title: Pentadecahedron
text: # Pentadecahedron
  
  Some pentadecahedrons :
  
  | Dual elongated triangular cupola | Elongated pentagonal dipyramid |
  | - | - |
  | Tridecagonal prism | Elongated heptagonal pyramid |
  
  A pentadecahedron (or pentakaidecahedron) is a polyhedron with 15 faces. No pentadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a pentadecahedron, for example the tetradecagonal pyramid, and tridecagonal prism. In the pentadecahedron, none of the shapes are regular polyhedra, in other words, the regular pentadecahedron does not exist, and the pentadecahedron cannot fill the space, or, a space-filling pentadecahedron does not exist.
  
  In chemistry, some clusters of atoms are in the form of pentadecahedrons. Calculations have shown that there is a unit cell of the pentadecahedron that is stable in the crystal.
  
  ## Convex
  
  There are 23,833,988,129 topologically distinct convex pentadecahedra, excluding mirror images, having at least 10 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
  
  ## Common pentadecahedrons
  
  | Name | Type | Symbol | Vertices | Sides | Faces | χ | Face type | Symmetry |
  | - | - | - | - | - | - | - | - | - |
  | Tridecagonal prism | prism | t{2,13} {13}x | 26 | 39 | 15 | 2 | 2 tridecagons 13 rectangles | $D_{13h},$ [13,2], (*13 2 2) |
  | Tetradecagonal pyramid | pyramid | ( )∨{14} | 15 | 28 | 15 | 2 | 1 tetradecagon 14 triangles | $C_{14v},$ [14], (*14 14) |
  | Elongated heptagonal pyramid | pyramid | | 15 | 28 | 15 | 2 | 7 triangles 7 rectangles 1 heptagon | $D_{7h},$ [7,2], (*227), order 28 |
  | Heptagonal truncated cone | truncated cone | | 15 | 28 | 15 | 2 | 7 triangles 7 kites 1 heptagon | $D_{7h},$ [7,2], (*227), order 28 |
  | Elongated pentagonal bipyramid | Bipyramid Johnson solid | | 12 | 25 | 15 | 2 | 10 triangles 5 squares | $D_{5h},$ [5,2], (*225) |
  