In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.[1] Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

According to the Oxford English Dictionary, the word tesseract was first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek téssara (τέσσαρα 'four') and aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. In this publication, as well as some of Hinton's later work, the word was occasionally spelled tessaract.[6]

The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points:

In this Cartesian frame of reference, the tesseract has radius 2 and is bounded by eight hyperplanes (xi = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom

It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.

Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:

A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. This view 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.

Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)

The cell-first parallel projection of the tesseract into three-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining six cells are projected onto the six square faces of the cube.

The face-first parallel projection of the tesseract into three-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the four remaining cells project to the side faces.

The edge-first parallel projection of the tesseract into three-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto six rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.

The vertex-first parallel projection of the tesseract into three-dimensional space has a rhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of dissecting a rhombic dodecahedron into four congruent rhombohedra, giving a total of eight possible rhombohedra, each a projected cube of the tesseract. This projection is also the one with maximal volume. One set of projection vectors are u=(1,1,-1,-1), v=(-1,1,-1,1), w=(1,-1,-1,1).

This configuration matrix represents the tesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole tesseract. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[7] For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.

Animation showing each individual cube within the B4 Coxeter plane projection of the tesseract.

The long radius (center to vertex) of the tesseract is equal to its edge length; thus its diagonal through the center (vertex to opposite vertex) is 2 edge lengths. 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. In particular, the tesseract is the only hypercube (other than a 0-dimensional point) with this property. The longest vertex-to-vertex diameter of an n-dimensional hypercube of unit edge length is n, so for the square it is 2, for the cube it is 3, and only for the tesseract it is 4, exactly 2 edge lengths.

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 itself can be decomposed into smaller polytopes. It is the convex hull of the compound of two demitesseracts (16-cells). The tesseract can also be triangulated into 4-dimensional simplices that share their vertices with the tesseract. It is known that there are 92487256 such triangulations[10] and that the fewest 4-dimensional simplices in any of them is 16.[11]

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

Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction. Notable examples include:

The word tesseract was later 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 of this article. See Tesseract (disambiguation).