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-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.

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dec 2017