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Tesseract

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From Wikipedia, the free encyclopedia

Four-dimensional analogue of the cube

This article is about the geometric shape. For other uses, seeTesseract (disambiguation).

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{4,3}
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,Hanner polytope
Uniform index	10

Look up tesseract in Wiktionary, the free dictionary.

Ingeometry, atesseract or4-cube is afour-dimensional hypercube, analogous to a two-dimensional square and a three-dimensionalcube.^[1] Just as the perimeter of the square consists of four edges and the surface of the cube consists of six squarefaces, thehypersurface of the tesseract consists of eight cubicalcells, meeting atright angles. The tesseract is one of the sixconvex regular 4-polytopes.

The tesseract is also called an8-cell,C₈, (regular)octachoron, orcubic prism. It is the four-dimensionalmeasure polytope, taken as a unit for hypervolume.^[2] Coxeter labels it theγ₄ polytope.^[3] The termhypercube without a dimension reference is frequently treated as a synonym for this specificpolytope.

TheOxford English Dictionary traces the wordtesseract toCharles Howard Hinton's 1888 bookA New Era of Thought. Hinton originally spelled the word astessaract,^[4] changing it totesseract in his 1904 bookThe Fourth Dimension. The term derives from the Greektéssara (τέσσαρα 'four') andaktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices.

Construction

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The construction of a tesseract can be visualized through the analogy of dimensions in the following steps:

One can take out two points with a certain length that form a line segment.
If another identical line segment is its length in a perpendicular direction from itself, it sweeps out and forms asquare (2-cube). The results have four points and four line segments, which are called vertices and edges, respectively.
Moving the square with the same length in the direction perpendicular to the plane it lies on generates acube (3-cube). The results have eight vertices, twelve edges, and six squares. The squares are called the faces.
Moving the cube with the same length again into the fourth-dimensional space generates a tesseract (4-cube).

For a tesseract, it is bounded by eight cubes that are known as thecells, each pair of which intersects, forming twenty-four square faces. Three cubes and three squares intersect at each edge. Four cubes, six squares, and four edges meet at every vertex. Collectively, the tesseract consists of eight cubes, twenty-four squares, thirty-two edges, and sixteen vertices. The tesseract, as well as both the square and the cube, is a member of thehypercube's family.^[5]

An animation of the shifting indimensions

TheDali cross is one of 261 tesseract nets, unfolded into eight cubes in three-dimensional space

An unfolding of apolytope is called anet. There are 261 distinct nets of the tesseract.^[6] The unfoldings of the tesseract can be counted by mapping the nets topaired trees (atree together with aperfect matching in itscomplement), which can tile 3-space.^[7] TheDali cross is one of the net examples, named after Spanish surrealist artistSalvador Dalí, whose paintingCorpus Hypercubus in 1954. It is constructed from eight cubes, whereby four cubes are stacked vertically, and four others are attached to the second-from-top cube of the stack.^[8]^[9]

Properties

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The eight cells of a tesseract may be regarded in three different ways as two interlocked rings of four cubes.^[10] As aregular polytope with threecubes folded together around every edge, it hasSchläfli symbol {4,3,3} withhyperoctahedral symmetry of order 384. Constructed as a 4Dhyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a 4-4duoprism, aCartesian product of twosquares, it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As anorthotope 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, thevertex figure of the tesseract is a regulartetrahedron. Thedual polytope of the tesseract is the16-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 anetwork topology to link multiple processors inparallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of twodemitesseracts (16-cells). It can also betriangulated into 4-dimensionalsimplices (irregular 5-cells) that share their vertices with the tesseract. It is known that there are92487256 such triangulations^[11] and that the fewest 4-dimensional simplices in any of them is 16.^[12]

The dissection of the tesseract into instances of itscharacteristic simplex (a particularorthoscheme with Coxeter diagram) is the most basic direct construction of the tesseract possible. Thecharacteristic 5-cell of the 4-cube is afundamental region of the tesseract's definingsymmetry group, the group which generates theB₄ polytopes. The tesseract's characteristic simplex directlygenerates the tesseract through the actions of the group, by reflecting itself in its own bounding facets (itsmirror walls).

Unit tesseract

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Aunit tesseract has side length1, and is typically taken as the basic unit forhypervolume in 4-dimensional space.The unit tesseract in aCartesian coordinate system for 4-dimensional space has two opposite vertices at coordinates[0, 0, 0, 0] and[1, 1, 1, 1], and other vertices with coordinates at all possible combinations of0s and1s. It is theCartesian product of the closedunit interval[0, 1] in each axis.

