View source for Tesseract

{{short description|Four-dimensional analogue of the cube}}
{{about|the geometric shape}}
[[File:Tesseract.glb|thumb|3D model of tesseract]]
{{Infobox polychoron
| Name=Tesseract<br />8-cell<br />(4-cube)
| Image_File=8-cell-simple.gif
| Type=[[Convex regular 4-polytope]]
| Family=[[Hypercubes]]
| Last=[[Omnitruncated 5-cell|9]]
| Index=10
| Next=[[Rectified tesseract|11]]
| Schläfli={4,3,3}<br />t<sub>0,3</sub>{4,3,2} or {4,3}×{&nbsp;}<br />t<sub>0,2</sub>{4,2,4} or {4}×{4}<br />t<sub>0,2,3</sub>{4,2,2} or {4}×{&nbsp;}×{&nbsp;}<br />t<sub>0,1,2,3</sub>{2,2,2} or {&nbsp;}×{&nbsp;}×{&nbsp;}×{&nbsp;}
| CD={{CDD|node_1|4|node|3|node|3|node}}<br />{{CDD|node_1|4|node|3|node|2|node_1}}<br />{{CDD|node_1|4|node|2|node_1|4|node}}<br />{{CDD|node_1|4|node|2|node_1|2|node_1}}<br />{{CDD|node_1|2|node_1|2|node_1|2|node_1}}
| Cell_List=8 [[cube|{4,3}]] [[File:Hexahedron.png|20px]]
| Face_List=24 [[Square (geometry)|{4}]]
| Edge_Count=32
| Vertex_Count=16
| Petrie_Polygon=[[octagon]]
| Coxeter_Group=B<sub>4</sub>, [3,3,4]
| Vertex_Figure=[[File:8-cell verf.svg|80px]]<br />[[Tetrahedron]]
| Dual=[[16-cell]]
| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]], [[Hanner polytope]]
}}
{{wikt | tesseract}}
[[File:8-cell net.png|thumb|The [[Dali cross|Dalí cross]], a [[Net (polyhedron)|net]] of a tesseract]]
[[File:Net of tesseract.gif|thumb|The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.]]

In [[geometry]], a '''tesseract''' or '''4-cube''' is a [[four-dimensional space|four-dimensional]] [[hypercube]], analogous to a two-[[dimension]]al [[square (geometry)|square]] and a three-dimensional [[cube]].<ref>{{Cite web|title= The Tesseract - a 4-dimensional cube|url= https://www.cut-the-knot.org/ctk/Tesseract.shtml|access-date= 2020-11-09|website= www.cut-the-knot.org}}</ref> Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square [[Face (geometry) |faces]], the [[hypersurface]] of the tesseract consists of eight cubical [[cell (geometry) |cells]], meeting at [[right angle]]s. The tesseract is one of the six [[convex regular 4-polytope]]s.

The tesseract is also called an '''8-cell''', '''C<sub>8</sub>''', (regular) '''octachoron''', or '''cubic prism'''. It is the four-dimensional '''measure polytope''', taken as a unit for hypervolume.<ref>{{Cite book |last=Elte |first=E. L. |author-link=Emanuel Lodewijk Elte |title=The Semiregular Polytopes of the Hyperspaces |date=2005 |publisher=University of Groningen |isbn=1-4181-7968-X |location=Groningen }}</ref> [[Harold Scott MacDonald Coxeter| Coxeter]] labels it the {{math|''γ''<sub>4</sub>}} polytope.{{Sfn|Coxeter|1973|pp=122-123|loc=§7.2. illustration Fig 7.2<small>C</small>}}  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 [[Ancient Greek| Greek]] {{lang|grc-Latn|téssara}} ({{wikt-lang|grc|τέσσαρα}} 'four') and {{lang|grc-Latn|aktís}} ({{wikt-lang|grc|ἀκτίς}} 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as ''tessaract''.<ref>
{{cite OED|term=tesseract|ID=199669}}</ref>

== Geometry ==
As a [[regular polytope]] with three [[cube]]s folded together around every edge, it has [[Schläfli symbol]] {4,3,3} with [[Hyperoctahedral group#By dimension|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}&nbsp;×&nbsp;{&nbsp;}, with symmetry order 96. As a 4-4 [[duoprism]], a [[Cartesian product]] of two [[Square (geometry)|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 {&nbsp;}&nbsp;×&nbsp;{&nbsp;}&nbsp;×&nbsp;{&nbsp;}&nbsp;×&nbsp;{&nbsp;} or {&nbsp;}<sup>4</sup>, 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.

A tesseract is bounded by eight three-dimensional [[hyperplane]]s. Each pair of non-parallel hyperplanes intersects to form 24 square faces. 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, a tesseract consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

===Coordinates===

A ''unit tesseract'' has side length {{math|1}}, and is typically taken as the basic unit for [[hypervolume]] in 4-dimensional space. ''The'' unit tesseract in a [[Cartesian coordinate system]] for 4-dimensional space has two opposite vertices at coordinates {{math|[0, 0, 0, 0]}} and {{math|[1, 1, 1, 1]}}, and other vertices with coordinates at all possible combinations of {{math|0}}s and {{math|1}}s. It is the [[Cartesian product]] of the closed [[unit interval]] {{math|[0, 1]}} in each axis.

Sometimes a unit tesseract is centered at the origin, so that its coordinates are the more symmetrical <math>\bigl({\pm\tfrac12}, \pm\tfrac12, \pm\tfrac12, \pm\tfrac12 \bigr).</math> This is the Cartesian product of the closed interval <math>\bigl[{-\tfrac12}, \tfrac12\bigr]</math> in each axis.

Another commonly convenient tesseract is the Cartesian product of the closed interval {{math|[&minus;1, 1]}} in each axis, with vertices at coordinates {{math|(±1, ±1, ±1, ±1)}}. This tesseract has side length 2 and hypervolume {{math|1=2<sup>4</sup> = 16}}.

===Net===
An unfolding of a [[polytope]] is called a [[Net (polyhedron)|net]]. There are 261 distinct nets of the tesseract.<ref>{{cite web|url=http://unfolding.apperceptual.com/|title=Unfolding an 8-cell|website=Unfolding.apperceptual.com|access-date=21 January 2018}}</ref> The unfoldings of the tesseract can be counted by mapping the nets to ''paired trees'' (a [[Tree (graph theory)|tree]] together with a [[perfect matching]] in its [[Complement graph|complement]]).

Each of the 261 nets can tile 3-space.<ref>[[Matt Parker|Parker, Matt]]. [https://whuts.org/ Which Hypercube Unfoldings Tile Space?] Retrieved 2025 May 11.</ref>

===Construction===
[[File:From Point to Tesseract (Looped Version).gif|thumb|An animation of the shifting in [[dimension]]s]]
The construction of [[hypercube]]s 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.
[[File:Dimension levels.svg|480px|A diagram showing how to create a tesseract from a point]]

The 8 cells of the tesseract may be regarded (three different ways) as two interlocked rings of four cubes.{{Sfn|Coxeter|1970|p=18}}

The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two [[Demihypercube|demitesseracts]] ([[Demitesseract|16-cells]]). It can also be [[Point-set triangulation|triangulated]] into 4-dimensional [[simplex|simplices]] ([[5-cell#Irregular 5-cells|irregular 5-cells]]) that share their vertices with the tesseract. It is known that there are {{val|92487256}} such triangulations<ref>{{citation
 | last1 = Pournin | first1 = Lionel
 | mr = 3038527
 | title = The flip-Graph of the 4-dimensional cube is connected
 | journal = [[Discrete & Computational Geometry]]
 | pages = 511–530
 | volume = 49
 | year = 2013
 | issue = 3
 | doi = 10.1007/s00454-013-9488-y| arxiv = 1201.6543| s2cid = 30946324
 }}
</ref> and that the fewest 4-dimensional simplices in any of them is 16.<ref>{{citation
 | last1 = Cottle | first1 = Richard W.
 | mr = 676709
 | title = Minimal triangulation of the 4-cube
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | pages = 25–29
 | volume = 40
 | year = 1982
 | doi = 10.1016/0012-365X(82)90185-6| doi-access = free
 }}</ref> 

The dissection of the tesseract into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplex]] (a particular [[orthoscheme]] with Coxeter diagram {{CDD|node|4|node|3|node|3|node}}) is the most basic direct construction of the tesseract possible. The '''[[5-cell#Orthoschemes|characteristic 5-cell of the 4-cube]]''' is a [[fundamental region]] of the tesseract's defining [[Coxeter group|symmetry group]], the group which generates the [[B4 polytope|B<sub>4</sub> 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'').

