10-orthoplex

10-orthoplex
Rank10
TypeRegular
Notation
Bowers style acronymKa
Coxeter diagramx3o3o3o3o3o3o3o3o4o ()
Schläfli symbol{3,3,3,3,3,3,3,3,4}
Elements
Xenna1024 decayotta
Yotta5120 enneazetta
Zetta11520 octaexa
Exa15360 heptapeta
Peta13440 hexatera
Tera8064 pentachora
Cells3360 tetrahedra
Faces960 triangles
Edges180
Vertices20
Vertex figurePentacosidodecayotton, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {\sqrt {5}}{10}}\approx 0.22361}$
Hypervolume${\displaystyle {\frac {1}{113400}}\approx 0.0000088183}$
Dixennal angle${\displaystyle \arccos \left(-{\frac {4}{5}}\right)\approx 143.13010^{\circ }}$
Height${\displaystyle {\frac {\sqrt {5}}{5}}\approx 0.44721}$
Central density1
Number of external pieces1024
Level of complexity1
Related polytopes
ArmyKa
RegimentKa
DualDekeract
ConjugateNone
Abstract & topological properties
Flag count3715891200
Euler characteristic0
OrientableYes
Properties
SymmetryB10, order 3715891200
Flag orbits1
ConvexYes
Net count26941775019280
NatureTame

The 10-orthoplex, also called the decacross, chiliaicositetraxennon, or ka, is a regular 10-polytope. It has 1024 regular 9-simplices as facets, joining 4 to an 7-simplex peak and 512 to a vertex in a 9-orthoplecial arrangement. It is the 10-dimensional orthoplex. It is also a 5-orthoplex duotegum and square pentategum.

Vertex coordinates

The vertices of a regular chiliaicositetraxennon of edge length 1, centered at the origin, are given by all permutations of:

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