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

o3o3o3o truncations

Name	OBSA	CD diagram	Picture
Pentachoron	pen	x3o3o3o ()
Truncated pentachoron	tip	x3x3o3o ()
Rectified pentachoron	rap	o3x3o3o ()
Decachoron	deca	o3x3x3o ()
Rectified pentachoron	rap	o3o3x3o ()
Truncated pentachoron	tip	o3o3x3x ()
Pentachoron	pen	o3o3o3x ()
Small rhombated pentachoron	srip	x3o3x3o ()
Great rhombated pentachoron	grip	x3x3x3o ()
Small rhombated pentachoron	srip	o3x3o3x ()
Great rhombated pentachoron	grip	o3x3x3x ()
Small prismatodecachoron	spid	x3o3o3x ()
Prismatorhombated pentachoron	prip	x3x3o3x ()
Prismatorhombated pentachoron	prip	x3o3x3x ()
Great prismatodecachoron	gippid	x3x3x3x ()

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".