Tesseract

The tesseract, or tes, also sometimes called the 8-cell or octachoron, is one of the 6 convex regular polychora. It has 8 cubes as cells, joining 3 to an edge and 4 to a vertex. It is the 4-dimensional hypercube.

It is also the uniform cubic prism (and thus also a segmentochoron designated K-4.20 on Richard Klitzing's list), uniform square duoprism, digonal duoantitegum, digonal diswirltegum, and the 8-3 gyrochoron. It is the first in an infinite family of isochoric tetrahedral swirlchora, the first in an infinite family of isogonal square dihedral swirlchora and also the first in an infinite family of isochoric square hosohedral swirlchora.

It is one of the three regular polychora that can tile 4D space, similar to hypercubes of any other dimension. The tiling is the tesseractic tetracomb.

The tesseract has the same circumradius as its edge length. This relates to the fact that it is the vertex figure of the Euclidean icositetrachoric tetracomb.

Naming
The name tesseract comes from the Greek ' (4) and ' (ray), referring to the four line segments meeting at each vertex. It was coined by Charles Howard Hinton. Other names include


 * Tessaract, an alternate spelling. Hinton spelled the word inconsistently, but "tesseract" is the spelling that is generally considered correct in the present day.
 * 8-cell or octachoron, referring to the number of cells. Octahedroid is sometimes also used.
 * 4-cube or sometimes tetracube, because it is the 4-dimensional hypercube.

Vertex coordinates
The vertices of a tesseract of edge length 1, centered at the origin, are given by:
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).$$

Solid angles

 * 1/4 ridge-angle
 * 1/8 edge-angle
 * 1/16 vertex-angle

Representations
A tesseract has the following Coxeter diagrams:


 * x4o3o3o (full symmetry)
 * x x4o3o (BC2×A1 symmetry, as cubic prism)
 * x4o x4o (BC2×BC2 symmetry, square duoprism)
 * x x x4o (BC2×A1×A1 symmetry, square prismatic prism)
 * x x x x (A1×A1×A1×A1 symmetry, 4D hypercuboid)
 * s4x2s4x
 * xx4oo3oo&#x (BC3 axial, cube atop cube)
 * xx xx4oo&#x (bases have BC2×A1 symmetry)
 * xx xx xx&#x (bases have A1×A1×A1 symmetry)
 * oqo xxx4ooo&#xt (BC2×A1 symmetry, square-first).
 * oqo xxx xxx&#xt (A1×A1×A1 axial, square-first)
 * xxxx oqoo3ooqo&#xt (A2×A1 axial, edge-first)
 * oqooo3ooqoo3oooqo&#xt (A3 axial, vertex-first, tetrahedral antitegum)
 * qo3oo3oq *b3oo&#zx (D4 subsymmetry, hull of 2 opposite demitesseracts/hexadecachora)
 * xx qo3oo3oq&#zx (A3×A1 symmetry, prism of hull of 2 tetrahedra)
 * xx4oo qo oq&#zx (as square/rhombic duoprism)
 * xx xx qo oq&#zx (as rectangular/rhombic duoprism)
 * qqoo ooqq qoqo oqoq&#zx (as rhombic/rhombic duoprism)

Variations
Besides the regular tesseract, there are various other polychora with 24 quadriateral faces and 8 hexahedral cells with lower symmetry. These include:


 * Cubic prism - 2 cubic bases and 6 square prism sides
 * Square duoprism - 2 sets of 4 square prisms
 * Square-rectangular duoprism - 4 identical cuboids, 2 pairs of different square prisms
 * Rectangular duoprism - 2 sets of 4 cuboids
 * Tesseractoid - 4 pairs of cuboids
 * Rhombic duoprism - 8 identical rhombic prisms
 * Tetrahedral antitegum - 8 identical triangular trapezohedra
 * 8-3 gyrochoron - least symetric isochoric version
 * Rectangular trapezoprismatic prism - 2 rectangle trapezoprisms, 2 cuboids, and 4 rectangle frustums, isogonal
 * Digonal duoantitegum - 8 identical stretched cubes
 * Tetraswirlic octachoron - as first tetrahedral swirltegum (no metrical variations)

Related polychora
A tesseract can be decomposed into 8 CRF cubic pyramids. If these cubic pyramids are attached to the cells of a tesseract the result is a regular icositetrachoron, as adjacent square pyramids merging into octahedra.

Two of the seven regular polychoron compounds are composed of tesseracts:


 * Great icositetrachoron (3)
 * Great stellated tetracontoctachoron (6)

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Cube (8): Hexadecachoron
 * Square (24): Icositetrachoron
 * Edge (32): Rectified tesseract