Sometimes a unit tesseract is centered at the origin, so that its coordinates are the more symmetrical ${\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.$ This is the Cartesian product of the closed interval ${\bigl [}{-{\tfrac {1}{2}}},{\tfrac {1}{2}}{\bigr ]}$ in each axis.

Another commonly convenient tesseract is the Cartesian product of the closed interval $[-1,1]$ in each axis, with vertices at coordinates $(\pm 1,\pm 1,\pm 1,\pm 1)$ . This tesseract has side length 2 and hypervolume $2^{4}=16$ .^[13]

Radial equilateral symmetry

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The radius of ahypersphere circumscribed about a regular polytope is the distance from the polytope's center to one of the vertices, and for the tesseract this radius is equal to its edge length; the diameter of the sphere, the length of the diagonal between opposite vertices of the tesseract, is twice the edge length. Only a few uniformpolytopes have this property, including the four-dimensional tesseract and24-cell, the three-dimensionalcuboctahedron, and the two-dimensionalhexagon. In particular, the tesseract is the only hypercube (other than a zero-dimensional point) that isradially equilateral. The longest vertex-to-vertex diagonal of an $n {\displaystyle n}$ -dimensional hypercube of unit edge length is ${\sqrt {n{\vphantom {t}}}},$ which for the square is ${\sqrt {2}},$ for the cube is ${\sqrt {3}},$ and only for the tesseract is ${\sqrt {4}}=2$ edge lengths.

An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates ${\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.$

Formulas

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Proof without words that ahypercube graph isnon-planar usingKuratowski's orWagner's theorems and finding eitherK₅ (top) orK_3,3 (bottom)subgraphs

For a tesseract with side lengths:

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

As a configuration

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Thisconfiguration 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 diagonal reduces to thef-vector (16,32,24,8).

The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[14] 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.

The bottom row defines they facets, here cubes, have f-vector (8,12,6). The next row left of diagonal is ridge elements (facet of cube), here a square, (4,4).

The upper row is the f-vector of thevertex figure, here tetrahedra, (4,6,4). The next row is vertex figure ridge, here a triangle, (3,3).

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

Projections

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

Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)

Therhombic 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 inPascal's triangle.

Thecell-first parallelprojection of the tesseract into three-dimensional space has acubical 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.

Theface-first parallel projection of the tesseract into three-dimensional space has acuboidal 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.

Theedge-first parallel projection of the tesseract into three-dimensional space has an envelope in the shape of ahexagonal 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.

Thevertex-first parallel projection of the tesseract into three-dimensional space has arhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways ofdissecting a rhombic dodecahedron into four congruentrhombohedra, giving a total of eight possible rhombohedra, each a projectedcube of the tesseract. This projection is also the one with maximal volume. One set of projection vectors areu = (1,1,−1,−1),v = (−1,1,−1,1),w = (1,−1,−1,1).

Animation showing each individual cube within the B₄ Coxeter plane projection of the tesseract

Orthographic projections
Coxeter plane	B₄	B₄ --> A₃	A₃
Graph
Dihedral symmetry	[8]	[4]	[4]
Coxeter plane	Other	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[2]	[6]	[4]

Orthographic projection Coxeter plane B₄ graph withhidden lines as dash lines, and the tesseract without hidden lines.

A 3D projection of a tesseract performing asimple 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 adouble rotation about two orthogonal planes in 4-dimensional space.

3D Projection of three tesseracts with and without faces

Perspective withhidden volume elimination. The red corner is the nearest in4D and has 4 cubical cells meeting around it.

Thetetrahedron forms theconvex hull of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex is projected toinfinity and the four edges to it are not shown.

Stereographic projection

(Edges are projected onto the3-sphere)

Stereoscopic 3D projection of a tesseract (parallel view)

Stereoscopic 3D DisarmedHypercube

Tessellation

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The tesseract, like allhypercubes,tessellates Euclidean space. The self-dualtesseractic honeycomb consisting of 4 tesseracts around each face hasSchläfli symbol{4,3,3,4}. Hence, the tesseract has adihedral angle of 90°.^[15]

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

Related polytopes and honeycombs

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The tesseract is 4th in a series ofhypercube:

