Regular pentacross (5orthoplex) 


Orthogonal projection inside Petrie polygon 

Type  Regular 5polytope 
Family  orthoplex 
Schläfli symbol  {3,3,3,4} {3^{2,1,1}} 
CoxeterDynkin diagrams  
Hypercells  32 {3^{3}} 
Cells  80 {3,3} 
Faces  80 {3} 
Edges  40 
Vertices  10 
Vertex figure  16cell 
Petrie polygon  decagon 
Coxeter groups  C_{5}, [3,3,3,4] D_{5}, [3^{2,1,1}] 
Dual  Penteract 
Properties  convex 
In fivedimensional geometry, a pentacross, also called a triacontakaiditeron, is a fivedimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 octahedron cells, 32 5cell hypercells.
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is the 5hypercube or penteract.
The name pentacross is derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
Contents 
There are two Coxeter groups associated with the pentacross, one regular, dual of the penteract with the C_{5} or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5cell facets, alternating, with the D_{5} or [3^{2,1,1}] Coxeter group.
Cartesian coordinates for the vertices of a pentacross, centered at the origin are
Precisely, the Perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of Pentacross. 10 sets of 4 edges forms 10 circles in the 4D Schlegel diagram: two of these circles are straight lines because contains the center of projection. 
Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
