In geometry, the **tesseract** is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an **8-cell**, **regular octachoron**, **cubic prism**, and **tetracube** (although this last term can also mean a polycube made of four cubes). It is the **four-dimensional hypercube**, or **4-cube** as a part of the dimensional family of hypercubes or “measure polytopes”.^{}

According to the *Oxford English Dictionary*, the word *tesseract* was coined and first used in 1888 by Charles Howard Hinton in his book *A New Era of Thought*, from the Greek τέσσερεις ακτίνες (“four rays”), referring to the four lines from each vertex to other vertices.^{[2]} In this publication, as well as some of Hinton’s later work, the word was occasionally spelled “tessaract”.

## Geometry

The tesseract can be constructed in a number of ways. 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 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.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.

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:

A tesseract is bounded by eight hyperplanes (*x*_{i} = ±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.

### Projections to 2 dimensions

The construction of a hypercube can be imagined the following way:

**1-dimensional:**Two points A and B can be connected to a line, giving a new line segment AB.**2-dimensional:**Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.**3-dimensional:**Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.**4-dimensional:**Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

It is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a 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.

Tesseracts are also bipartite graphs, just as a path, square, cube and tree are.

### Parallel projections to 3 dimensions

The The The The |

## Image gallery

The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space (view animation). An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract. |
Stereoscopic 3D projection of a tesseract (parallel view ) |

### Alternative projections

A 3D projection of an 8-cell performing a double rotation about two orthogonal planes |
Perspective with hidden volume elimination. The red corner is the nearest in 4D and has 4 cubical cells meeting around it. |

The tetrahedron forms the convex hull of the tesseract’s vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex is projected to infinity and the four edges to it are not shown. |
Stereographic projection (Edges are projected onto the 3-sphere) |

### 2D orthographic projections

Coxeter plane | B_{4} |
B_{3} / D_{4} / A_{2} |
B_{2} / D_{3} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | Other | F_{4} |
A_{3} |

Graph | |||

Dihedral symmetry | [2] | [12/3] | [4] |

## Related uniform polytopes

Name | {3}×{}×{} | {4}×{}×{} |
{5}×{}×{} | {6}×{}×{} | {7}×{}×{} | {8}×{}×{} | {p}×{}×{} |
---|---|---|---|---|---|---|---|

Coxeter diagrams |
|||||||

Image | |||||||

Cells | 3 {4}×{} 4 {3}×{} |
4 {4}×{} 4 {4}×{} |
5 {4}×{} 4 {5}×{} |
6 {4}×{} 4 {6}×{} |
7 {4}×{} 4 {7}×{} |
8 {4}×{} 4 {8}×{} |
p {4}×{} 4 {p}×{} |

Name | tesseract |
rectified tesseract |
truncated tesseract |
cantellated tesseract |
runcinated tesseract |
bitruncated tesseract |
cantitruncated tesseract |
runcitruncated tesseract |
omnitruncated tesseract |
---|---|---|---|---|---|---|---|---|---|

Coxeter diagram |
= |
= |
|||||||

Schläfli symbol |
{4,3,3} | t_{1}{4,3,3}r{4,3,3} |
t_{0,1}{4,3,3}t{4,3,3} |
t_{0,2}{4,3,3}rr{4,3,3} |
t_{0,3}{4,3,3} |
t_{1,2}{4,3,3}2t{4,3,3} |
t_{0,1,2}{4,3,3}tr{4,3,3} |
t_{0,1,3}{4,3,3} |
t_{0,1,2,3}{4,3,3} |

Schlegel diagram |
|||||||||

B_{4} |
|||||||||

Name | 16-cell | rectified 16-cell |
truncated 16-cell |
cantellated 16-cell |
runcinated 16-cell |
bitruncated 16-cell |
cantitruncated 16-cell |
runcitruncated 16-cell |
omnitruncated 16-cell |

Coxeter diagram |
= |
= |
= |
= |
= |
= |
|||

Schläfli symbol |
{3,3,4} | t_{1}{3,3,4}r{3,3,4} |
t_{0,1}{3,3,4}t{3,3,4} |
t_{0,2}{3,3,4}rr{3,3,4} |
t_{0,3}{3,3,4} |
t_{1,2}{3,3,4}2t{3,3,4} |
t_{0,1,2}{3,3,4}tr{3,3,4} |
t_{0,1,3}{3,3,4} |
t_{0,1,2,3}{3,3,4} |

Schlegel diagram |
|||||||||

B_{4} |

It is in a sequence of regular 4-polytopes and honeycombs with tetrahedral vertex figures.

Space | S^{3} |
H^{3} |
||||||
---|---|---|---|---|---|---|---|---|

Form | Finite | Paracompact | Noncompact | |||||

Name | {3,3,3} | {4,3,3} |
{5,3,3} | {6,3,3} | {7,3,3} | {8,3,3} | … {∞,3,3} | |

Image | ||||||||

Coxeter diagrams |
1 | |||||||

4 | ||||||||

6 | ||||||||

12 | ||||||||

24 | ||||||||

Cells {p,3} |
{3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |

It is in a sequence of regular 4-polytope and honeycombs with cubic cells.

Space | S^{3} |
E^{3} |
H^{3} |
||||
---|---|---|---|---|---|---|---|

Form | Finite | Affine | Compact | Paracompact | Noncompact | ||

Name |
{4,3,3} |
{4,3,4} |
{4,3,5} |
{4,3,6} |
{4,3,7} |
{4,3,8} |
… {4,3,∞} |

Image | |||||||

Vertex figure |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |