Editing Tesseract (section)

==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]]
|}