View source for Tesseract

{{about|the geometrical shape|the cube used in the [[Marvel Cinematic Universe]] also known as Tesseract|Cosmic Cube|other uses|Tesseract (disambiguation)}}
{{Infobox polychoron
| Name=Tesseract<br />8-cell<br />4-cube
| Image_File=Schlegel wireframe 8-cell.png
| Image_Caption=[[Schlegel diagram]]
| Type=[[Convex regular 4-polytope]]
| Family=[[Hypercube]]
| 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.png|80px]]<br />[[Tetrahedron]]
| Dual=[[16-cell]]
| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
}}
[[File:8-cell net.png|thumb|The [[Dali cross]], a [[Net (polyhedron)|net]] of a tesseract]]

In [[geometry]], the '''tesseract''' is the [[Four-dimensional space|four-dimensional]] analogue of the [[cube]]; the tesseract is to the cube as the cube is to the [[Square (geometry)|square]]. Just as the surface of the cube consists of six square [[Face (geometry)|faces]], the [[hypersurface]] of the tesseract consists of eight cubical [[cell (geometry)|cells]]. The tesseract is one of the six [[convex regular 4-polytope]]s.

The tesseract is also called an '''eight-cell''', '''C<sub>8</sub>''', (regular) '''octachoron''', '''octahedroid''',<ref>Matila Ghyka, ''The geometry of Art and Life'' (1977), p.68</ref> '''cubic prism''', and '''tetracube'''.<ref>This term can also mean a [[polycube]] made of four cubes</ref> It is the '''four-dimensional hypercube''', or '''4-cube''' as a part of the dimensional family of [[hypercube]]s or '''measure polytopes'''.<ref>[[E. L. Elte]], ''The Semiregular Polytopes of the Hyperspaces'', (1912)</ref> Coxeter labels it the <math>\gamma_4</math> polytope.{{Sfn|Coxeter|1973|pp=122-123|loc=§7.2. illustration Fig 7.2<small>C</small>}}

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 [[Ancient Greek|Greek]] {{lang|grc|τέσσερεις ἀκτίνες}} (''téssereis aktines'', "four rays"), referring to the four lines from each vertex to other vertices.<ref>{{cite web|url=http://www.oed.com/;jsessionid=046D859BEE893EBE8940E85E9D2D7F80?authRejection=true&url=%2Fview%2FEntry%2F199669%3FredirectedFrom%3Dtesseract|title=Home : Oxford English Dictionary|website=Oed.com|accessdate=21 January 2018}}</ref> 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 [[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|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 called the regular [[hexadecachoron]], or 16-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:
: <math>\{(x_1,x_2,x_3,x_4) \in \mathbb R^4 \,:\, -1 \leq x_i \leq 1 \}</math>

A tesseract is bounded by eight [[hyperplane]]s (''x''<sub>i</sub> = ±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.

===Projections to two dimensions===
The construction of [[Hypercube|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 can be connected to become a square, with the corners marked as ABCD.
* '''3-dimensional:''' Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
* '''4-dimensional:''' Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a tesseract, with the corners marked as ABCDEFGHIJKLMNOP.

[[File:8-cell.gif|thumb|A 3D projection of an 8-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]] about a plane which bisects the figure from front-left to back-right and top to bottom]]
{| class="wikitable"
|[[File:Dimension levels.svg|thumb|385x385px|A diagram showing how to create a tesseract from a point]]
|[[File:From Point to Tesseract (Looped Version).gif|thumb|168x168px|An animation of the shifting in [[dimension]]s as shown above]]
|}

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.{{clear}}

===Parallel projections to 3 dimensions===
[[File:Hypercubeorder binary.svg|thumb|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]].]]
{| class=wikitable
|[[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)]]

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 ''u''=(1,1,-1,-1), ''v''=(-1,1,-1,1), ''w''=(1,-1,-1,1). 
|}

=== 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 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}}

<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>

== Image gallery ==
{| class=wikitable width=720
|- align=left valign=top
|[[File:Net of tesseract.gif|right|169x169px|3-D net of a tesseract]]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 (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|accessdate=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]]).
| [[File:3D stereographic projection tesseract.PNG|360px]]<BR> [[Stereoscopy|Stereoscopic]] 3D projection of a tesseract (parallel view)
|}
[[File:Hypercube Disarmed.PNG|320px|3D Crossed Eyes (no glasses needed) Disarmed Hypercube]]

===Alternative projections===
{| class=wikitable width=640
|- align=center valign=top
|[[File:8-cell-orig.gif|240x240px]]<BR>A 3D projection of a tesseract performing a [[SO(4)#Geometry of 4D rotations|double rotation]] about two orthogonal planes
|[[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]])
|}

