Tesseract 8-cell 4-cube | |
---|---|
Type | Convex regular 4-polytope |
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 | 16-cell |
Properties | convex, isogonal, isotoxal, isohedral |
Uniform index | 10 |
In geometry, the tesseract is the four-dimensional analogue 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 4-polytopes.
The tesseract is also called an eight-cell, C_{8}, (regular) octachoron, octahedroid,^{[1]} cubic prism, and tetracube.^{[2]} It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or "measure polytopes".^{[3]}
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.^{[4]} In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract".
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 4-4 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 regular hexadecachoron, or sixteen-cell, with Schläfli symbol {3,3,4}, with which it can be combined to form the compound of tesseract and 16-cell.
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:
A tesseract is bounded by eight hyperplanes (x_{i} = ±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.
The construction of a hypercube can be imagined the following way:
It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into 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.
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 u=(1,1,-1,-1), v=(-1,1,-1,1), w=(1,-1,-1,1). |
The elements of a regular polytope can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements that are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[5]}^{[6]}
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.^{[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). | Stereoscopic 3D projection of a tesseract (parallel view) |
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 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) |
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] |
Orthogonal | Perspective |
---|---|
_{4}{4}_{2}, with 16 vertices and 8 4-edges, with the 8 4-edges 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 4-4 duoprism in 4-dimensional space. _{4}{4}_{2} has 16 vertices, and 8 4-edges. 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 4-edges are considered distinct.^{[8]}
The tesseract, along with all hypercubes, tessellates Euclidean space. The self-dual 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°.^{[9]}
As a uniform duoprism, the tesseract exists in a sequence of uniform duoprisms: {p}×{4}.
The regular tesseract, along with the 16-cell, exists in a set of 15 uniform 4-polytopes with the same symmetry. The tesseract {4,3,3} exists in a sequence of regular 4-polytopes and honeycombs, {p,3,3} with tetrahedral vertex figures, {3,3}. The tesseract is also in a sequence of regular 4-polytope and honeycombs, {4,3,p} with cubic cells.
Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and fiction. Notable examples include:
Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided 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)..