Pentachoron

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Regular pentachoron (5-cell) (4-simplex)
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Vertices	5
Edges	10
Faces	10 {3}
Cell	5 (3.3.3)
Vertex figure	(tetrahedron)
Schläfli symbol	{3,3,3}
Coxeter-Dynkin diagram
Petrie polygon	pentagon
Coxeter group	A₄, [3,3,3]
Dual	Self-dual
Properties	convex
Uniform index	' 1 2

Vertex figure: tetrahedron

In geometry, the pentachoron is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the 5-cell, pentatope, or hyperpyramid. It is a 4-simplex, the simplest possible convex regular 4-polytope (four-dimensional analogue of a polyhedron), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions.

The regular pentachoron is bounded by regular tetrahedra, and is one of the six regular convex polychora, represented by Schläfli symbol {3,3,3}.

[edit] Geometry

The pentachoron is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism.

[edit] Construction

The pentachoron can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant with all the other vertices of the tetrahedron. (The pentachoron is essentially a 4-dimensional pyramid with a tetrahedral base.)

The Cartesian coordinates of the vertices of an origin-centered pentachoron having edge length 2 are:

$\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$

$\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)$

$\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)$

$\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)$

[edit] Projections

Projections to 2 dimensions
	One of the possible projections of the pentachoron into 2 dimensions is the pentagram inscribed inside a pentagon, as seen here in its orthogonal projection inside its Petrie polygon.
	Four orthographic projections, showing various viewpoints of the pentatope.

Projections to 3 dimensions
	Stereographic projection wireframe (edge projected onto a 3-sphere).
	A 3D projection of a 5-cell performing a double rotation about two orthogonal planes.
	The vertex-first projection of the pentachoron into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the pentachoron 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.
	The edge-first projection of the pentachoron into 3 dimensions has a triangular dipyramidal 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.
	The face-first projection of the pentachoron 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 projects 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.
	The cell-first projection of the pentachoron 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.

[edit] Alternative names

5-cell
4-simplex
Pentatope
Pentahedroid (Henry Parker Manning)
Pen (Jonathan Bowers: for pentachoron)
Hyperpyramid

[edit] Related polychora

The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group:

Name	Coxeter-Dynkin and Schläfli symbols	Cells	Faces	Edges	Vertices
5-cell	{3,3,3}	5	10	10	5
truncated 5-cell	t_0,1{3,3,3}	10	30	40	20
rectified 5-cell	t₁{3,3,3}	10	30	30	10
cantellated 5-cell	t_0,2{3,3,3}	20	80	90	30
cantitruncated 5-cell	t_0,1,2{3,3,3}	20	80	120	60
runcitruncated 5-cell	t_0,1,3{3,3,3}	30	120	150	60
bitruncated 5-cell	t_1,2{3,3,3}	10	40	60	30
runcinated 5-cell	t_0,3{3,3,3}	30	70	60	20
omnitruncated 5-cell	t_0,1,2,3{3,3,3}	30	150	240	120