 9332443.See in particular Lemma 3.9, p. 924, which states a generalization of this boundary connectivity property to higher-dimensional polycubes.
^Barequet, Ronnie; Barequet, Gill; Rote, Günter (2010), "Formulae and growth rates of high-dimensional polycubes",Combinatorica,30(3): 257–275,CiteSeerX10.1.1.217.7661,doi:10.1007/s00493-010-2448-8,MR 2728490,S2CID 18571788.
^Aloupis, Greg;Bose, Prosenjit K.;Collette, Sébastien;Demaine, Erik D.;Demaine, Martin L.;Douïeb, Karim;Dujmović, Vida;Iacono, John;Langerman, Stefan;Morin, Pat(2011), "Common unfoldings of polyominoes and polycubes",Computational geometry, graphs and applications(PDF),Lecture Notes in Comput. Sci., vol. 7033, Springer, Heidelberg, pp. 44–54,doi:10.1007/978-3-642-24983-9_5,hdl:1721.1/73836,MR 2927309.

External links

Wooden hexacube puzzle by Kadon
Polycube Symmetries
Polycube solverProgram (with Lua source code) to fill boxes with polycubes usingAlgorithm X.

[1] Weisstein, Eric W. "Polycube." From MathWorld

[symmetry-2] Lunnon, W. F. (1972), "Symmetry of Cubical and General Polyominoes", in Read, Ronald C. (ed.),Graph Theory and Computing,New York: Academic Press, pp. 101–108,ISBN 978-1-48325-512-5

[3] Polycubes, at The Poly Pages

[4] Kevin Gong's enumeration of polycubes

[5] "Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, FrancePDF

[6] "Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013PDF

[7] Aarts, Ronald M."Pentacube".From MathWorld.

[8] Kemp, Martin (1 January 1998), "Dali's dimensions",Nature,391(27): 27,Bibcode:1998Natur.391...27K,doi:10.1038/34063

[9] Fowler, David (2010), "Mathematics in Science Fiction: Mathematics as Science Fiction",World Literature Today,84(3): 48–52,doi:10.1353/wlt.2010.0188,JSTOR 27871086,S2CID 115769478,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)..

[hut-10] Diaz, Giovanna;O'Rourke, Joseph(2015),Hypercube unfoldings that tile $\mathbb {R} ^{3}$ and $\mathbb {R} ^{2}$ ,arXiv:1512.02086,Bibcode:2015arXiv151202086D.

[pucc-11] Langerman, Stefan;Winslow, Andrew (2016),"Polycube unfoldings satisfying Conway's criterion"(PDF),19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016).

[12] Turney, Peter (1984), "Unfolding the tesseract",Journal of Recreational Mathematics,17(1): 1–16,MR 0765344.

[13] Bagchi, Amitabha; Bhargava, Ankur; Chaudhary, Amitabh;Eppstein, David;Scheideler, Christian (2006), "The effect of faults on network expansion",Theory of Computing Systems,39(6): 903–928,arXiv:cs/0404029,doi:10.1007/s00224-006-1349-0,MR 2279081,S2CID 9332443.See in particular Lemma 3.9, p. 924, which states a generalization of this boundary connectivity property to higher-dimensional polycubes.

[14] Barequet, Ronnie; Barequet, Gill; Rote, Günter (2010), "Formulae and growth rates of high-dimensional polycubes",Combinatorica,30(3): 257–275,CiteSeerX10.1.1.217.7661,doi:10.1007/s00493-010-2448-8,MR 2728490,S2CID 18571788.

[15] Aloupis, Greg;Bose, Prosenjit K.;Collette, Sébastien;Demaine, Erik D.;Demaine, Martin L.;Douïeb, Karim;Dujmović, Vida;Iacono, John;Langerman, Stefan;Morin, Pat(2011), "Common unfoldings of polyominoes and polycubes",Computational geometry, graphs and applications(PDF),Lecture Notes in Comput. Sci., vol. 7033, Springer, Heidelberg, pp. 44–54,doi:10.1007/978-3-642-24983-9_5,hdl:1721.1/73836,MR 2927309.

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v t e Polyforms
Polyominoes	Domino Tromino Tetromino Pentomino Hexomino Heptomino Octomino Nonomino Decomino
Higher dimensions	Polyominoid Polycube
Others	Polyabolo Polydrafter Polyhex Polyiamond Pseudo-polyomino Polystick
Gamesandpuzzles	Blokus Soma cube Snake cube Tangram Hexastix Tantrix Tetris
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