Editing Polycube (section)

==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]] ]]