In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube.

The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism.

[3] The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.

The Oxford English Dictionary traces the word tesseract to Charles Howard Hinton's 1888 book A New Era of Thought.

The term derives from the Greek téssara (τέσσαρα 'four') and aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices.

[4] As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol {4,3,3} with hyperoctahedral symmetry of order 384.

Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96.

As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64.

As an orthotope it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }4, with symmetry order 16.

This is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

Each pair of non-parallel hyperplanes intersects to form 24 square faces.

All in all, a tesseract consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

Another commonly convenient tesseract is the Cartesian product of the closed interval [−1, 1] in each axis, with vertices at coordinates (±1, ±1, ±1, ±1).

The 8 cells of the tesseract may be regarded (three different ways) as two interlocked rings of four cubes.

The radius of a hypersphere circumscribed about a regular polytope is the distance from the polytope's center to one of the vertices, and for the tesseract this radius is equal to its edge length; the diameter of the sphere, the length of the diagonal between opposite vertices of the tesseract, is twice the edge length.

Only a few uniform polytopes have this property, including the four-dimensional tesseract and 24-cell, the three-dimensional cuboctahedron, and the two-dimensional hexagon.

The rows and columns correspond to vertices, edges, faces, and cells.

The next row left of diagonal is ridge elements (facet of cube), here a square, (4,4).

The cell-first parallel projection of the tesseract into three-dimensional space has a cubical envelope.

The face-first parallel projection of the tesseract into three-dimensional space has a cuboidal envelope.

The edge-first parallel projection of the tesseract into three-dimensional space has an envelope in the shape of a hexagonal prism.

The vertex-first parallel projection of the tesseract into three-dimensional space has a rhombic dodecahedral envelope.

The tetrahedron forms the convex hull of the tesseract's vertex-centered central projection.

(Edges are projected onto the 3-sphere) The tesseract, like all hypercubes, tessellates Euclidean space.

[10] The tesseract's radial equilateral symmetry makes its tessellation the unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions.

The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).

The tesseract {4,3,3} exists in a sequence of regular 4-polytopes and honeycombs, {p,3,3} with tetrahedral vertex figures, {3,3}.

The tesseract is also in a sequence of regular 4-polytope and honeycombs, {4,3,p} with cubic cells.

[11] Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction.

Notable examples include: The word tesseract has been adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube; see Tesseract (disambiguation).