Editing 5-cell (section)

== Properties ==
The 5-cell is the 4-dimensional [[simplex]], the simplest possible [[4-polytope]]. In other words, the 5-cell is a [[polychoron]] analogous to a [[tetrahedron]] in high dimension.{{sfn|Miyazaki|Ishii|2021|p=[https://books.google.com/books?id=d2VXEAAAQBAJ&pg=PA46 46]}} It is formed by any five points which are not all in the same [[hyperplane]] (as a tetrahedron is formed by any four points which are not all in the same plane, and a [[triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[Polytope compound|compound]] of 120 regular 5-cells.

The 5-cell is [[Self-dual polytope|self-dual]], meaning its dual polytope is 5-cell itself.{{sfn|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA41 41]}} Its maximal intersection with 3-dimensional space is the [[triangular prism]]. Its dichoral angle is <math display=inline> \arccos(-1/4) \approx 75.52^\circ </math>.{{sfn|Akiyama|Hitotumatu|Sato|2012}}

It is the first in the sequence of 6 convex regular 4-polytopes, in order of volume at a given radius or number of vertexes.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions}}

The convex hull of two 5-cells in dual configuration is the [[Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]].

=== As a configuration ===
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref>
[[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]]
{| class=wikitable
|-
!Element||''k''-face||f<sub>''k''</sub>
!f<sub>0</sub>
!f<sub>1</sub>
!f<sub>2</sub>
!f<sub>3</sub>
!''k''-figs
|- align=right
|align=left bgcolor=#ffffe0 |<!--. . . .-->{{CDD|node_x|2|node_x|2|node_x|2|node_x}}||( )
|rowspan=1|f<sub>0</sub>
|bgcolor=#e0ffe0|5
|  bgcolor=#e0e0e0|4
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|4
|[[Tetrahedron|{3,3}]]
|- align=right
|align=left bgcolor=#ffffe0 |<!--x . . .-->{{CDD|node_1|2|node_x|2|node_x|2|node_x}}||{ }
|rowspan=1|f<sub>1</sub>
|  bgcolor=#e0e0e0|2
|bgcolor=#e0ffe0|10
|  bgcolor=#e0e0e0|3
|  bgcolor=#ffffff|3
|[[Triangle|{3}]]
|- align=right
|align=left bgcolor=#ffffe0 |<!--x3o . .-->{{CDD|node_1|3|node|2|node_x|2|node_x}}||[[triangle|{3}]]
|rowspan=1|f<sub>2</sub>
|  bgcolor=#ffffff|3
|  bgcolor=#e0e0e0|3
|bgcolor=#e0ffe0|10
|  bgcolor=#e0e0e0|2
|{ }
|- align=right
|align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{CDD|node_1|3|node|3|node|2|node_x}}||[[Tetrahedron|{3,3}]]
|rowspan=1|f<sub>3</sub>
|  bgcolor=#e0e0e0|4
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|4
|bgcolor=#e0ffe0|5
|( )
|}
All these elements of the 5-cell are enumerated in [[Branko Grünbaum]]'s [[Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane.



=== Geodesics and rotations ===
[[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[SO(4)#Double rotations|double rotation]].]]
The 5-cell has only [[digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no vertices of the 5-cell.

Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[Clifford torus]] is depicted in its rectangular (wrapping) form.

<gallery caption="[[Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]">
File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane
File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane
File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio
File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation
File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation
</gallery>

===Projections===
[[Image:Stereographic polytope 5cell.png|240px|thumb|[[Stereographic projection]] wireframe (edge projected onto a [[3-sphere]])]]

The A<sub>4</sub> Coxeter plane projects the 5-cell into a regular [[pentagon]] and [[pentagram]]. The A<sub>3</sub> Coxeter plane projection of the 5-cell is that of a [[square pyramid]]. The A<sub>2</sub> Coxeter plane projection of the regular 5-cell is that of a [[triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered.

{{4-simplex Coxeter plane graphs|t0|150}}

{|class="wikitable" width=640
!colspan=2|Projections to 3 dimensions
|- valign=top align=center
|[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.
|[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope.
|- valign=top align=center
|[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.
|[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.
|}