Simplices

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

Metric n-Simplex > s.a. Tetrahedron.
* Triangle: If the triangle is isosceles, the area is A = (l² sin )/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

* Tetrahedron: If the edge lengths are a_ij, the volume is given by the Cayley determinant

One gets V² > 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].

Geometric n-Simplex
$ Def: Given n + 1 independent points a₀, ..., a_n R^N, an n-simplex is

= {x 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 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 = {x | x = _i t_i a_i}, for some set of independent points a_i.

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
> Related to individual simplices: see join; Polytope; quantum tetrahedra in canonical quantum gravity.
> Related to sets of simplices: see cell complex [simplicial]; curvature; principal fiber bundle [with simplicial base space]; tiling.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 21 jun 2008