Editing Cube (section)

=== Space-filling and honeycombs ===
[[Hilbert's third problem]] asked whether every two equal-volume polyhedra could always be dissected into polyhedral pieces and reassembled into each other. If it were, then the volume of any polyhedron could be defined axiomatically as the volume of an equivalent cube into which it could be reassembled. [[Max Dehn]] solved this problem in an invention [[Dehn invariant]], answering that not all polyhedra can be reassembled into a cube.{{r|gruber}} It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.{{r|zeeman}}

[[File:Partial cubic honeycomb.png|thumb|upright=0.6|[[Cubic honeycomb]]]]
The cube has a Dehn invariant of zero. This indicates the cube is applied for [[Honeycomb (geometry)|honeycomb]]. More strongly, the cube is a [[Space-filling polyhedron|space-filling tile]] in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap.{{r|lm}} The cube is a [[plesiohedron]], a special kind of space-filling polyhedron that can be defined as the [[Voronoi cell]] of a symmetric [[Delone set]].{{r|erdahl}} The plesiohedra include the [[parallelohedra]], which can be [[Translation (geometry)|translated]] without rotating to fill a space in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid.{{r|alexandrov}} Every three-dimensional parallelohedron is [[zonohedron]], a [[centrally symmetric]] polyhedron whose faces are [[Zonogon|centrally symmetric polygons]].{{r|shephard}} In the case of cube, it can be represented as the [[Cell (geometry)|cell]]. Some honeycombs have cubes as the only cells; one example is [[cubic honeycomb]], the only regular honeycomb in Euclidean three-dimensional space, having four cubes around each edge.{{r|twelveessay|ns}}