# Tesseract

Last updated
Tesseract
8-cell
(4-cube)
Type Convex regular 4-polytope
Schläfli symbol {4,3,3}
t0,3{4,3,2} or {4,3}×{ }
t0,2{4,2,4} or {4}×{4}
t0,2,3{4,2,2} or {4}×{ }×{ }
t0,1,2,3{2,2,2} or { }×{ }×{ }×{ }
Coxeter diagram

Cells 8 {4,3}
Faces 24 {4}
Edges 32
Vertices 16
Vertex figure
Tetrahedron
Petrie polygon octagon
Coxeter group B4, [3,3,4]
Dual 16-cell
Properties convex, isogonal, isotoxal, isohedral
Uniform index 10

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. [1] 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.

## Contents

The tesseract is also called an 8-cell, C8, (regular) octachoron, octahedroid, [2] cubic prism, and tetracube. [3] It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of hypercubes or measure polytopes. [4] Coxeter labels it the ${\displaystyle \gamma _{4}}$ polytope. [5] 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 ( 'four') and from aktís ( 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract. [6]

## Geometry

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 the 16-cell with Schläfli symbol {3,3,4}, 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.

### Coordinates

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:

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

In this Cartesian frame of reference, the tesseract has radius 2 and 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.

### Net

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).

### Construction

The construction of hypercubes can be imagined the following way:

• 1-dimensional: Two points A and B can be connected to become a line, giving a new line segment AB.
• 2-dimensional: Two parallel line segments AB and CD separated by a distance of AB can be connected to become a square, with the corners marked as ABCD.
• 3-dimensional: Two parallel squares ABCD and EFGH separated by a distance of AB can be connected to become a cube, with the corners marked as ABCDEFGH.
• 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP separated by a distance of AB can be connected to become a tesseract, with the corners marked as ABCDEFGHIJKLMNOP. However, this parallel positioning of two cubes such that their 8 corresponding pairs of vertices are each separated by a distance of AB can only be achieved in a space of 4 or more dimensions.

The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two demitesseracts (16-cells). It can also be triangulated into 4-dimensional simplices (irregular 5-cells) that share their vertices with the tesseract. It is known that there are 92487256 such triangulations [8] and that the fewest 4-dimensional simplices in any of them is 16. [9]

The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) 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 (its mirror walls).

The long radius (center to vertex) of the tesseract is equal to its edge length; thus its diagonal through the center (vertex to opposite vertex) is 2 edge lengths. 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 (other than a 0-dimensional point) that is radially equilateral. The longest vertex-to-vertex diameter of an n-dimensional hypercube of unit edge length is n, so for the square it is 2, for the cube it is 3, and only for the tesseract it is 4, exactly 2 edge lengths.

### Formulas

For a tesseract with side length s:

• Hypervolume: ${\displaystyle H=s^{4}}$
• Surface volume: ${\displaystyle SV=8s^{3}}$
• Face diagonal: ${\displaystyle d_{\mathrm {2} }={\sqrt {2}}s}$
• Cell diagonal: ${\displaystyle d_{\mathrm {3} }={\sqrt {3}}s}$
• 4-space diagonal: ${\displaystyle d_{\mathrm {4} }=2s}$

### 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 nondiagonal numbers say how many of the column's element occur in or at the row's element. [10] For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.

${\displaystyle {\begin{bmatrix}{\begin{matrix}16&4&6&4\\2&32&3&3\\4&4&24&2\\8&12&6&8\end{matrix}}\end{bmatrix}}}$

## Projections

It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.

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).

Orthographic projections
Coxeter plane B4B4 --> A3A3
Graph
Dihedral symmetry [8][4][4]
Coxeter planeOtherB3 / D4 / A2B2 / D3
Graph
Dihedral symmetry[2][6][4]
 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.
 3D Projection of three tesseracts with and without faces 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)
 Stereoscopic 3D projection of a tesseract (parallel view) Stereoscopic 3D Disarmed Hypercube

## Tessellation

The tesseract, like 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°. [11]

The tesseract's radial equilateral symmetry makes its tessellation the unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions.

The tesseract is 4th in a series of hypercube:

The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).

Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

Schläfli symbol {3, 3, 3}{3, 3, 4}{4, 3, 3}{3, 4, 3}{3, 3, 5}{5, 3, 3}
Coxeter mirrors
Mirror dihedrals𝝅/2𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2
Graph
Vertices581624120600
Edges102432967201200
Faces10 triangles32 triangles24 squares96 triangles1200 triangles720 pentagons
Cells5 tetrahedra16 tetrahedra8 cubes24 octahedra600 tetrahedra120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed120 in 120-cell675 in 120-cell2 16-cells3 8-cells25 24-cells10 600-cells
Great polygons 2 𝝅/2 squares x 34 𝝅/2 rectangles x 34 𝝅/3 hexagons x 412 𝝅/5 decagons x 650 𝝅/15 dodecagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Isocline polygons 1 2 2 4 20
Long radius${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$
Edge length${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$${\displaystyle {\sqrt {2}}\approx 1.414}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius${\displaystyle {\tfrac {1}{4}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area${\displaystyle 10\left({\sqrt {\tfrac {8}{9}}}\right)\approx 9.428}$${\displaystyle 32\left({\sqrt {\tfrac {3}{16}}}\right)\approx 13.856}$${\displaystyle 24}$${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$${\displaystyle 1200\left({\tfrac {\sqrt {3}}{8\phi ^{2}}}\right)\approx 99.238}$${\displaystyle 720\left({\tfrac {25+10{\sqrt {5}}}{8\phi ^{4}}}\right)\approx 621.9}$
Volume${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$${\displaystyle 8}$${\displaystyle 24\left({\sqrt {\tfrac {2}{9}}}\right)\approx 11.314}$${\displaystyle 600\left({\tfrac {1}{3\phi ^{3}{\sqrt {8}}}}\right)\approx 16.693}$${\displaystyle 120\left({\tfrac {2+\phi }{2\phi ^{3}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$${\displaystyle {\tfrac {2}{3}}\approx 0.667}$${\displaystyle 1}$${\displaystyle 2}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

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.

OrthogonalPerspective
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 ${\displaystyle \mathbb {C} ^{2}}$ 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. [12]

Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction. Notable examples include:

• "And He Built a Crooked House", Robert Heinlein‘s 1940 science fiction story featuring a building in the form of a four-dimensional hypercube. [13] This 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).
• Crucifixion (Corpus Hypercubus) , a 1954 oil painting by Salvador Dalí featuring a four-dimensional hypercube unfolded into a three-dimensional Latin cross. [14]
• The Grande Arche, a monument and building near Paris, France, completed in 1989. According to the monument's engineer, Erik Reitzel, the Grande Arche was designed to resemble the projection of a hypercube. [15]
• Fez , a video game where one plays a character who can see beyond the two dimensions other characters can see, and must use this ability to solve platforming puzzles. Features "Dot", a tesseract who helps the player navigate the world and tells how to use abilities, fitting the theme of seeing beyond human perception of known dimensional space. [16]

The word tesseract was later adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube of this article. See Tesseract (disambiguation).

## Notes

1. "The Tesseract - a 4-dimensional cube". www.cut-the-knot.org. Retrieved 2020-11-09.
2. Matila Ghyka, The geometry of Art and Life (1977), p.68
3. This term can also mean a polycube made of four cubes
4. Elte, E. L. (1912). The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen. ISBN   1-4181-7968-X.
5. Coxeter 1973, pp. 122–123, §7.2. illustration Fig 7.2C.
6. . Oxford English Dictionary (Online ed.). Oxford University Press. 199669. (Subscription or participating institution membership required.)
7. "Unfolding an 8-cell". Unfolding.apperceptual.com. Retrieved 21 January 2018.
8. Pournin, Lionel (2013), "The flip-Graph of the 4-dimensional cube is connected", Discrete & Computational Geometry , 49 (3): 511–530, arXiv:, doi:10.1007/s00454-013-9488-y, MR   3038527, S2CID   30946324
9. Cottle, Richard W. (1982), "Minimal triangulation of the 4-cube", Discrete Mathematics , 40: 25–29, doi:, MR   0676709
10. Coxeter 1973, p. 12, §1.8 Configurations.
11. Coxeter 1973, p. 293.
12. Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991).
13. Fowler, David (2010), "Mathematics in Science Fiction: Mathematics as Science Fiction", World Literature Today, 84 (3): 48–52, JSTOR   27871086
14. Kemp, Martin (1 January 1998), "Dali's dimensions", Nature , 391 (27): 27, Bibcode:1998Natur.391...27K, doi:, S2CID   5317132
15. Ursyn, Anna (2016), "Knowledge Visualization and Visual Literacy in Science Education", Knowledge Visualization and Visual Literacy in Science Education, Information Science Reference, p. 91, ISBN   9781522504818
16. "Dot (Character) - Giant Bomb". Giant Bomb. Retrieved 21 January 2018.

## Related Research Articles

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

In geometry, a hypercube is an n-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to .

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension n.

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid [sic?].

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.

In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as n for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n(n−1)-demicubes, and 2n(n−1)-simplex facets are formed in place of the deleted vertices.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.

## References

Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds