3cube

3cube commonly refers to the three-dimensional cube itself, or in graph theory the 3-dimensional hypercube, denoted Q3. In geometry, the 3-cube is a regular polyhedron with six square faces, twelve edges, and eight vertices. It is the three-dimensional analogue of the square (2-cube) and the line segment (1-cube). In graph theory, the 3-cube graph represents the state space of three binary variables, connecting configurations that differ in a single bit.

Construction and structure: The 3-cube can be modeled by the set of points with coordinates (x,y,z) where x,y,z are 0 or 1. Vertices are all eight combinations, and edges join pairs that differ in exactly one coordinate. This yields a highly symmetric structure: the full symmetry group is isomorphic to the octahedral group, with 48 symmetries. The graph is bipartite, with partitions based on the parity of the sum x+y+z.

Properties and applications: As a geometric object, the 3-cube is a standard example of a regular polyhedron and a unit cube in Euclidean space. In computing and discrete mathematics, the 3-cube graph models three-bit state spaces and underpins Gray code orderings, path and routing problems, and error-detecting codes. It is also used as a simple blueprint for interconnection networks and parallel computing topologies due to its modest size and high connectivity.

See also: 2-cube, 4-cube, hypercube, cube, Qn. History and nomenclature: The term 3-cube is sometimes used interchangeably with cube or Q3.