Simplex
A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to
dimensions. The
boundary of a
-simplex has
0-faces (polytope vertices),
1-faces
(polytope edges), and
-faces, where
is a binomial
coefficient. An
-dimensional simplex can be denoted using
the Schläfli symbol
.
The simplex is so-named because it represents the simplest possible polytope in any
given space.
The content (i.e., hypervolume) of a simplex can be computed using the Cayley-Menger determinant.
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In one dimension, the simplex is the line segment
. In two dimensions, the simplex
is the convex
hull of the equilateral triangle. In
three dimensions, the simplex
is the convex
hull of the tetrahedron. The simplex in four
dimensions (the pentatope) is a regular tetrahedron
in which a point
along the fourth
dimension through the center of
is chosen so
that
. The regular simplex in
dimensions with
is denoted
. If
,
, ...,
are
points in
such that
, ...,
are linearly independent,
then the convex hull of these points is an
-simplex.
The above figures show the graphs for the
-simplexes with
to 7. Note that the graph of an
-simplex is the
complete graph of
vertices.
The
-simplex has graph
spectrum
(Cvetkovic et al. 1998,
p. 72; Buekenhout and Parker 1998).



simplex


