(Redirected from 15-orthoplex)
Rank15
TypeRegular
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o3o3o4o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Elements
Doka860160 tridecahenda
Daka2795520 hendecaxenna
Xenna3075072 decayotta
Yotta2562560 enneazetta
Zetta1647360 octaexa
Exa823680 heptapeta
Peta320320 hexatera
Tera96096 pentachora
Cells21840 tetrahedra
Faces3640 triangles
Edges420
Vertices30
Vertex figure14-orthoplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {\sqrt {30}}{30}}\approx 0.18257}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{10216206000}}\approx 1.3843\times 10^{-10}}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {13}{15}}r\right)\approx 150.07357^{\circ }}$
Height${\displaystyle {\frac {\sqrt {30}}{15}}\approx 0.36515}$
Central density1
Number of external pieces32768
Level of complexity1
Related polytopes
Army*
Regiment*
ConjugateNone
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB15, order 42849873690624000
ConvexYes
NatureTame

The trismyriadischiliaheptacosihexacontaoctatedakon, also called the pentadecacross or 15-orthoplex, is a regular polytedakon. It has 32768 regular pentadecatradaka as facets, joining 4 to a peak and 16384 to a vertex in a myriahexachiliatriacosioctacontatetratradakal arrangement. It is the 15-dimensional orthoplex. As such it is a triacontaditeron triotegum and octahedron pentategum.

## Vertex coordinates

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

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