=== Radial equilateral symmetry ===
The radius of a [[hypersphere]] 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 uniform [[polytopes]] have this property, including the four-dimensional tesseract and [[24-cell#Radially equilateral honeycomb|24-cell]], the three-dimensional [[Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[hexagon]]. In particular, the tesseract is the only hypercube (other than a zero-dimensional point) that is ''radially equilateral''. The longest vertex-to-vertex diagonal of an <math>n</math>-dimensional hypercube of unit edge length is <math>\sqrt{n\vphantom{t}},</math> which for the square is <math>\sqrt2,</math> for the cube is <math>\sqrt3,</math> and only for the tesseract is <math>\sqrt4 = 2</math> edge lengths.

An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates <math>\bigl({\pm\tfrac12}, \pm\tfrac12, \pm\tfrac12, \pm\tfrac12\bigr).</math>

=== Properties {{anchor|Formulas}} ===
{{tesseract_graph_nonplanar_visual_proof.svg|150px|thumb|right}}
For a tesseract with side length {{Mvar|s}}:

* [[Hypervolume]] (4D): <math>H=s^4</math>
* Surface "volume" (3D): <math>SV=8s^3</math>
*[[Face diagonal]]: <math>d_\mathrm{2}=\sqrt{2} s</math>
*[[Space diagonal|Cell diagonal]]: <math>d_\mathrm{3}=\sqrt{3} s</math>
*4-space diagonal: <math>d_\mathrm{4}=2s</math>

=== As a configuration ===
This [[Regular 4-polytope#As configurations|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 diagonal reduces to the [[f-vector]] (16,32,24,8).

The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations|p=12}} 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 the [[vertex figure]], here tetrahedra, (4,6,4). The next row is vertex figure ridge, here a triangle, (3,3).

<math>\begin{bmatrix}\begin{matrix}16 & 4 & 6 & 4 \\ 2 & 32 & 3 & 3 \\ 4 & 4 & 24 & 2 \\ 8 & 12 & 6 & 8 \end{matrix}\end{bmatrix}</math>

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

[[File:Orthogonal projection envelopes tesseract.png|thumb|left|Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)]]

[[File:Hypercubeorder binary.svg|thumb|right|The [[rhombic 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&nbsp;4&nbsp;6&nbsp;4&nbsp;1—the fourth row in [[Pascal's triangle]].]]

The ''cell-first'' parallel [[graphical projection|projection]] of the tesseract into three-dimensional space has a [[cube|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 [[cuboid]]al 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 dodecahedron|rhombic dodecahedral]] envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of [[dissection (geometry)|dissecting]] a rhombic dodecahedron into four congruent [[rhombohedron|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 {{nowrap|1=''u'' = (1,1,−1,−1)}}, {{nowrap|1=''v'' = (−1,1,−1,1)}}, {{nowrap|1=''w'' = (1,−1,−1,1)}}.

{{clear|left}}
[[File:Orthogonal Tesseract Gif.gif|thumb|right|Animation showing each individual cube within the B<sub>4</sub> Coxeter plane projection of the tesseract]]

{| class=wikitable
|+ [[Orthographic projection]]s
|- align=center
![[Coxeter plane]]
!B<sub>4</sub>
!B<sub>4</sub> --> A<sub>3</sub>
!A<sub>3</sub>
|- align=center
!Graph
|[[File:4-cube t0.svg|150px]]
|[[File:4-4 duoprism-isotoxal.svg|150px]]
|[[File:4-cube t0 A3.svg|150px]]
|- align=center
![[Dihedral symmetry]]
|[8]
|[4]
|[4]
|- align=center
!Coxeter plane
!Other
!B<sub>3</sub> / D<sub>4</sub> / A<sub>2</sub>
!B<sub>2</sub> / D<sub>3</sub>
|- align=center
!Graph
|[[File:4-cube column graph.svg|150px]]
|[[File:4-cube t0 B3.svg|150px]]
|[[File:4-cube t0 B2.svg|150px]]
|- align=center
!Dihedral symmetry
|[2]
|[6]
|[4]
|}

