title: Dodecahedral pyramid
text: # Dodecahedral pyramid
  
  Dodecahedral pyramid :
  
  | Schlegel diagram |||
  | - | - | - |
  | Type | Polyhedral pyramid | Polyhedral pyramid |
  | Schläfli symbol | ( ) ∨ {5,3} | ( ) ∨ {5,3} |
  | Cells | 13 | 1 {5,3} 12 ( ) ∨ {5} |
  | Faces | 42 | 30v{ } 12 {5} |
  | Edges | 50 | 50 |
  | Vertices | 21 | 21 |
  | Dual | icosahedral pyramid | icosahedral pyramid |
  | Symmetry group | $H_3,$ [5,3,1], order 120 | $H_3,$ [5,3,1], order 120 |
  | Properties | convex | convex |
  
  In 4-dimensional geometry, the dodecahedral pyramid is bounded by one dodecahedron on the base and 12 pentagonal pyramid cells which meet at the apex. Since a dodecahedron's circumradius is greater than its edge length, the pentagonal pyramids require tall isosceles triangle faces.
  
  The dual to the dodecahedral pyramid is an icosahedral pyramid, seen as an icosahedral base, and 20 regular tetrahedra meeting at an apex.
  
  