View source for Demicross

{{Infobox polytope|suppress_missing_measures=true|suppress_missing_off=true|suppress_missing_image=true
|name = {{mvar|n}}-demicross
|dim = {{mvar|n}}
|type = [[Uniform]]
|coxeter = {{CDD|label3-2|branch|3ab|branch|3b|nodeb|3b}}...{{CDD|3b|nodeb|3b|nodeb_1}}/2 (o3o3/2o3o3*a3o...o3x)
|vertices = 2{{mvar|n}}
|facets = {{mvar|n}} ({{mvar|n}} &minus; 1)-[[orthoplices]], <math>2^{n - 1}</math> ({{mvar|n}} &minus; 1)-[[simplices]]
|verf = {{mvar|n}}&minus;1-demicross
|circum = <math>\frac{\sqrt2}{2}</math>
|inrad = <math>\frac{1}{\sqrt{2n}}</math> (for simplex facets)
|surface = <math>\frac{\left(n+\sqrt{n}\right)\sqrt{2^{n-1}}}{\left(n-1\right)!}</math>
|height = <math>\frac{2}{\sqrt{2n}}</math>
|dih = <math>\arccos\left(\frac{1}{\sqrt{n}}\right)</math>
|conjugate = None
|flags = <math>\left(n-1\right)!\times2^{n-1}\left(n+1\right)</math>
|orientable = No
|symmetry = D<sub>{{mvar|n}}</sub>, order <math>n!\times2^{n-1}</math>
|conv = No
|hull = {{mvar|n}}-[[orthoplex]]
|nat = Tame
}}
The '''demicrosses''' are a series of [[self-intersecting]] [[uniform]] [[polytope]]s that generalize the 3D [[tetrahemihexahedron]] to higher dimensions. They are [[semiregular]] (having only regular facets) and [[quasiregular]] (representable with [[Coxeter-Dynkin diagram]]s 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 [[facet]]ing of the ''n''-[[orthoplex]] (the result having lower symmetry), retaining elements of every rank up to and including its [[ridge]]s. 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 [[hemipolytope]]s.

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==
{| class="wikitable mw-collapsible" style="border: none; background: none;"
|+{{nowrap|Demicrosses by dimension}}
!Rank
!Name
|-
|3
|[[Tetrahemihexahedron]]
|-
|4
|[[Tesseractihemioctachoron]]
|-
|5
|[[Hexadecahemidecateron]]
|-
|6
|[[Triacontidihemidodecapeton]]
|-
|7
|[[Hexacontatetrahemitetradecaexon]]
|-
|8
|[[Hecatonicosoctahemihexadecaexon]]
|-
|9
|[[Diacosipentacontahexahemioctadecayotton]]
|}

==Vertex coordinates==
Coordinates for the vertices of an ''n''-demicross with edge length 1 are given by all permutations of:
*<math>\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,...,\,0\right)</math>,
where the last ''n''–1 entries are zeros.

==External links==
* Klitzing, Richard. [https://bendwavy.org/klitzing/explain/analog.htm#demicross "Demicross hO<sub>n</sub>"].

[[Category:Quasiregular polytopes]]
[[Category:2-orbit polytopes]]