{{-}}
{{multiple image
| class=wikitable
 | footer = Orthographic projection Coxeter plane B<sub>4</sub> graph with [[hidden lines]] as dash lines, and the tesseract without hidden lines.
 | image1 = Tesseract_With_Hidden_Dash_Lines.jpg
 | image2 = Tesseract_Without_Hidden_Lines.jpg
 | total_width = 300px
}}

{{-}}
{| class="wikitable" width=480
|- align=center valign=top
|rowspan=2|[[File:8-cell.gif]]<BR>A 3D projection of a tesseract performing a [[SO(4)#Geometry of 4D rotations|simple rotation]] about a plane in 4-dimensional space. The plane bisects the figure from front-left to back-right and top to bottom.
|[[File:8-cell-orig.gif]]<BR>A 3D projection of a tesseract performing a [[SO(4)#Geometry of 4D rotations|double rotation]] about two orthogonal planes in 4-dimensional space.
|}
{{-}}
{| class=wikitable width=640
|- align=center valign=top
|[[File:Animation of three four dimensional cube.webm|thumb|3D Projection of three tesseracts with and without faces]]
|[[File:Tesseract-perspective-vertex-first-PSPclarify.png|200px]]<BR>Perspective with '''hidden volume elimination'''. The red corner is the nearest in [[Four-dimensional space|4D]] and has 4 cubical cells meeting around it.
|}

{| class=wikitable width=640
|- align=center valign=top
|[[File:Tesseract tetrahedron shadow matrices.svg|200px|right]]
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 [[point at infinity|infinity]] and the four edges to it are not shown.
|[[File:Stereographic polytope 8cell.png|200px]]<BR>[[Stereographic projection]]<BR>
(Edges are projected onto the [[3-sphere]])
|}

{| class=wikitable
|- align=left valign=top
|[[File:3D stereographic projection tesseract.PNG|360px]]<BR>[[Stereoscopy|Stereoscopic]] 3D projection of a tesseract (parallel view)
|-
|[[File:Hypercube Disarmed.PNG|360px]]<BR>[[Stereoscopy|Stereoscopic]] 3D Disarmed [[Hypercube]]
|}

== Tessellation ==
The tesseract, like all [[hypercubes]], [[Tessellation|tessellates]] [[Euclidean space]]. The self-dual [[tesseractic honeycomb]] consisting of 4 tesseracts around each face has [[Ludwig Schläfli|Schläfli]] symbol '''{4,3,3,4}'''. Hence, the tesseract has a [[dihedral angle]] of 90°.{{Sfn|Coxeter|1973|p=293}}

The tesseract's [[#Radial equilateral symmetry|radial equilateral symmetry]] makes its tessellation the [[Tesseractic honeycomb#Sphere packing|unique regular body-centered cubic lattice]] of equal-sized spheres, in any number of dimensions.

== Related polytopes and honeycombs ==
The tesseract is 4th in a series of [[hypercube]]:
{{Hypercube polytopes}}

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}}

As a uniform [[duoprism]], the tesseract exists in a [[Uniform 4-polytope#Polygonal prismatic prisms: .5Bp.5D .C3.97 .5B .5D .C3.97 .5B .5D|sequence of uniform duoprisms]]: {''p''}×{4}.

The regular tesseract, along with the [[16-cell]], exists in a set of 15 [[Truncated tesseract#Related uniform polytopes in tesseract symmetry|uniform 4-polytopes with the same symmetry]]. The tesseract {4,3,3} exists in a [[Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures|sequence of regular 4-polytopes and honeycombs]], {''p'',3,3} with [[tetrahedron|tetrahedral]] [[vertex figure]]s, {3,3}. The tesseract is also in a [[Order-5 cubic honeycomb#Related polytopes and honeycombs with cubic cells|sequence of regular 4-polytope and honeycombs]], {4,3,''p''} with [[cube|cubic]] [[cell (geometry)|cells]].

