Trismyriadischiliaheptacosihexacontoctatedakon

Trismyriadischiliaheptacosihexacontoctatedakon
Rank	15
Type	Regular
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o3o4o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Elements
Tedaka	32768 pentadecatradaka
Tradaka	245760 tetradecadoka
Doka	860160 tridecahenda
Henda	1863680 dodecadaka
Daka	2795520 hendecaxenna
Xenna	3075072 decayotta
Yotta	2562560 enneazetta
Zetta	1647360 octaexa
Exa	823680 heptapeta
Peta	320320 hexatera
Tera	96096 pentachora
Cells	21840 tetrahedra
Faces	3640 triangles
Edges	420
Vertices	30
Vertex figure	14-orthoplex, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {\sqrt {30}}{30}}\approx 0.18257$
Hypervolume	${\frac {\sqrt {2}}{10216206000}}\approx 1.3843\times 10^{-10}$
Dihedral angle	$\arccos \left(-{\frac {13}{15}}r\right)\approx 150.07357^{\circ }$
Height	${\frac {\sqrt {30}}{15}}\approx 0.36515$
Central density	1
Number of external pieces	32768
Level of complexity	1
Related polytopes
Army	*
Regiment	*
Dual	Pentadekeract
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B15, order 42849873690624000
Convex	Yes
Nature	Tame

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:

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