Pentadecahedron

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Polyhedron with 15 faces

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.^[1]

In chemistry, some clusters of atoms are in the form of pentadecahedrons.^[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 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)

References

^ Parker), 麥特‧帕克(Matt (2020-06-11). 數學大觀念2：從掐指一算到穿越四次元的數學魔術 (in Chinese (Taiwan)). 貓頭鷹. ISBN 978-986-262-426-5.
^ Montejano, JM and Rodríguez, JL and Gutierrez-Wing, C and Miki, M and José-Yacamán, M (2004). "Crystallography and Shape of Nanoparticles and Clusters" (PDF). Encyclopedia of Nanoscience and Nanotechnology X: 1–44. Archived from the original (PDF) on 2022-08-28. Retrieved 2023-06-26.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Lagunov, VA and Sinani, AB (1998). "Formation of a bistructure of a solid in a computer experiment". Physics of the Solid State. 40 (10). Springer: 1742–1747. doi:10.1134/1.1130648. S2CID 121047989.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Counting polyhedra