Pentachoron

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Regular pentachoron (5-cell)
Schlegel diagram (vertices and edges)
Type	Regular polychoron
Family	simplex
Cells	5 (3.3.3)
Faces	10 {3}
Edges	10
Vertices	5
Vertex figure	(3.3.3)
Schläfli symbol	{3,3,3}
Coxeter-Dynkin diagram
Symmetry group	A4, [3,3,3]
Dual	self-dual
Properties	convex

The pentachoron, (also called a 5-cell, pentatope, or hyperpyramid) is a 4-simplex, the simplest convex polychoron (a type of four-dimensional geometric figure). It consists of 5 tetrahedral cells and is an analog of the triangle (2-simplex) and tetrahedron (3-simplex).

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

Contents 1 Geometry 2 Images 3 Construction 4 Projections 5 Alternate names 6 Related polychora 7 References 8 External links

[edit] Geometry

The pentachoron consists of five cells, all tetrahedra, and is self-dual. Its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism.

[edit] Images

Schlegel diagram wireframe projected onto a 3-sphere

A 3D projection of a 5-cell performing a double rotation about two orthogonal planes.

Four orthographic projections

[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. (Essentially, the pentachoron is a 4-dimensional pyramid with a tetrahedral base.)

[edit] Projections

One of the possible projections of the pentachoron into 2 dimensions is the pentagram inscribed inside a pentagon.

Both the vertex-first and cell-first parallel projection of the pentachoron into 3 dimensions have a tetrahedral projection_envelope. The closest or farthest vertex of the pentachoron, respectively, projects to the center of the tetrahedron. The farthest/closest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the center.

The edge-first and face-first projections of the pentachoron into 3 dimensions have a triangular dipyramidal envelope. Two of the cells project to the upper and lower halves of the dipyramid, while the remaining 3 project to 3 non-regular tetrahedral volumes arranged around the central axis of the dipyramid at 120 degrees to each other.

[edit] Alternate 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	Picture	Coxeter-Dynkin and Schläfli symbols	Cells	Faces	Edges	Vertices
5-cell		{3,3,3}	5	10	10	5
truncated 5-cell		t0,1{3,3,3}	10	30	40	20
rectified 5-cell		t1{3,3,3}	10	30	30	10
cantellated 5-cell		t0,2{3,3,3}	20	80	90	30
cantitruncated 5-cell		t0,1,2{3,3,3}	20	80	120	60
runcitruncated 5-cell		t0,1,3{3,3,3}	30	120	150	60
bitruncated 5-cell		t1,2{3,3,3}	10	40	60	30
runcinated 5-cell		t0,3{3,3,3}	30	70	60	20
omnitruncated 5-cell		t0,1,2,3{3,3,3}	30	150	240	120

[edit] References

• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.

[edit] External links

• Eric W. Weisstein, Pentatope at MathWorld.
• Olshevsky, George, Pentachoron at Glossary for Hyperspace.
  • (1) Convex uniform polychora based on the pentachoron
• Der 5-Zeller (5-cell) Marco Möller's Regular polytopes in R⁴ (German)
• Jonathan Bowers, Regular polychora

Convex regular 4-polytopes
pentachoron	tesseract	16-cell	24-cell	120-cell	600-cell
{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}