Tesseract

In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.
The tesseract is also called an 8-cell, C₈, octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for hypervolume. Coxeter labels it the polytope. 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 and aktís, referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract.

Construction

The construction of a tesseract can be visualized through the analogy of dimensions in the following steps:

One can take out two points with a certain length that form a line segment.
If another identical line segment is its length in a perpendicular direction from itself, it sweeps out and forms a square. The results have four points and four line segments, which are called vertices and edges, respectively.
Moving the square with the same length in the direction perpendicular to the plane it lies on generates a cube. The results have eight vertices, twelve edges, and six squares. The squares are called the faces.
Moving the cube with the same length again into the fourth-dimensional space generates a tesseract.

For a tesseract, it is bounded by eight cubes that are known as the cells, each pair of which intersects, forming twenty-four square faces. Three cubes and three squares intersect at each edge. Four cubes, six squares, and four edges meet at every vertex. Collectively, the tesseract consists of eight cubes, twenty-four squares, thirty-two edges, and sixteen vertices. The tesseract, as well as both the square and the cube, is a member of the hypercube's family.
An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to paired trees, which can tile 3-space. The Dali cross is one of the net examples, named after Spanish surrealist artist Salvador Dalí, whose painting Corpus Hypercubus in 1954. It is constructed from eight cubes, whereby four cubes are stacked vertically, and four others are attached to the second-from-top cube of the stack.

Properties

The eight cells of a tesseract may be regarded in three different ways as two interlocked rings of four cubes. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol 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 × , with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, 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 16-cell with Schläfli symbol, with which it can be combined to form the compound of tesseract and 16-cell.
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 tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two demitesseracts. It can also be triangulated into 4-dimensional simplices that share their vertices with the tesseract. It is known that there are such triangulations and that the fewest 4-dimensional simplices in any of them is 16.
The dissection of the tesseract into instances of its characteristic simplex is the most basic direct construction of the tesseract possible. The characteristic 5-cell of the 4-cube is a fundamental region of the tesseract's defining symmetry group, the group which generates the B₄ polytopes. The tesseract's characteristic simplex directly generates the tesseract through the actions of the group, by reflecting itself in its own bounding facets.

Unit tesseract

A unit tesseract has side length, and is typically taken as the basic unit for hypervolume in 4-dimensional space. The unit tesseract in a Cartesian coordinate system for 4-dimensional space has two opposite vertices at coordinates and, and other vertices with coordinates at all possible combinations of s and s. It is the Cartesian product of the closed unit interval in each axis.
Sometimes a unit tesseract is centered at the origin, so that its coordinates are the more symmetrical This is the Cartesian product of the closed interval in each axis.
Another commonly convenient tesseract is the Cartesian product of the closed interval in each axis, with vertices at coordinates. This tesseract has side length 2 and hypervolume.

Radial equilateral symmetry

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. In particular, the tesseract is the only hypercube that is radially equilateral. The longest vertex-to-vertex diagonal of an -dimensional hypercube of unit edge length is which for the square is for the cube is and only for the tesseract is edge lengths.
An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates

Formulas

For a tesseract with side length :

Hypervolume :
Surface "volume" :
Face diagonal:
Cell diagonal:
4-space diagonal:
As a configuration

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 diagonal reduces to the f-vector.
The nondiagonal numbers say how many of the column's element occur in or at the row's element. For example, the 2 in the first column of the second row indicates that there are 2 vertices in each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.
The bottom row defines they facets, here cubes, have f-vector. The next row left of diagonal is ridge elements, here a square,.
The upper row is the f-vector of the vertex figure, here tetrahedra,. The next row is vertex figure ridge, here a triangle,.

Projections

It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.
File:Hypercubeorder binary.svg|thumb|right|The rhombic dodecahedron forms the convex hull of the tesseract's vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1—the fourth row in Pascal's triangle.
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,,.

Coxeter plane	B₄	B₄ --> A₃	A₃
Graph
Dihedral symmetry
Coxeter plane	Other	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry

A 3D projection of a tesseract performing a simple rotation about a plane in 4-dimensional space. The plane bisects the figure from front-left to back-right and top to bottom.	A 3D projection of a tesseract performing a double rotation about two orthogonal planes in 4-dimensional space.
A 3D projection of a tesseract performing a simple rotation about a plane in 4-dimensional space. The plane bisects the figure from front-left to back-right and top to bottom.	-

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

Stereoscopic 3D projection of a tesseract

Stereoscopic 3D Disarmed Hypercube