Editing Cube (section)

== Representation ==
=== As a graph ===
{{main|Hypercube graph}}
[[File:Cube skeleton.svg|upright=0.8|thumb|The graph of a cube]]

According to [[Steinitz's theorem]], the [[Graph (discrete mathematics)|graph]] can be represented as the [[Skeleton (topology)|skeleton]] of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties: [[Planar graph|planar]] (the edges of a graph are connected to every vertex without crossing other edges), and [[k-vertex-connected graph|3-connected]] (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).{{r|grunbaum-2003|ziegler}} The skeleton of a cube can be represented as the graph, and it is called the '''cubical graph''', a [[Platonic graph]]. It has the same number of vertices and edges as the cube, twelve vertices and eight edges.{{r|rudolph}} The cubical graph is also classified as a [[prism graph]], resembling the skeleton of a cuboid.{{r|ps}}

The cubical graph is a special case of [[hypercube graph]] or {{nowrap|1=<math>n</math>-}}cube&mdash;denoted as <math> Q_n </math>&mdash;because it can be constructed by using the operation known as the [[Cartesian product of graphs]]: it involves two graphs connecting the pair of vertices with an edge to form a new graph.{{r|hh}} In the case of the cubical graph, it is the product of two <math> Q_2 </math>; roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph is <math> Q_3 </math>.{{r|cz}} Like any hypercube graph, it has a [[Cycle (graph theory)|cycle]] visits [[Hamiltonian path|every vertex exactly once]],{{r|ly}} and it is also an example of a [[unit distance graph]].{{r|hp}}

The cubical graph is [[bipartite graph|bipartite]], meaning every [[Independent set (graph theory)|independent set]] of four vertices can be [[Disjoint set|disjoint]] and the edges connected in those sets.{{r|berman-graph}} However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not [[complete bipartite graph|complete]].{{r|aw}} It is an example of both [[crown graph]] and [[bipartite Kneser graph]].{{r|kl|berman-graph}}

=== In orthogonal projection ===
An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an [[Orthographic projection|orthogonal projection]]. A polyhedron is considered ''equiprojective'' if, for some position of the light, its orthogonal projection is a regular polygon. The cube is equiprojective because, if the light is parallel to one of the four lines joining a vertex to the opposite vertex, its projection is a [[regular hexagon]].{{r|hhlnqr}}

=== As a configuration matrix ===
The cube can be represented as [[Platonic solid#As a configuration|configuration matrix]]. A configuration matrix is a [[Matrix (mathematics)|matrix]] in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The [[Main diagonal|diagonal]] of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. As mentioned above, the cube has eight vertices, twelve edges, and six faces; each element in a matrix's diagonal is denoted as 8, 12, and 6. The first column of the middle row indicates that there are two vertices in (i.e., at the extremes of) each edge, denoted as 2; the middle column of the first row indicates that three edges meet at each vertex, denoted as 3. The following matrix is:{{r|coxeter}}
<math display="block"> \begin{bmatrix}\begin{matrix}8 & 3 & 3 \\ 2 & 12 & 2 \\ 4 & 4 & 6 \end{matrix}\end{bmatrix}</math>