{| class=wikitable style="float:right;" width=320
!Orthogonal||Perspective
|-
|[[File:4-generalized-2-cube.svg|160px]]
|[[File:Complex polygon 4-4-2-stereographic3.svg|160px]]
|-
|colspan=2|<sub>4</sub>{4}<sub>2</sub>, 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]] <sub>4</sub>{4}<sub>2</sub>, {{CDD|4node_1|4|node}}, in <math>\mathbb{C}^2</math> has a real representation as a tesseract or 4-4 [[duoprism]] in 4-dimensional space. <sub>4</sub>{4}<sub>2</sub> has 16 vertices, and 8 4-edges. Its symmetry is <sub>4</sub>[4]<sub>2</sub>, order 32. It also has a lower symmetry construction, {{CDD|4node_1|2|4node_1}}, or <sub>4</sub>{}×<sub>4</sub>{}, with symmetry <sub>4</sub>[2]<sub>4</sub>, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.<ref>Coxeter, H. S. M., ''Regular Complex Polytopes'', second edition, Cambridge University Press, (1991).</ref>

{{Clear}}

==In popular culture==
Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction. Notable examples include:
<!-- Do not add examples without sources. Also, do not add examples that use the word "tesseract" but are not about hypercubes.
     In particular, do not add "A Wrinkle in Time" or "Interstellar", as their uses of "tesseract" are not about hypercubes. -->
* "[[And He Built a Crooked House]]", [[Robert A. Heinlein|Robert Heinlein]]'s 1940 science fiction story featuring a building in the form of a four-dimensional hypercube.<ref>{{citation
|title=Mathematics in Science Fiction: Mathematics as Science Fiction
|first=David
|last=Fowler
|journal=World Literature Today
|volume=84
|issue=3
|year=2010
|pages=48–52
|doi=10.1353/wlt.2010.0188
|jstor=27871086|s2cid=115769478
}}</ref> 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]].<ref>{{citation|title=Dali's dimensions|first=Martin|last=Kemp|journal=[[Nature (journal)|Nature]]|volume=391|issue=27|date=1 January 1998|pages=27|doi=10.1038/34063|bibcode=1998Natur.391...27K|s2cid=5317132|doi-access=free}}</ref>
* 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.<ref>{{citation|last=Ursyn|first=Anna|title=Knowledge Visualization and Visual Literacy in Science Education|publisher=Information Science Reference|year=2016|isbn=9781522504818|pages=91|contribution-url=https://books.google.com/books?id=-JBJDAAAQBAJ&pg=PA91|contribution=Knowledge Visualization and Visual Literacy in Science Education}}</ref>
* ''[[Fez (video game)|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.<ref>{{cite web|url=http://www.giantbomb.com/dot/3005-23100/|title=Dot (Character) - Giant Bomb|website=Giant Bomb|access-date=21 January 2018}}</ref>
The word ''tesseract'' 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; see [[Tesseract (disambiguation)]].
<!-- Do not add examples without sources. Also, do not add examples that use the word "tesseract" but are not about hypercubes. The last bullet directs readers to the page that will help them find other, non-hypercube, per this article, links. -->

==Notes==
{{Reflist}}

== References ==
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) | pages=122–123}}
* F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss (1995) ''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', Wiley-Interscience Publication {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [[Mathematische Zeitschrift]] 46 (1940) 380–407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* {{Citation | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }}
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss (2008) ''The Symmetries of Things'', {{isbn|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1<sub>n1</sub>)
* [[Thorold Gosset|T. Gosset]] (1900) ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan.
* {{cite journal
 | last = Hall | first = T. Proctor | authorlink = T. Proctor Hall
 | year = 1893
 | jstor = 2369565
 | title = The projection of fourfold figures on a three-flat
 | journal = [[American Journal of Mathematics]]
 | volume = 15
 | issue = 2 | pages = 179–189
| doi = 10.2307/2369565 }}
* [[Norman Johnson (mathematician)|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 vier-dimensionalen Körper'', Waren.

== External links ==
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x4o3o3o - tes}}
* [http://mrl.nyu.edu/~perlin/demox/Hyper.html ken perlin's home page] A way to visualize hypercubes, by [[Ken Perlin]]
* [https://www.math.union.edu/~dpvc/math/4D/ Some Notes on the Fourth Dimension] includes animated tutorials on several different aspects of the tesseract, by [http://www.math.union.edu/~dpvc/ Davide P. Cervone]
* [http://www.fano.co.uk/hypermodel/tesseract.html Tesseract animation with hidden volume elimination]
{{Hypercube polytopes}}
{{Regular 4-polytopes}}
{{Polytopes}}

[[Category:Algebraic topology]]
[[Category:Regular 4-polytopes]]
[[Category:Cubes]]