Tesseract
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 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4polytopes.
The tesseract is also called an 8cell, C_{8}, (regular) octachoron, octahedroid,^{}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".^{}
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 or "four rays"), referring to the four lines from each vertex to other vertices.^{} 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 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.^{} 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] 
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°.^{}
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}×{} 
Net 
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  16cell  rectified 16cell 
truncated 16cell 
cantellated 16cell 
runcinated 16cell 
bitruncated 16cell 
cantitruncated 16cell 
runcitruncated 16cell 
omnitruncated 16cell 
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 4polytopes 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  
Cells {p,3} 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
It is in a sequence of regular 4polytope 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,∞} 
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^{}
 The Grande Arche, a monument and building near Paris, France said to resemble the projection of a hypercube^{}
 "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).^{}
Notes
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 [1]
 (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
 Weisstein, Eric W., "Tesseract", MathWorld.
 Olshevsky, George, Tesseract at Glossary for Hyperspace.
 2. Convex uniform polychora based on the tesseract (8cell) and hexadecachoron (16cell)  Model 10, George Olshevsky.
 Richard Klitzing, 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

Fundamental convex regular and uniform polytopes in dimensions 2–10  

Family  A_{n}  B_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 