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orthoplex

The orthoplex, commonly called the cross-polytope, is a regular convex polytope that is dual to the hypercube. In n dimensions it is the regular n-orthoplex. It can be constructed as the convex hull of the 2n points ±e1, ..., ±en, where ei are the standard basis vectors in R^n. Its vertices lie on the coordinate axes, at distance 1 from the origin.

The orthoplex has 2n vertices, 2n(n−1) edges, and 2^n facets, each of which is an (n−1)-simplex. It

In relation to other polytopes, the orthoplex is dual to the hypercube, whose vertices are at all

Specific cases illustrate its form: in two dimensions the orthoplex is a diamond, equivalently a square rotated

is
a
regular
polytope
for
all
n
≥
2,
with
symmetry
group
the
hyperoctahedral
group
of
order
2^n
n!.
sign
combinations
of
{0,1}
coordinates;
conversely,
facets
of
the
cross-polytope
correspond
to
vertices
of
the
hypercube.
It
is
also
the
unit
ball
of
the
L1
norm:
{x
∈
R^n
:
sum_i
|x_i|
≤
1};
its
circumscribed
radius
is
1
and
its
inradius
is
1/√n.
45
degrees.
In
three
dimensions
it
is
the
octahedron;
in
four
dimensions
it
is
the
16-cell.
The
cross-polytope
appears
in
various
areas
of
geometry,
optimization,
and
computational
geometry
as
a
canonical
dual
to
the
hypercube
and
as
a
representative
of
L1
geometry.