=== 2D orthographic projections===
{| class=wikitable
|+ [[Orthographic projection]]s
|- align=center
![[Coxeter plane]]
!B<sub>4</sub>
!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 t0.svg|150px]]
|[[File:4-cube t0 B3.svg|150px]]
|[[File:4-cube t0 B2.svg|150px]]
|- align=center
![[Dihedral symmetry]]
|[8]
|[6]
|[4]
|- align=center
!Coxeter plane
!Other
!F<sub>4</sub>
!A<sub>3</sub>
|- align=center
!Graph
|[[File:4-cube column graph.svg|150px]]
|[[File:4-cube t0 F4.svg|150px]]
|[[File:4-cube t0 A3.svg|150px]]
|- align=center
!Dihedral symmetry
|[2]
|[12/3]
|[4]
|}

== Radial equilateral symmetry ==
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 [[polytopes]] have this property, including the four-dimensional tesseract and [[24-cell#Radially equilateral honeycomb|24-cell]], the three-dimensional [[Cuboctahedron#Geometric relations|cuboctahedron]], and the two-dimensional [[hexagon]]. In particular, the tesseract is the only hypercube with this property.<ref>Strictly, the hypercubes of 0 dimensions (a point) and 1 dimension (a line segment) are also radially equilateral.</ref> The longest vertex-to-vertex diameter of an ''n''-dimensional hypercube of unit edge length is {{radic|''n''}}, so for the square it is {{radic|2}}, for the cube it is {{radic|3}}, and only for the tesseract it is {{radic|4}}, exactly 2 edge lengths.

== Tessellation ==
The tesseract, like all [[hypercubes]], 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 makes its tessellation the [[Tesseractic honeycomb#Sphere packing|unique regular body-centered cubic lattice]] of equal-sized spheres, in any number of dimensions.

== Related complex polygon==
{| class=wikitable align=right width=320
!Orthogonal||Perspective
|-
|[[File:4-generalized-2-cube.svg|160px]]
|[[File:Complex polygon 4-4-2-stereographic3.png|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}}

== Related polytopes and honeycombs ==
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]].

==In popular culture==
Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and fiction. Notable examples include:
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        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|jstor=27871086|quote=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).}}.</ref>
* ''[[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|doi=10.1038/34063}}</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.fi/books?id=-JBJDAAAQBAJ&pg=PA91#v=onepage&q&f=false|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 you navigate the world and tells you 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|accessdate=21 January 2018}}</ref>
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)]].
<!-- Do not add examples without sources. Also do not add examples that use the word "tesseract" but are not about hypercubes. Last bullet directs readers to the page that will help them find other, non-hypercube per this article, links. -->

==See also==
* [[Mathematics and art]]

==Notes==
{{Reflist}}

== References ==
* {{Cite book | ref=harv | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | isbn= | title-link=Regular Polytopes (book) | pp=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]
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass (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.
* [[T. Proctor Hall]] (1893) [https://www.jstor.org/stable/2369565 "The projection of fourfold figures on a three-flat"], [[American Journal of Mathematics]] 15:179–89.
* [[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 ==
* {{MathWorld|title=Tesseract|urlname=Tesseract}}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x4o3o3o - tes}}
* [http://eusebeia.dyndns.org/4d/8-cell.html The Tesseract] Ray traced images with hidden surface elimination. This site provides a good description of methods of visualizing 4D solids.
* [http://www.polytope.de/c8.html Der 8-Zeller (8-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
* [https://web.archive.org/web/20070928165237/http://www.polychora.de/wiki/index.php?title=TES WikiChoron: Tesseract]
* [http://uoregon.edu/~koch/hypersolids/hypersolids.html HyperSolids] is an open source program for the [[Apple Macintosh]] (Mac OS X and higher) which generates the five regular solids of three-dimensional space and the six regular hypersolids of four-dimensional space.
* [https://web.archive.org/web/20130326090312/http://www.cs.sjsu.edu/~rucker/hypercube.htm Hypercube 98] A [[Microsoft Windows|Windows]] program that displays animated hypercubes, by [[Rudy Rucker]]
* [http://mrl.nyu.edu/~perlin/demox/Hyper.html ken perlin's home page] A way to visualize hypercubes, by [[Ken Perlin]]
* [http://www.math.union.edu/~dpvc/math/4D/ Some Notes on the Fourth Dimension] includes very good 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]

{{Regular 4-polytopes}}
{{Polytopes}}

[[Category:Algebraic topology]]
[[Category:Four-dimensional geometry]]
[[Category:Polychora| 008]]