Tesseract 8cell (4cube)  

Type  Convex regular 4polytope 
Schläfli symbol  {4,3,3} t_{0,3}{4,3,2} or {4,3}×{ } t_{0,2}{4,2,4} or {4}×{4} t_{0,2,3}{4,2,2} or {4}×{ }×{ } t_{0,1,2,3}{2,2,2} or { }×{ }×{ }×{ } 
Coxeter diagram  
Cells  8 {4,3} 
Faces  24 {4} 
Edges  32 
Vertices  16 
Vertex figure  Tetrahedron 
Petrie polygon  octagon 
Coxeter group  B_{4}, [3,3,4] 
Dual  16cell 
Properties  convex, isogonal, isotoxal, isohedral, Hanner polytope 
Uniform index  10 
In geometry, a tesseract is the fourdimensional 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 4polytopes.
The tesseract is also called an 8cell, C_{8}, (regular) octachoron, octahedroid,^{[2]} cubic prism, and tetracube.^{[3]} It is the fourdimensional hypercube, or 4cube as a member of the dimensional family of hypercubes or measure polytopes.^{[4]} Coxeter labels it the polytope.^{[5]} 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 from aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract.^{[6]}
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 44 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 the 16cell with Schläfli symbol {3,3,4}, with which it can be combined to form the compound of tesseract and 16cell.
Each edge of a regular tesseract is of the same length. 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.
The standard tesseract in Euclidean 4space 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 (x_{i} = ±1). Each pair of nonparallel 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.
An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract.^{[7]} The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement).
The construction of hypercubes can be imagined the following way:
The 8 cells of the tesseract may be regarded (three different ways) as two interlocked rings of four cubes.^{[8]}
The tesseract can be decomposed into smaller 4polytopes. It is the convex hull of the compound of two demitesseracts (16cells). It can also be triangulated into 4dimensional simplices (irregular 5cells) that share their vertices with the tesseract. It is known that there are 92487256 such triangulations^{[9]} and that the fewest 4dimensional simplices in any of them is 16.^{[10]}
The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct construction of the tesseract possible. The characteristic 5cell of the 4cube is a fundamental region of the tesseract's defining symmetry group, the group which generates the B_{4} polytopes. The tesseract's characteristic simplex directly generates the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its mirror walls).
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 fourdimensional tesseract and 24cell, the threedimensional cuboctahedron, and the twodimensional hexagon. In particular, the tesseract is the only hypercube (other than a 0dimensional point) that is radially equilateral. The longest vertextovertex diameter of an ndimensional 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.
In unitradius coordinates the unitedgelength tesseract's coordinates are:
For a tesseract with side length s:
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.^{[11]} 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.
It is possible to project tesseracts into three and twodimensional spaces, similarly to projecting a cube into twodimensional space.
The cellfirst parallel projection of the tesseract into threedimensional 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 facefirst parallel projection of the tesseract into threedimensional 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 edgefirst parallel projection of the tesseract into threedimensional 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 vertexfirst projection. The two remaining cells project onto the prism bases.
The vertexfirst parallel projection of the tesseract into threedimensional 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).
Coxeter plane  B_{4}  B_{4} > A_{3}  A_{3} 

Graph  
Dihedral symmetry  [8]  [4]  [4] 
Coxeter plane  Other  B_{3} / D_{4} / A_{2}  B_{2} / D_{3} 
Graph  
Dihedral symmetry  [2]  [6]  [4] 
A 3D projection of a tesseract performing a simple rotation about a plane in 4dimensional space. The plane bisects the figure from frontleft to backright and top to bottom. 
A 3D projection of a tesseract performing a double rotation about two orthogonal planes in 4dimensional space. 
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 vertexcentered 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 3sphere) 
Stereoscopic 3D projection of a tesseract (parallel view) 
Stereoscopic 3D Disarmed Hypercube 
The tesseract, like all hypercubes, tessellates Euclidean space. The selfdual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4}. Hence, the tesseract has a dihedral angle of 90°.^{[12]}
The tesseract's radial equilateral symmetry makes its tessellation the unique regular bodycentered cubic lattice of equalsized spheres, in any number of dimensions.
The tesseract is 4th in a series of hypercube:
Line segment  Square  Cube  4cube  5cube  6cube  7cube  8cube 
The tesseract (8cell) is the third in the sequence of 6 convex regular 4polytopes (in order of size and complexity).
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon  1 octagon  2 octagons  2 dodecagons  4 30gons  20 30gons  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
As a uniform duoprism, the tesseract exists in a sequence of uniform duoprisms: {p}×{4}.
The regular tesseract, along with the 16cell, exists in a set of 15 uniform 4polytopes with the same symmetry. The tesseract {4,3,3} exists in a sequence of regular 4polytopes and honeycombs, {p,3,3} with tetrahedral vertex figures, {3,3}. The tesseract is also in a sequence of regular 4polytope and honeycombs, {4,3,p} with cubic cells.
Orthogonal  Perspective 

_{4}{4}_{2}, with 16 vertices and 8 4edges, with the 8 4edges shown here as 4 red and 4 blue squares 
The regular complex polytope _{4}{4}_{2}, , in has a real representation as a tesseract or 44 duoprism in 4dimensional space. _{4}{4}_{2} has 16 vertices, and 8 4edges. Its symmetry is _{4}[4]_{2}, order 32. It also has a lower symmetry construction, , or _{4}{}×_{4}{}, with symmetry _{4}[2]_{4}, order 16. This is the symmetry if the red and blue 4edges are considered distinct.^{[13]}
Since their discovery, fourdimensional 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 fourdimensional hypercube; see Tesseract (disambiguation).