Editing Tesseract (section)

== 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>