Simplex

A simplex (plural simplices or simplexes) is the simplest possible non-degenerate polytope in each respective dimension. The n-dimensional simplex, or simply n-simplex, consists of n+1 vertices, with each n of them joined in the unique manner by a simplex of the lower dimension. Alternatively, one may construct an n-simplex as the pyramid of the (n-1)-simplex. The pyramid product of an m-dimensional simplex and an n-dimensional simplex is an (m+n+1)-dimensional simplex. All possible realizations of simplexes are convex.

n-simplex
Rank	n
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3...3o (...)
Schläfli symbol	{3,3,3, ..., ,3}
Elements
Vertices	$n+1$
Vertex figure	(n-1)-simplex, edge length 1
Measures (edge length 1)
Circumradius	$\sqrt{\frac{n}{2n+2}}$
Inradius	$\frac{1}{\sqrt{2n\left(n+1\right)}}$
Volume	$\frac{\sqrt{(n+1)/2^n}}{n!}$
Height	$\sqrt{\frac{n+1}{2n}}$
Central density	1
Number of external pieces	$n+1$
Level of complexity	1
Related polytopes
Dual	n-simplex
Conjugate	None
Abstract & topological properties
Flag count	$n!$
Euler characteristic	0 if n even 2 if n odd
Orientable	Yes
Properties
Symmetry	An, order $n!$
Convex	Yes
Nature	Tame

A tetrahedron, with a triangle, a dyad, and a point outlined. These are all examples of simplexes.

Every simplex can be made regular. As such, the simplexes comprise one of the three infinite families of regular polytopes that exist in every dimension, the other two being the hypercubes and the orthoplexes.

In even dimensions greater than 2, simplexes can be thought of as step prisms, such as the pentachoron (5-2 step prism) and the heptapeton (7-2-3 step prism). In odd dimensions greater than 1, simplexes can be thought of as disphenoids of an $\tfrac{n-1}2$-dimensional simplex, such as the tetrahedron (digonal disphenoid) and the hexateron (triangular disphenoid).

Every simplex is self-dual.

Elements

All of the elements of a simplex are simplexes themselves. The number of d-dimensional elements in an n-dimensional simplex is given by the binomial coefficient $C(n+1, d+1)$. This is because any choice of d+1 vertices defines a unique d-dimensional simplex. In particular, an n-dimensional simplex has n+1 vertices and n+1 facets (each shaped like an (n-1)-dimensional simplex), and its vertex figure is also the simplex of the previous dimension.

In total, an n-simplex has 2ⁿ elements, including the nullitope and the bulk of the polytope.

Examples

Excluding the degenerate nullitope and the point, the simplexes up to 10D are the following:

Simplexes by dimension

Dimension	Name	Picture		Dimension	Name	Picture
1	Dyad			6	Heptapeton
2	Triangle			7	Octaexon
3	Tetrahedron			8	Enneazetton
4	Pentachoron			9	Decayotton
5	Hexateron			10	Hendecaxennon

Vertex coordinates

Coordinates for the vertices of an n-simplex of edge length 1, centered at the origin, are given by:

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

Alternatively, there are simpler coordinates but the simplex is not centered at the origin:

• all permutations of $\left(\frac{\sqrt{2}}{2},\,0,\,0,\,...,\,0\right)$
• $(a,\,a,\,a,\,...,\,a)$ where $a = \frac{\left(1-\sqrt{n+1}\right)\sqrt{2}}{2n}$.

Much simpler coordinates can be given in n+1 dimensions, as all permutations of

• $\left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,...,\,0\right)$.

Measures

• The circumradius of a regular n-dimensional simplex of unit edge length is given by $\sqrt{\frac{n}{2n+2}}$.
• Its inradius is given by $\frac{1}{\sqrt{2n(n+1)}}$.
• Its height from a vertex to the opposite facet is given by $\sqrt{\frac{n+1}{2n}}$.
• Its hypervolume is given by $\frac{\sqrt{\frac{n+1}{2^n}}}{n!}$.
• The angle between two facet hyperplanes is $\arccos\left({\frac{1}{n}}\right)$.

External links

• Wikipedia Contributors. "Simplex".