Petrie polygon orthographic projections

Line segment	Square	Cube	4-cube	5-cube	6-cube	7-cube	8-cube	9-cube	10-cube

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 dihedrals	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/4⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/4⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/4⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/5⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/5⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠
Graph
Vertices	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Tori	15-tetrahedron	28-tetrahedron	24-cube	46-octahedron	2030-tetrahedron	1210-dodecahedron
Inscribed	120 in 120-cell	675 in 120-cell	2 16-cells	3 8-cells	25 24-cells	10 600-cells
Great polygons		2squares x 3	4 rectangles x 4	4hexagons x 4	12decagons x 6	100irregular hexagons x 4
Petrie polygons	1pentagon x 2	1octagon x 3	2octagons x 4	2dodecagons x 4	430-gons x 6	2030-gons x 4
Long radius	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$
Edge length	${\sqrt {\tfrac {5}{2}}}\approx 1.581$	${\sqrt {2}}\approx 1.414$	$1 {\displaystyle 1}$	$1 {\displaystyle 1}$	${\tfrac {1}{\phi }}\approx 0.618$	${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius	${\tfrac {1}{4}}$	${\tfrac {1}{2}}$	${\tfrac {1}{2}}$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area	$10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$	$32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$	$24 {\displaystyle 24}$	$96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$	$1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$	$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume	$5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$	$16\left({\tfrac {1}{3}}\right)\approx 5.333$	$8 {\displaystyle 8}$	$24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$	$600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$	$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$	${\tfrac {2}{3}}\approx 0.667$	$1 {\displaystyle 1}$	$2 {\displaystyle 2}$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$

As a uniformduoprism, the tesseract exists in asequence of uniform duoprisms: {p}×{4}.

The regular tesseract, along with the16-cell, exists in a set of 15uniform 4-polytopes with the same symmetry. The tesseract {4,3,3} exists in asequence of regular 4-polytopes and honeycombs, {p,3,3} withtetrahedral vertex figures, {3,3}. The tesseract is also in asequence of regular 4-polytope and honeycombs, {4,3,p} withcubic cells.

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

Theregular complex polytope₄{4}₂,, in $\mathbb {C} ^{2}$ has a real representation as a tesseract or 4-4duoprism in 4-dimensional space.₄{4}₂ has 16 vertices, and 8 4-edges. Its symmetry is₄[4]₂, order 32. It also has a lower symmetry construction,, or₄{}×₄{}, with symmetry₄[2]₄, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.^[16]

In popular culture

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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.^[17] This andMartin Gardner's "The No-Sided Professor", published in 1946, are among the first in science fiction to introduce readers to theMoebius band, theKlein bottle, and the hypercube (tesseract).
Crucifixion (Corpus Hypercubus), a 1954 oil painting by Salvador Dalí featuring a four-dimensional hypercube unfolded into a three-dimensionalLatin cross.^[9]
TheGrande 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.^[18]
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.^[19]

The wordtesseract has been 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; seeTesseract (disambiguation).

References

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^"The Tesseract - a 4-dimensional cube".www.cut-the-knot.org. Retrieved9 November 2020.
^Elte, E. L. (2005).The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen.ISBN 1-4181-7968-X.
^Coxeter 1973, pp. 122–123, §7.2. illustration Fig 7.2C.
^"tesseract".Oxford English Dictionary (Online ed.). Oxford University Press. 199669. (Subscription orparticipating institution membership required.)
^Hall 1893, pp. 179–189.
^"Unfolding an 8-cell".Unfolding.apperceptual.com. Retrieved21 January 2018.
^Parker, Matt.Which Hypercube Unfoldings Tile Space? Retrieved 2025 May 11.
^Langerman, Stefan; Winslow, Andrew (2016).Polycube Unfoldings Satisfying Conway's Criterion(PDF). 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG³ 2016). Tokyo.
^^a ^bKemp, Martin (1 January 1998)."Dali's Dimensions".Nature.391 (27): 27.Bibcode:1998Natur.391...27K.doi:10.1038/34063.
^Coxeter 1970, p. 18.
^Pournin, Lionel (2013), "The flip-Graph of the 4-dimensional cube is connected",Discrete & Computational Geometry,49 (3):511–530,arXiv:1201.6543,doi:10.1007/s00454-013-9488-y,MR 3038527,S2CID 30946324
^Cottle, Richard W. (1982), "Minimal triangulation of the 4-cube",Discrete Mathematics,40:25–29,doi:10.1016/0012-365X(82)90185-6,MR 0676709
^Petrov, Miroslav S.; Todorov, Todor D.; Walters, Gage S.; Willams, David M.; Witherden, Freddie D. (2022). "Enabling four-dimensional conformal hybrid meshing with cubic pyramids".Numerical Algorithms.91:671–709.arXiv:2101.05884.doi:10.1007/s11075-022-01278-y.
^Coxeter 1973, p. 12, §1.8 Configurations.
^Coxeter 1973, p. 293.
^Coxeter, H. S. M.,Regular Complex Polytopes, second edition, Cambridge University Press, (1991).
^Fowler, David (2010), "Mathematics in Science Fiction: Mathematics as Science Fiction",World Literature Today,84 (3):48–52,doi:10.1353/wlt.2010.0188,JSTOR 27871086,S2CID 115769478
^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 978-1-5225-0481-8
^"Dot (Character) - Giant Bomb".Giant Bomb. Retrieved21 January 2018.

