Polycube

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Shape made from cubes joined together

title: "Polycube" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polyforms", "discrete-geometry"] description: "Shape made from cubes joined together" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Polycube" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Shape made from cubes joined together ::

thumb|upright|All 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total thumb|right|A puzzle involving arranging nine L tricubes into a 3×3×3 cube

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.

Enumerating polycubes

thumb|right|A [[Chirality (mathematics)|chiral]] pentacube

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:

::data[format=table] | n | Name of n-polycube | Number of one-sided n-polycubes (reflections counted as distinct) | Number of free n-polycubes (reflections counted together) | |---|---|---|---| | 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 ), one-sided polycubes, and free polycubes have been enumerated up to n=22. Specific families of polycubes have also been investigated.

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 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).

Properties of pentacubes

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.

Octacube and hypercube unfoldings

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/60/8-cell_net.png" caption="The Dalí cross"] ::

The tesseract (four-dimensional hypercube) has eight cubes as its facets, and just as the cube can be unfolded into a hexomino, the tesseract can be unfolded into an octacube. One unfolding, in particular, mimics the well-known unfolding of a cube into a Latin 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-dimensional double cross shape. Salvador Dalí used this shape in his 1954 painting Crucifixion (Corpus Hypercubus) and it is described in Robert A. Heinlein's 1940 short story "And He Built a Crooked House". In honor of Dalí, this octacube has been called the Dalí cross. It can tile space.

More generally (answering a question posed by Martin Gardner in 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.{{citation | last = Turney | first = Peter | issue = 1 | journal = Journal of Recreational Mathematics | mr = 765344 | pages = 1–16 | title = Unfolding the tesseract | volume = 17 | year = 1984}}. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/ea/distances_between_double_cube_corners.svg" caption="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 root]]s of every [[natural number]]"] ::

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 a manifold. 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.{{citation | last1 = Bagchi | first1 = Amitabha | last2 = Bhargava | first2 = Ankur | last3 = Chaudhary | first3 = Amitabh | last4 = Eppstein | first4 = David | author4-link = David Eppstein | last5 = Scheideler | first5 = Christian | doi = 10.1007/s00224-006-1349-0 | issue = 6 | journal = Theory of Computing Systems | mr = 2279081 | pages = 903–928 | title = The effect of faults on network expansion | volume = 39 | year = 2006| arxiv = cs/0404029| s2cid = 9332443

Every k-cube with {{math|k It is an open 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.

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.{{citation | last1 = Barequet | first1 = Ronnie | last2 = Barequet | first2 = Gill | last3 = Rote | first3 = Günter | doi = 10.1007/s00493-010-2448-8 | issue = 3 | journal = Combinatorica | mr = 2728490 | pages = 257–275 | title = Formulae and growth rates of high-dimensional polycubes | volume = 30 | year = 2010| s2cid = 18571788 | citeseerx = 10.1.1.217.7661

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.{{citation | last1 = Aloupis | first1 = Greg | last2 = Bose | first2 = Prosenjit K. | author2-link = Jit Bose | last3 = Collette | first3 = Sébastien | last4 = Demaine | first4 = Erik D. | author4-link = Erik Demaine | last5 = Demaine | first5 = Martin L. | author5-link = Martin Demaine | last6 = Douïeb | first6 = Karim | last7 = Dujmović | first7 = Vida | author7-link = Vida Dujmović | last8 = Iacono | first8 = John | author8-link = John Iacono | last9 = Langerman | first9 = Stefan | author9-link = Stefan Langerman | last10 = Morin | first10 = Pat | author10-link = Pat Morin | contribution = Common unfoldings of polyominoes and polycubes | doi = 10.1007/978-3-642-24983-9_5 | mr = 2927309 | pages = 44–54 | publisher = Springer, Heidelberg | series = Lecture Notes in Comput. Sci. | title = Computational geometry, graphs and applications | volume = 7033 | year = 2011| hdl = 1721.1/73836 | isbn = 978-3-642-24982-2 | url = http://cg.scs.carleton.ca/%7Evida/pubs/papers/Cubigami.pdf

References

[https://mathworld.wolfram.com/Polycube.html Weisstein, Eric W. "Polycube." From MathWorld]
Lunnon, W. F.. (1972). "Graph Theory and Computing". Academic Press.
[http://recmath.org/PolyPages/PolyPages/index.htm?Polycubes.html Polycubes, at The Poly Pages]
[https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p26/pdf "Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, France] PDF
[https://arxiv.org/abs/1311.4836 "Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013] PDF
[[Ronald Aarts. Aarts, Ronald M.]] [https://mathworld.wolfram.com/Pentacube.html "Pentacube"]. From MathWorld.
Kemp, Martin. (1 January 1998). "Dali's dimensions". [[Nature (journal).
Fowler, David. (2010). "Mathematics in Science Fiction: Mathematics as Science Fiction". World Literature Today.
(2015). "Hypercube unfoldings that tile \mathbb{R}^3 and \mathbb{R}^2".
(2016). "19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016)".

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