Simplices  

Regular or Standard n-Simplex
* Angle between adjacent faces: α = arccos 1/n.

Metric n-Simplex > s.a. Tetrahedron.
* Triangle: If the triangle is isosceles, the area is A = (l2 sin α)/2; In general, if the side lengths are a, b, c, the area is given by Heron's formula (from Hero of Alexandria), and can be written as

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

* Tetrahedron: If the edge lengths are aij, 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 V2 > 0 if |aijaik| < ajk < aij + aik, for each face.
@ References: Luo m.GT/04 [volume of spherical and hyperbolic simplices]; Conway 13.

Geometric n-Simplex
$ Def: Given n + 1 independent points a0, ..., an ∈ \(\mathbb R\)N, an n-simplex σ is

σ = {x ∈ \(\mathbb R\)N | ∃ {ti}i = 0, ..., n , ti ≥ 0, ∑i ti = 1, such that x = ∑i ti ai} .

* Properties: The simplex σ is a compact, convex set, intersection of all the convex sets containing the vertices ai.
* Barycentric coordinates: The numbers ti, i = 1, ..., n, satisfying ti > 0 and ∑i ti = 1, such that we can write the n-simplex as σ = {x | x = ∑i ti ai}, for some set of independent points ai.
@ 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
$ Def: A map from the standard n-simplex to a topological space X (need not be invertible).

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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