Editing Tesseract (section)

===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'').