Simplex (EntityClass, 14)
From Hi.gher. Space
m |
|||
| Line 1: | Line 1: | ||
| - | A '''simplex''' is an n-dimensional [[polytope]] with n+1 faces and n+1 vertices. | + | A '''simplex''' is an n-dimensional [[polytope]] with n+1 faces 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 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]]. | 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. | [[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 [[Tamfang naming scheme]], simplices are denoted by the ''pyro-'' prefix, meaning the classical element of "fire". | ||
{{Simplices}} | {{Simplices}} | ||
Revision as of 22:42, 29 May 2010
A simplex is an n-dimensional polytope with n+1 faces 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 Tamfang naming scheme, simplices are denoted by the pyro- prefix, meaning the classical element of "fire".
| Simplices |
| triangle • tetrahedron • pyrochoron • pyroteron • pyropeton |
