Hexateron

The hexateron, hix or triangular disphenoid, also commonly called the 5-simplex, is the simplest possible non-degenerate polyteron. The full symmetry version has 6 regular pentachora as cells, joining 5 to a vertex, and is one of the 3 regular polytera. It is the 5-dimensional simplex.

Hexateron
(OFF file)
Rank	5
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Hix
Coxeter diagram	x3o3o3o3o ()
Schläfli symbol	{3,3,3,3}
Tapertopic notation	14
Elements
Tera	6 pentachora
Cells	15 tetrahedra
Faces	20 triangles
Edges	15
Vertices	6
Vertex figure	Pentachoron, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{15}}{6} ≈ 0.64550$
Edge radius	$\frac{\sqrt6}{6} ≈ 0.40825$
Face radius	$\frac{\sqrt3}{6} ≈ 0.28868$
Cell radius	$\frac{\sqrt6}{12} ≈ 0.20412$
Inradius	$\frac{\sqrt{15}}{30} ≈ 0.12910$
Hypervolume	$\frac{\sqrt3}{480} ≈ 0.0036084$
Diteral angle	$\arccos\left(\frac15\right) ≈ 78.46301^\circ$
Heights	Point atop pen: $\frac{\sqrt{15}}{5} ≈ 0.77460$
	Dyad atop perp tet: $\frac{\sqrt6}{4} ≈ 0.61237$
	Trig atop perp trig: $\frac{\sqrt3}{3} ≈ 0.57735$
Central density	1
Number of external pieces	6
Level of complexity	1
Related polytopes
Army	Hix
Regiment	Hix
Dual	Hexateron
Conjugate	None
Abstract & topological properties
Flag count	720
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

It can be viewed as a segmentoteron in three ways: as a pentachoric pyramid, as a dyad atop perpendicular tetrahedron, and as a triangle atop perpendicular triangle. This makes it the triangular member of an infinite family of isogonal polygonal disphenoids.

Vertex coordinates

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

• $\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$
• $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$
• $\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$
• $\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30}\right),$
• $\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6}\right).$

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

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

Representations

A regular hexateron has the following Coxeter diagrams:

• x3o3o3o3o (full symmetry)
• ox3oo3oo3oo&#x (A4 axial, pentachoric pyramid)
• xo ox3oo3oo&#x (A3×A1 symmetry, tetrahedral scalene)
• xo3oo ox3oo&#x (A2×A2 axial, triangular disphenoid)
• oxo3ooo3ooo&#x (A3 symmetry, tetrahedral pyramidal pyramid)
• oxo oox3ooo&#x A2×A1 symmetry, triangular scalene pyramid)
• xoo oxo oox&#x (A1×A1×A1 symmetry, digonal trisphenoid)
• ooox ooxo&#x (A1×A1 symmetry, disphenoidal pyramidal pyramid)
• ooox3oooo&#x (A2 symmetry, triangular symmetry only)
• oooox&#x (A1 symmetrry only)
• oooooo&#x (no symmetry, fully irregular)

Variations

The regular hexateron has 2 subsymmetrical forms that remain isogonal:

• Triangular disphenoid - triangle atop an orthogonal triangle, facets and vertex figures are triangular scalenes
• Digonal trisphenoid - Cells and vertex figures are disphenoidal pyramids

External links

• Bowers, Jonathan. "Category 1: Primary Polytera" (#1).

• Klitzing, Richard. "hix".

• Wikipedia Contributors. "5-simplex".
• Hi.gher.Space Wiki Contributors. "Pyroteron".

• Hartley, Michael. "{3,3,3,3}*720".