Difference between revisions of "Simplex"
From Encyclopedia of Mathematics
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| + | An $ n $- | ||
| + | dimensional polytope (cf. [[Polyhedron|Polyhedron]]) that is the convex hull of $ n + 1 $ | ||
| + | points (the vertices of the simplex) which do not lie in any $ ( n - 1) $- | ||
| + | dimensional plane. When $ n = 0, 1, 2, $ | ||
| + | or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce [[Barycentric coordinates|barycentric coordinates]] in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see [[Choquet simplex|Choquet simplex]]; [[Simplex (abstract)|Simplex (abstract)]]). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its $ ( n - 1) $- | ||
| + | dimensional faces. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Grünbaum, "Convex polytopes" , Wiley (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. McMullen, G.C. Shephard, "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press (1971)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Grünbaum, "Convex polytopes" , Wiley (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. McMullen, G.C. Shephard, "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press (1971)</TD></TR></table> | ||
Latest revision as of 08:14, 6 June 2020
An $ n $-
dimensional polytope (cf. Polyhedron) that is the convex hull of $ n + 1 $
points (the vertices of the simplex) which do not lie in any $ ( n - 1) $-
dimensional plane. When $ n = 0, 1, 2, $
or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce barycentric coordinates in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see Choquet simplex; Simplex (abstract)). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its $ ( n - 1) $-
dimensional faces.
Comments
References
| [a1] | B. Grünbaum, "Convex polytopes" , Wiley (1967) |
| [a2] | P. McMullen, G.C. Shephard, "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press (1971) |
How to Cite This Entry:
Simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplex&oldid=14558
Simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplex&oldid=14558
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article