In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
It is represented by Schläfli symbol {4,3,3,3} or {4,33}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge.
Applying an alternation operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.
The rows and columns correspond to vertices, edges, faces, cells, and 4-faces.
The Cartesian coordinates of the vertices of a 5-cube centered at the origin and having edge length 2 are while this 5-cube's interior consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1 for all i. n-cube Coxeter plane projections in the Bk Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.
The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope.
The 10 interior vertices have the convex hull of a pentagonal antiprism.
The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron.
The 5-cube has Coxeter group symmetry B5, abstract structure
The Schläfli symbol for the 5-cube, {4,3,3,3}, matches the Coxeter notation symmetry [4,3,3,3].
All hypercubes have lower symmetry forms constructed as prisms.
The 5-cube has 7 prismatic forms from the lowest 5-orthotope, { }5, and upwards as orthogonal edges are constrained to be of equal length.
The edges of a prism can be partitioned into the number of edges in an element times the number of vertices in all the other elements.