Editing Polycube

{{Short description|Shape made from cubes joined together}}{{Redirects|Tetracube|the four-dimensional object|tesseract}}
[[image:tetracube_categories.svg|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]]
[[image:9L cube puzzle solution.svg|thumb|right|A puzzle involving arranging nine L tricubes into a 3×3×3 cube]]

A '''polycube''' is a solid figure formed by joining one or more equal [[cube (geometry)|cubes]] face to face.   Polycubes are the three-dimensional analogues of the planar [[polyomino]]es.  The [[Soma cube]], the [[Bedlam cube]], the [[Diabolical cube]], the [[Slothouber–Graatsma puzzle]], and the [[Conway puzzle]] are examples of [[packing problem]]s based on polycubes.<ref>[http://mathworld.wolfram.com/Polycube.html Weisstein, Eric W. "Polycube." From MathWorld]</ref>

==Enumerating polycubes==
[[image:AGK-pentacube.png|thumb|right|A [[Chirality (mathematics)|chiral]] pentacube]]

Like [[polyomino]]es, polycubes can be enumerated in two ways, depending on whether [[Chirality (mathematics)|chiral]] pairs of polycubes (those equivalent by [[Reflection symmetry|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.<ref name=symmetry>{{citation|last=Lunnon |first=W. F. |contribution=Symmetry of Cubical and General Polyominoes |editor-last=Read |editor-first=Ronald C. |title=Graph Theory and Computing |place=New York |publisher=Academic Press |year=1972 |pages=101–108 |isbn=978-1-48325-512-5}}</ref> 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:<ref>[http://recmath.org/PolyPages/PolyPages/index.htm?Polycubes.html Polycubes, at The Poly Pages]</ref>

{| class=wikitable
|-
!''n''
!Name of ''n''-polycube
!Number of one-sided ''n''-polycubes<br>(reflections counted as distinct)<br>{{OEIS|id=A000162}}
!Number of free ''n''-polycubes<br>(reflections counted together)<br>{{OEIS|id=A038119}}
|-
|1
|monocube
|align=right|1
|align=right|1
|-
|2
|dicube
|align=right|1
|align=right|1
|-
|3
|tricube
|align=right|2
|align=right|2
|-
|4
|tetracube
|align=right|8
|align=right|7
|-
|5
|pentacube
|align=right|29
|align=right|23
|-
|6
|hexacube
|align=right|166
|align=right|112
|-
|7
|heptacube
|align=right|1023
|align=right|607
|-
|8
|octacube
|align=right|6922
|align=right|3811
|}

Fixed polycubes (both reflections and rotations counted as distinct {{OEIS|id=A001931}}), one-sided polycubes, and free polycubes have been enumerated up to ''n''=22. Specific families of polycubes have also been investigated.<ref>[http://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</ref><ref>[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</ref>

==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).<ref name=symmetry/>

==Properties of pentacubes==
12 pentacubes are flat and correspond to the [[pentomino]]es. 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.<ref>[[Ronald Aarts|Aarts, Ronald M.]] [http://mathworld.wolfram.com/Pentacube.html "Pentacube"]. From MathWorld.</ref>

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==
[[File:8-cell net.png|thumb|The Dalí cross]]
The [[tesseract]] (four-dimensional [[hypercube]]) has eight cubes as its [[facet (geometry)|facets]], and just as the cube can be [[net (polyhedron)|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 [[Two-barred cross|double cross]] shape. [[Salvador Dalí]] used this shape in his 1954 painting ''[[Crucifixion (Corpus Hypercubus)]]''<ref>{{citation|title=Dali's dimensions|first=Martin|last=Kemp|journal=[[Nature (journal)|Nature]]|volume=391|issue=27|date=1 January 1998|page=27|doi=10.1038/34063|bibcode=1998Natur.391...27K|doi-access=free}}</ref> and it is described in [[Robert A. Heinlein]]'s 1940 short story "[[And He Built a Crooked House]]".<ref>{{citation|title=Mathematics in Science Fiction: Mathematics as Science Fiction|first=David|last=Fowler|journal=World Literature Today|volume=84|issue=3|year=2010|pages=48–52|doi=10.1353/wlt.2010.0188 |jstor=27871086|s2cid=115769478 |quote=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).}}.</ref> In honor of Dalí, this octacube has been called the ''Dalí cross''.<ref name="hut">{{citation|first1=Giovanna|last1=Diaz|first2=Joseph|last2=O'Rourke|author2-link=Joseph O'Rourke (professor)|title=Hypercube unfoldings that tile <math>\mathbb{R}^3</math> and <math>\mathbb{R}^2</math>|year=2015|arxiv=1512.02086|bibcode=2015arXiv151202086D}}.</ref><ref name="pucc">{{citation|contribution=Polycube unfoldings satisfying Conway's criterion|first1=Stefan|last1=Langerman|author1-link=Stefan Langerman|first2=Andrew|last2=Winslow|title=19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016)|year=2016|contribution-url=http://andrewwinslow.com/papers/polyunfold-jcdcggg16.pdf}}.</ref> It can [[Honeycomb (geometry)|tile space]].<ref name="hut"/>

More generally (answering a question posed by [[Martin Gardner]] in 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.<ref name="hut"/><ref>{{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}}.</ref>
[[File:distances_between_double_cube_corners.svg|thumb|Unlike in three dimensions in which distances between [[Vertex (geometry)|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.<ref>{{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
 }}. See in particular Lemma 3.9, p.&nbsp;924, which states a generalization of this boundary connectivity property to higher-dimensional polycubes.</ref> That is, in this case the boundary forms a [[polyominoid]].

{{unsolved|mathematics|Can every polycube with a connected boundary be [[Net (polyhedron)|unfolded]] to a polyomino? If so, can every such polycube be unfolded to a polyomino that tiles the plane?}}
Every {{mvar|k}}-cube with {{math|''k'' < 7}} as well as the Dalí cross (with {{math|1=''k'' = 8}}) can be [[Net (polyhedron)|unfolded]] to a polyomino that tiles the plane.
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.<ref name="pucc"/>

==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.<ref>{{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
 }}.</ref> This is different from the similarly-named notions of a [[dual polyhedron]], and of the [[dual 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.<ref>{{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
 }}.</ref>

==See also==
*[[Herzberger Quader]]
*[[Tripod packing]]

==References==
{{reflist}}

== External links ==
*[http://www.gamepuzzles.com/polycub4.htm#HX Wooden hexacube puzzle by Kadon]
*[http://sicherman.net/csym/index.html Polycube Symmetries]
*[https://github.com/mlepage/polycube-solver Polycube solver] Program (with Lua source code) to fill boxes with polycubes using [[Algorithm X]].
*[http://kevingong.com/Polyominoes/Enumeration.html Kevin Gong's enumeration of polycubes]

{{Polyforms}}

[[Category:Polyforms]]
[[Category:Discrete geometry]]