# Tesseract

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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".[1]

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{{#invoke:Category handler|main}} ("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:

${\displaystyle \{(x_{1},x_{2},x_{3},x_{4})\in \mathbb {R} ^{4}\,:\,-1\leq x_{i}\leq 1\}}$

A tesseract 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.

### Projections to 2 dimensions

A diagram showing how to create a tesseract from a point

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.
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- 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 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.
 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 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube. The face-first parallel projection of the tesseract into 3-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces. The edge-first parallel projection of the tesseract into 3-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 6 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 3-dimensional space has a rhombic dodecahedral envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a total of 8 possible parallelepipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelepipeds. This projection is also the one with maximal volume.

## 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.[3] 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). 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

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane Other F4 A3
Graph
Dihedral symmetry [2] [12/3] [4]

## Related uniform polytopes

Convex p-gonal prismatic prisms
Name {3}×{}×{} {4}×{}×{} {5}×{}×{} {6}×{}×{} {7}×{}×{} {8}×{}×{} {p}×{}×{}
Coxeter
diagrams
Template:CDD Template:CDD
Template:CDD
Template:CDD Template:CDD
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Template:CDD
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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}×{}

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

{p,3,3}
Space S3 H3
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 Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
4 Template:CDD Template:CDD Template:CDD Template:CDD
6 Template:CDD Template:CDD Template:CDD Template:CDD
12 Template:CDD Template:CDD Template:CDD Template:CDD
24 Template:CDD Template:CDD
Cells
{p,3}
Template:CDD

{3,3}
Template:CDD

{4,3}
Template:CDD
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{5,3}
Template:CDD

{6,3}
Template:CDD
Template:CDD
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{7,3}
Template:CDD

{8,3}
Template:CDD
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{∞,3}
Template:CDD
Template:CDD
Template:CDD

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

{4,3,p}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name
Template:CDD
{4,3,3}
Template:CDD
{4,3,4}
Template:CDD
Template:CDD
{4,3,5}
Template:CDD
{4,3,6}
Template:CDD
Template:CDD
{4,3,7}
Template:CDD
{4,3,8}
Template:CDD
Template:CDD
... {4,3,∞}
Template:CDD
Template:CDD
Image
Vertex
figure
Template:CDD

{3,3}
Template:CDD

{3,4}
Template:CDD
Template:CDD

{3,5}
Template:CDD

{3,6}
Template:CDD
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{3,7}
Template:CDD

{3,8}
Template:CDD
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{3,∞}
Template:CDD
Template:CDD

## References

• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• T. Gosset (1900) On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan.
• T. Proctor Hall (1893) "The projection of fourfold figures on a three-flat", American Journal of Mathematics 15:179–89.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Victor Schlegel (1886) Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren.