Demicross

n -demicross
Rankn
TypeUniform
Notation
Coxeter diagram.../2 (o3o3/2o3o3*a3o...o3x)
Elements
Facetsn  (n  − 1)-orthoplices, ${\displaystyle 2^{n-1}}$ (n  − 1)-simplices
Vertices2n
Vertex figuren −1-demicross
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}}$
Inradius${\displaystyle {\frac {1}{\sqrt {2n}}}}$ (for simplex facets)
Surface area${\displaystyle {\frac {\left(n+{\sqrt {n}}\right){\sqrt {2^{n-1}}}}{\left(n-1\right)!}}}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{\sqrt {n}}}\right)}$
Height${\displaystyle {\frac {2}{\sqrt {2n}}}}$
Related polytopes
ConjugateNone
Convex hulln -orthoplex
Abstract & topological properties
Flag count${\displaystyle \left(n-1\right)!\times 2^{n-1}\left(n+1\right)}$
OrientableNo
Properties
SymmetryDn , order ${\displaystyle n!\times 2^{n-1}}$
ConvexNo
NatureTame

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.

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.

Examples

Demicrosses by dimension
Rank Name
3 Tetrahemihexahedron
4 Tesseractihemioctachoron
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,...,\,0\right)}$,