Demicross

n-demicross
Rank	n
Type	Uniform
Space	Spherical
Notation
Coxeter diagram	.../2 (o3o3/2o3o3*a3o...o3x)
Elements
Vertices	2n
Vertex figure	n−1-demicross
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2}$
Inradius	$\frac{1}{\sqrt{2n}}$ (for simplex facets)
Surface area	$\frac{\left(n+\sqrt{n}\right)\sqrt{2^{n-1}}}{\left(n-1\right)!}$
Dihedral angle	$\arccos\left(\frac{1}{\sqrt{n}}\right)$
Height	$\frac{2}{\sqrt{2n}}$
Related polytopes
Conjugate	None
Abstract & topological properties
Flag count	$\left(n-1\right)!\times2^{n-1}\left(n+1\right)$
Orientable	No
Properties
Symmetry	Dn, order $n!\times2^{n-1}$
Convex	No
Nature	Tame

The demicross series is a series of quasiregular, semiregular polytopes. A d-dimensional demicross shares all its vertices, edges, and all elements up to ridges (dimension d-2) with the n-dimensional orthoplex. However, it only uses half of the orthoplex's facets, along with d (d-1)-orthoplexes which pass through the center of the polytope, making all demicrosses hemipolytopes. Its vertex figure is the demicross of the previous dimension.

In all dimensions greater than or equal to 3, the demicross is uniform. A semi-uniform bowtie formed from faceting a square could be considered to be the 2D demicross.

Vertex coordinates

Coordinates for the vertices of an n-demicross with edge length 1 are given by all permutations of:

• $\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,...,\,0\right)$,

where the last n–1 entries are zeros.

External links

• Klitzing, Richard. "Demicross hO_n".