Cross Polytope
The cross polytope
is the regular polytope
in
dimensions corresponding to the convex
hull of the points formed by permuting the coordinates (
, 0, 0, ...,
0). A cross-polytope (also called an orthoplex) is denoted
and has
vertices and Schläfli
symbol
. The cross polytope
is named because its
vertices are located equidistant from
the origin along the Cartesian axes in Euclidean space,
which each such axis perpendicular to all others. A cross polytope is bounded by
-simplexes,
and is a dipyramid erected (in both directions) into the
th dimension, with
an
-dimensional cross polytope as its
base.
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In one dimension, the cross polytope is the line segment
. In two dimensions, the cross polytope
is the filled square
with vertices
,
,
,
. In three
dimensions, the cross polytope
is the convex
hull of the octahedron with vertices
,
,
,
,
,
. In four
dimensions, the cross polytope
is the 16-cell, depicted in the above figure by projecting onto
one of the four mutually perpendicular three-spaces within the four-space obtained
by dropping one of the four vertex components (R. Towle).
The skeleton of
is isomorphic
with the circulant graph
,
also known as the cocktail party graph
.
For all dimensions, the dual of the cross polytope is the hypercube (and vice versa). Consequently, the number of
-simplices contained
in an
-cross polytope is
.



cross polytope


