Pentadecahedron

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, a regular pentadecahedron does not exist, and the pentadecahedron cannot fill space; a space-filling pentadecahedron does not exist.^[1]

Examples of pentadecahedra
Dual elongated triangular cupola	Elongated pentagonal bipyramid
Tridecagonal prism	Elongated heptagonal pyramid

In chemistry, some clusters of atoms are in the form of pentadecahedra.^[2] Calculations have shown that there is a unit cell of the pentadecahedron that is stable in the crystal.^[3]

Convex

There are 23,833,988,129 topologically distinct convex pentadecahedra, excluding mirror images, having at least 10 vertices.^[4] (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 pentadecahedra

Name	Type	Image	Symbol	Vertices	Sides	Faces	χ	Face type	Symmetry
Tridecagonal prism	prism		t{2,13} {13}x{}	26	39	15	2	2 tridecagons 13 rectangles	D13h, [13,2], (*13 2 2)
Tetradecagonal pyramid	pyramid		( )∨{14}	15	28	15	2	1 tetradecagon 14 triangles	C14v, [14], (*14 14)
Elongated heptagonal pyramid	pyramid			15	28	15	2	7 triangles 7 rectangles 1 heptagon	D7h, [7,2], (*227), order 28
Heptagonal truncated cone	truncated cone			15	28	15	2	7 triangles 7 kites 1 heptagon	D7h, [7,2], (*227), order 28
Elongated pentagonal bipyramid	Bipyramid Johnson solid			12	25	15	2	10 triangles 5 squares	D5h, [5,2], (*225)