Simplices |

**Regular or Standard n-Simplex**

*

**Metric n-Simplex** > s.a. Tetrahedron.

*

\[ 16\,A^2 = (a+b+c)\, (a+b-c)\, (a-b+c)\, (-a+b+c) = -\left| \matrix{0 & a^2 & b^2 & 1 \cr

a^2 & 0 & c^2 & 1 \cr b^2 & c^2 & 0 & 1 \cr
1 & 1 & 1 & 0} \right|. \]

* __Tetrahedron__: If the edge
lengths are *a*_{ij}, the volume
is given by the Cayley determinant

\[ V^2 = {1\over288} \left|\matrix{0 & a_{12}^2 & a_{23}^2 & a_{31}^2 & 1 \cr

a_{12}^2 & 0 & a_{24}^2 & a_{41}^2 & 1 \cr a_{23}^2 & a_{24}^2 & 0 & a_{42}^2 & 1 \cr

a_{31}^2 & a_{41}^2 & a_{42}^2 & 0 & 1 \cr 1 & 1 & 1 & 1 & 0 }\right|. \]

One gets *V*^{2} > 0 if |*a*_{ij}
− *a*_{ik}| < *a*_{jk}
< *a*_{ij} + *a*_{ik},
for each face.

@ __References__: Luo m.GT/04 [volume of spherical and hyperbolic simplices];
Conway 13;
Anderson a1712 [two new versions of Heron's formula].

**Geometric n-Simplex**

$

*σ* =
{*x* ∈ \(\mathbb R\)^{N}
| ∃ {*t*_{i}}_{i
= 0, ..., n} , *t*_{i} ≥ 0,
∑_{i}* t*_{i}
= 1, such that *x* = ∑_{i}
*t*_{i} *a*_{i}} .

* __Properties__: The simplex
*σ* is a compact, convex set, intersection of all the convex sets
containing the vertices *a*_{i}.

* __Barycentric coordinates__: The
numbers *t*_{i}, *i* = 1, ...,
*n*, satisfying *t*_{i} > 0 and
∑_{i} *t*_{i}
= 1, such that we can write the *n*-simplex as *σ* =
{*x* | *x* = ∑_{i}
*t*_{i} *a*_{i}},
for some set of independent points *a*_{i}.

@ __General references__: Mäkelä a1011
[as a variable replacing the metric for Riemannian manifolds].

@ __Lorentzian case__: Tate & Visser JHEP(12)-a1110
[realizability conditions for a set of edge lengths].

> __Online resources__:
see Wikipedia page.

**Singular n-Simplex**

$

**Triangle** > s.a. Triangulations.

* __Incenter__: The location
of the incenter of a triangle with vertices at **P**,
**Q** and **R** is

**I** = (|*QR*| **P** + |*RP*| **Q**
+ |*PQ*| **R** )/(|*QR*| + |*RP*| + |*PQ*|) .

* __Angles__: Use the law of
cosines to find an internal angle if the three side lengths are known.

* __Pseudo-triangle__:
A simple polygon with exactly three convex vertices;
> s.a. Triangulations.

**Other Concepts**

@ __Physics-related__:
Bendjoudi & Mebarki ChPL(16)-a1610 [quantum triangle];
> s.a. gravity theories; lattice gravity;
regge calculus; spin-foam models [quantum tetrahedra].

> __Related to individual simplices__:
see join; Polytope.

> __Related to sets of simplices__:
see cell complex [simplicial]; curvature;
principal fiber bundle [with simplicial base space]; tiling.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 27 apr 2019