Tesseract
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 (4.4.4) 
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 
Uniform index  10 
In geometry, the tesseract is the fourdimensional 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 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,^{[1]} cubic prism, and tetracube (although this last term can also mean a polycube made of four cubes). It is the fourdimensional hypercube, or 4cube as a part of the dimensional family of hypercubes or "measure polytopes".^{[2]}
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 (téssereis aktines, "four rays"), referring to the four lines from each vertex to other vertices.^{[3]} 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 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 called the hexadecachoron, or 16cell, with Schläfli symbol {3,3,4}.
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:
A tesseract 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.
Projections to 2 dimensions
The construction of a hypercube can be imagined the following way:
 1dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
 2dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
 3dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
 4dimensional: 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 twodimensional spaces, as projecting a cube is possible on a twodimensional space.
Projections on the 2Dplane 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 lowerdimensional 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.
Parallel projections to 3 dimensions
The cellfirst parallel projection of the tesseract into 3dimensional 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 facefirst parallel projection of the tesseract into 3dimensional 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 edgefirst parallel projection of the tesseract into 3dimensional 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 vertexfirst projection. The two remaining cells project onto the prism bases. The vertexfirst parallel projection of the tesseract into 3dimensional 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. An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract.^{[4]} 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 a tesseract 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 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) 
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 complex polygon
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.^{[5]}
Tessellation
The tesseract, along with 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°.^{[6]}
Related polytopes and honeycombs
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.
In popular culture
Since their discovery, fourdimensional hypercubes have been a popular theme in art, architecture, and fiction. Notable examples include:
 Crucifixion (Corpus Hypercubus) – Oil painting by Salvador Dalí featuring a fourdimensional hypercube unfolded into a threedimensional Latin cross^{[7]}
 The Grande Arche – A monument and building near Paris, France said to resemble the projection of a hypercube^{[8]}
 "And He Built a Crooked House" – A science fiction story featuring a building in the form of a fourdimensional hypercube written by Robert Heinlein (1940)^{[9]}
Notes
 ↑ Matila Ghyka, The geometry of Art and Life (1977), p.68
 ↑ E. L. Elte, The Semiregular Polytopes of the Hyperspaces, (1912)
 ↑ http://www.oed.com/view/Entry/199669?redirectedFrom=tesseract#eid
 ↑ "Unfolding an 8cell".
 ↑ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991).
 ↑ Coxeter, Regular polygons, p.293
 ↑ Kemp, Martin (1 January 1998), "Dali's dimensions", Nature, 391 (27), doi:10.1038/34063
 ↑ Du Sautoy, Marcus. "A 4 Dimensional Cube in Paris". The Number Mysteries. Archived from the original on 20140428. Retrieved 17 June 2012.
 ↑ Fowler, David (2010), "Mathematics in Science Fiction: Mathematics as Science Fiction", World Literature Today, 84 (3): 48–52, JSTOR 27871086,
Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The NoSided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube (tesseract).
.
References
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (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 ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 T. Gosset (1900) On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan.
 T. Proctor Hall (1893) "The projection of fourfold figures on a threeflat", 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 vierdimensionalen Körper, Waren.
External links
 Olshevsky, George. "Tesseract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Klitzing, Richard. "4D uniform polytopes (polychora) x4o3o3o  tes".
 The Tesseract Ray traced images with hidden surface elimination. This site provides a good description of methods of visualizing 4D solids.
 Der 8Zeller (8cell) Marco Möller's Regular polytopes in R^{4} (German)
 WikiChoron: Tesseract
 HyperSolids is an open source program for the Apple Macintosh (Mac OS X and higher) which generates the five regular solids of threedimensional space and the six regular hypersolids of fourdimensional space.
 Hypercube 98 A Windows program that displays animated hypercubes, by Rudy Rucker
 ken perlin's home page A way to visualize hypercubes, by Ken Perlin
 Some Notes on the Fourth Dimension includes very good animated tutorials on several different aspects of the tesseract, by Davide P. Cervone
 Tesseract animation with hidden volume elimination