Demicross

The demicrosses are a series of self-intersecting uniform polytopes that generalize the 3D tetrahemihexahedron to higher dimensions. They are semiregular (having only regular facets) and quasiregular (representable with Coxeter-Dynkin diagrams with a single ringed node). The vertex figure of a demicross is the demicross of the previous dimension.

n -demicross
Rank	n
Type	Uniform
Notation
Coxeter diagram	.../2 (o3o3/2o3o3*a3o...o3x)
Elements
Facets	n  (n  − 1)-orthoplices, $2^{n-1}$ (n  − 1)-simplices
Vertices	2n
Vertex figure	n −1-demicross
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}$
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
Convex hull	n -orthoplex
Abstract & topological properties
Flag count	$\left(n-1\right)!\times 2^{n-1}\left(n+1\right)$
Orientable	No
Properties
Symmetry	Dn , order $n!\times 2^{n-1}$
Convex	No
Nature	Tame

The n-demicross is formed as a faceting of the n-orthoplex (the result having lower symmetry), retaining elements of every rank up to and including its ridges. Only the facets are changed: half of the (n - 1)-simplex facets are removed so that no two of them share a ridge, and n (n - 1)-orthoplices are added that pass through the center of the polytope. Thus, the demicrosses are hemipolytopes.

An isogonal bowtie formed from faceting a square could be considered to be the 2D demicross, but it isn't uniform so is conventionally excluded.

Vertex coordinates

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

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

where the last n–1 entries are zeros.

External links

• Klitzing, Richard. "Demicross hO_n".