Sources

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Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1_n1".The Symmetries of Things. p. 409.ISBN 978-1-56881-220-5.
(Paper 22) Coxeter, H. S. M.,Regular and Semi Regular Polytopes I,Mathematische Zeitschrift 46 (1940) 380–407, MR 2,10]
Coxeter, H. S. M. (1970). "Twisted Honeycombs".Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics.4. Providence, Rhode Island: American Mathematical Society.
Coxeter, H. S. M. (1973).Regular Polytopes (3 ed.). New York: Dover. pp. 122–123.
(Paper 23) Coxeter, H. S. M.,Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) Coxeter, H. S. M.,Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Gosset, Thorold (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions".Messenger of Mathematics. Macmillan.
Hall, T. Proctor (1893). "The projection of fourfold figures on a three-flat".American Journal of Mathematics.15 (2).doi:10.2307/2369565.JSTOR 2369565.
Johnson, Norman W. (1966).The Theory of Uniform Polytopes and Honeycombs (Ph.D.).
Johnson, Norman W. (1991).Uniform Polytopes (manuscript).
Schlegel, Victor (1886). "Projections-Modelle der sechs regelmässigen vier-dimensionalen Körper und des vier-dimensionalen vierseitigen Prismas". Written at Hagen in Westfalen. In Schilling, Martin (ed.).Catalog mathematischer Modelle für den höheren mathematischen Unterricht (in German) (6 ed.). Halle an der Saale (published 1903). pp. 31–34.OCLC 609855972. Retrieved14 February 2026 – via Internet Archive.
Schlegel, Victor (May 1886). Written at Waren. Knoblauch, C. H. (ed.)."Ueber Entwickelung und Stand der n-dimensionalen Geometrie, mit besonderer Berücksichtigung der vierdimensionalen".Leopoldina (in German).22 (9–10). Halle an der Saale [Dresden]: [E. Blochmann & Sohn]:92–96,108–110,133–135,149–152,160–163.OCLC 9670930. Retrieved14 February 2026 – via Internet Archive.
Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (1995).Kaleidoscopes: Selected Writings of H. S. M. Coxeter. New York: Wiley-Interscience Publication.ISBN 0-471-01003-0. Retrieved14 February 2026 – via Google Books.

External links

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Klitzing, Richard."4D uniform polytopes (polychora) x4o3o3o - tes".
ken perlin's home page A way to visualize hypercubes, byKen Perlin
Some Notes on the Fourth Dimension includes animated tutorials on several different aspects of the tesseract, byDavide P. Cervone
Tesseract animation with hidden volume elimination

Petrie polygon orthographic projections

Line segment	Square	Cube	4-cube	5-cube	6-cube	7-cube	8-cube	9-cube	10-cube

Regular 4-polytopes

Convex

5-cell	8-cell	16-cell	24-cell	120-cell	600-cell
{3,3,3} pentachoron 4-simplex	{4,3,3} tesseract 4-cube	{3,3,4} hexadecachoron 4-orthoplex	{3,4,3} icositetrachoron octaplex	{5,3,3} hecatonicosachoron dodecaplex	{3,3,5} hexacosichoron tetraplex

Star

icosahedral 120-cell	small stellated 120-cell	great 120-cell	grand 120-cell	great stellated 120-cell	grand stellated 120-cell	great grand 120-cell	great icosahedral 120-cell	grand 600-cell	great grand stellated 120-cell
{3,5,⁠5/2⁠} icosaplex	{⁠5/2⁠,5,3} stellated dodecaplex	{5,⁠5/2⁠,5} great dodecaplex	{5,3,⁠5/2⁠} grand dodecaplex	{⁠5/2⁠,3,5} great stellated dodecaplex	{⁠5/2⁠,5,⁠5/2⁠} grand stellated dodecaplex	{5,⁠5/2⁠,3} great grand dodecaplex	{3,⁠5/2⁠,5} great icosaplex	{3,3,⁠5/2⁠} grand tetraplex	{⁠5/2⁠,3,3} great grand stellated dodecaplex

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

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