Simplex (EntityClass, 14)
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| - | A '''simplex''' is an n-dimensional [[ | + | <[#ontology [kind class] [cats Regular Polytope Tapertope Pyramid]]> |
| + | 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 [[wikipedia:Pascal's triangle|Pascal's triangle]]. | ||
| - | Simplices are special because they are always [[convex]] and are never [[self-intersecting]]. Any polytope can | + | 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-[[dual]]s. 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 == |
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| + | 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 ([[wikipedia:Simplex#Volume|see Wikipedia article]]). | ||
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| + | 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. | ||
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| + | {{Simplices| }} | ||
Latest revision as of 14:08, 15 March 2014
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 |
| triangle • tetrahedron • pyrochoron • pyroteron • pyropeton |
