title: Dodecahedral cupola
text: # Dodecahedral cupola
  
  Dodecahedral cupola :
  
  | Schlegel diagram |||
  | - | - | - |
  | Type | Polyhedral cupola | Polyhedral cupola |
  | Schläfli symbol | {5,3} v rr{5,3} | {5,3} v rr{5,3} |
  | Cells | 64 | 1 rr{5,3} 1 {5,3} 30×{3} 12×{5} 20 {3,3} |
  | Faces | 194 | 80 triangles 90 squares 24 pentagons |
  | Edges | 210 | 210 |
  | Vertices | 80 | 80 |
  | Dual | | |
  | Symmetry group | [5,3,1], order 120 | [5,3,1], order 120 |
  | Properties | convex, regular-faced | convex, regular-faced |
  
  In 4-dimensional geometry, the dodecahedral cupola is a polychoron bounded by a rhombicosidodecahedron, a parallel dodecahedron, connected by 30 triangular prisms, 12 pentagonal prisms, and 20 tetrahedra.
  
  ## Related polytopes
  
  The dodecahedral cupola can be sliced off from a runcinated 120-cell, on a hyperplane parallel to a dodecahedral cell. The cupola can be seen in a pentagonal centered orthogonal projection of the runcinated 120-cell:
  
  | Runcinated 120-cell | Dodecahedron (cupola top) |
  | - | - |
  | Runcinated 120-cell | Rhombicosidodecahedron (cupola base) |
  