Pentachoron

Pentachoron
(OFF file)
Rank	4
Type	Regular
Notation
Bowers style acronym	Pen
Coxeter diagram	x3o3o3o ()
Schläfli symbol	{3,3,3}
Tapertopic notation	13
Elements
Cells	5 tetrahedra
Faces	10 triangles
Edges	10
Vertices	5
Vertex figure	Tetrahedron, edge length 1
Edge figure	tet 3 tet 3 tet 3
Petrie polygons	12 pentagonal-pentagrammic coils
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{5}}\approx 0.63246$
Edge radius	${\frac {\sqrt {15}}{10}}\approx 0.38730$
Face radius	${\frac {\sqrt {15}}{15}}\approx 0.25820$
Inradius	${\frac {\sqrt {10}}{20}}\approx 0.15811$
Hypervolume	${\frac {\sqrt {5}}{96}}\approx 0.023292$
Dichoral angle	$\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
Heights	Point atop tet: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
	Dyad atop perp trig: ${\frac {\sqrt {15}}{6}}\approx 0.64550$
Central density	1
Number of external pieces	5
Level of complexity	1
Related polytopes
Army	Pen
Regiment	Pen
Dual	Pentachoron
Conjugate	None
Abstract & topological properties
Flag count	120
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	Yes
Nature	Tame

The pentachoron, or pen, also commonly called the 5-cell or the 4-simplex, is the simplest possible non-degenerate polychoron. The full symmetry version has 5 regular tetrahedra as cells, joining 3 to an edge and 4 to a vertex, and is one of the 6 convex regular polychora. It is the 4-dimensional simplex.

In addition, it can also be considered to be the regular-faced pyramid of the tetrahedron, or the pyramid product of a triangle and a dyad. This makes it the simplest segmentochoron as well, and it is designated K-4.1 in Richard Klitzing's list of convex segmentochora. It is also the 5-2 step prism and gyrochoron.

Gallery

• 

  Rotating pentachoron

• 

  Segmentochoron display, point atop tet

• 

  Net

• 

  Cross-sections

• 

  Cross-section animation

• 

  Wireframe, cell, net

Vertex coordinates

The vertices of a regular pentachoron of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right)$,
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right)$,
• $\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}}\right)$,
• $\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}}\right)$.

Much simpler coordinates can be given in five dimensions, as all permutations of:

• $\left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0\right)$.

A further set of coordinates, derived from the 5-2 step prism construction, is given by:

• $\left({\frac {1}{\sqrt {5}}},\,{\frac {1}{\sqrt {5}}},\,0,\,0\right)$,
• $\left({\frac {5-{\sqrt {5}}}{20}},\,{\frac {-5-{\sqrt {5}}}{20}},\,{\sqrt {{\frac {1}{8}}+{\frac {\sqrt {5}}{40}}}},\,{\sqrt {{\frac {1}{8}}-{\frac {\sqrt {5}}{40}}}}\right)$,

together with reflections through the $x=y$ and the $z=w$ hyperplanes.

Representations

A pentachoron has the following Coxeter diagrams:

• x3o3o3o () (full symmetry)
• ox3oo3oo&#x (A3 axial, as tetrahedral pyramid)
• xo ox3oo&#x (A2×A1 axial, as triangle-dyad disphenoid)
• oox3ooo&#x (A2 axial, as triangular scalene)
• oxo oox&#x (A1×A1 axial, as disphenoidal pyramid)
• ooox&#x (bilateral symmetry only)
• ooooo&#x (no symmetry)

Variations

Besides the regular pentachoron, various other types of pentachora exist, some of which occur in vertex figures. These include:

• Tetrahedral pyramid - 1 regular tetrahedron, 4 triangular pyramids, tetrahedral symmetry
• Triangular scalene - 2 triangular pyramids, 3 digonal disphenoids, triangular prismatic axial symmetry
• Triangular pyramidal pyramid - 2 different triangular pyramids, 3 sphenoids, triangular symmetry
• Tetragonal disphenoidal pyramid - 1 tetragonal disphenoid, 4 sphenoids, digonal antiprismatic axial symmetry
• Rhombic disphenoidal pyramid - 1 rhombic disphenoid, 4 irregular tetrahedra, chiral digonal prismatic axial symmetry
• Digonal disphenoidal pyramid - 1 digonal disphenoid, 2 pairs of sphenoids, digonal axial symmetry
• Sphenoidal pyramid - 2 identical irregular tetrahedra, 3 different sphenoids
• Phyllic disphenoidal pyramid - 1 phyllic dissphenoid, 2 pairs of identical irregular tetrahedra, bilateral symmetry
• Irregular pentachoron - no symmetry, 5 different irregular tetrahedra
• 5-2 step prism (or gyrochoron) noble, 5 phyllic disphenoids

Related polychora

Two pentachora can be attached at a common cell to form the tetrahedral tegum.

Uniform polychoron compounds composed of pentachora include:

• Stellated decachoron (2)
• Gyrosimplexifissal icosachoron (4)
• Chirosimplexifissal icosipentachoron (5)
• Simplexifissial disicosipentachoron (10)
• Compound of 12 pentachora (12)
• Medial hexacosichoron (120)
• Compound of 720 pentachora (720)
• An infinite family of gyrochoron-symmetric compounds

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Tetrahedron (5): Pentachoron
• Triangle (10): Rectified pentachoron
• Edge (10): Rectified pentachoron

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#1).

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Bowers, Jonathan. "Pennic and Decaic Isogonals".

• Klitzing, Richard. "Pen".

• Quickfur. "The Pentachoron".

• Wikipedia contributors. "5-cell".
• Hartley, Michael. "{3,3,3}*120".