Simplex (EntityClass, 14)

From Higher Dimensions Database

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A simplex is an n-dimensional polytope with n+1 facets and n+1 vertices. Each simplex's element counts, treated as a list and including a single "-1D element" and a single element of its own dimension, read identical to a row of Pascal's triangle.

Simplices are special because they are always convex and are never self-intersecting. Any polytope can be defined as the union of a set of simplices of its own dimension, or as the space bounded by the union of a set of simplices of one dimension less. For these reasons, simplices are often used as "building blocks" in CGI.

Regular simplices are all also self-duals. The only other regular polytope to exhibit this behavior, bar 2D shapes, is the icositetrachoron in 4D.

Under the elemental naming scheme, simplices are denoted by the pyro- prefix, meaning the classical element of "fire".

Hypervolume

Without loss of generality, if one vertex of an n-simplex is chosen to be the origin, and the coordinates of the other n vertices are written as columns of an n×n matrix, then the n-hypervolume of the simplex is the determinant of the matrix divided by n factorial (see Wikipedia article).

Combined with the ability to represent any polytope as a union of simplices mentioned above, this could be used to write an algorithm to determine the hypervolumes of any polytope given its coordinates.


Simplices
triangletetrahedronpyrochoronpyroteronpyropeton

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