Heptadecahedron

A heptadecahedron (or heptakaidecahedron) is a polyhedron with 17 faces. No heptadecahedron is regular; hence, the name is ambiguous. There are heptadecahedra which are nearly spherical, like those seen in some chemical structures,^[1][2] however their faces are not composed of regular polygons. There also exist heptadecahedra made up of regular polygons, such as the pentagonal rotunda and augmented sphenocorona, but their symmetry is low. In addition, there are numerous topologically distinct forms of a heptadecahedron; for example, the hexadecagonal pyramid and pentadecagonal prism.

In industry, heptadecahedra have many uses. For example, specially-designed heptadecahdra can help reduce empty space and make stacking more convenient.^[3][4]

• 
  The heptadecahedron that tiles space in the Voronoi diagram of the Laves graph.
• 
  Pentagonal rotunda, the sixth Johnson solid ${\displaystyle J_{6}}$
• 
  Triaugmented hexagonal prism, the fifty-seventh Johnson solid ${\displaystyle J_{57}}$
• 
  Augmented sphenocorona, the eighty-seventh Johnson solid ${\displaystyle J_{87}}$

Convex heptadecahedra

There are 6,415,851,530,241 topologically distinct convex heptadecahedra, excluding mirror images, having at least 11 vertices.^[5] (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.)

The infinite Laves graph has convex heptadecahedral Voronoi cells. Because of the symmetries of the graph, these heptadecahedra are plesiohedra and form an isohedral tessellation of three-dimensional space.^[6]

Other convex polyhedra with 17 faces are the Archimedean solid of a cuboctahedron and four Johnson solids of pentagonal rotunda, triangular orthobicupola, triaugmented hexagonal prism, and augmented sphenocorona.^[7]

Non-convex heptadecahedra

There are also many non-convex heptadecahedra, such as star-shaped prisms, cones, or truncated prisms.

Common heptadecahedra

Pentadecagonal prism

Main article: Pentadecagonal prism

 

Pentadecagonal prism

A pentadecagonal prism is a prism with a pentadecagon base. It consists of 17 faces, 45 edges, and 30 vertices. A regular pentadecagonal prism is one whose faces are all regular polygons. Each vertex is a common vertex of 2 squares and 1 pentadecagon, and its vertex configuration is ${\displaystyle 4{.}4{.}15}$ , and therefore has the property of being isogonal and can be classified as a semi-regular heptadecahedron.

In Schläfli notation, it can be denoted as {15}×{} or t{2,15}, and its Coxeter-Dynkin diagram is given by      ; its Wythoff symbol is 2 15 | 2, and in Conway polyhedron notation it is represented by P15.

If the pentadecagonal prism has base side length ${\displaystyle s}$  and height ${\displaystyle h}$ , then its volume ${\displaystyle V}$  and surface area ${\displaystyle S}$  is given by:^[8]

${\displaystyle V={\frac {15hs^{2}\cot {\frac {\pi }{15}}}{4}}\approx 17.6424hs^{2}}$ 

${\displaystyle S=15s\left(h+{\frac {s\cot {\frac {\pi }{15}}}{2}}\right)\approx 15s\left(h+2.35232s\right)}$ 

Hexadecagonal pyramid

Main article: Hexadecagonal pyramid

 

Hexadecagonal pyramid

A hexagonal pyramid is a pyramid with a hexagon base. It consists of 17 faces, 32 edges, and 17 vertices. Its dual polyhedron is itself.^[9] A regular hexagonal pyramid is one whose base is a regular hexagon. In Schläfli notation, it can be denoted as {}∨{16}.

If the hexagonal pyramid has base side length ${\displaystyle s}$  and height ${\displaystyle h}$ , then its volume ${\displaystyle V}$  and surface area ${\displaystyle S}$  is given by:^[9]

${\displaystyle V={\frac {4hs^{2}\cot {\frac {\pi }{16}}}{3}}\approx 6.70312hs^{2}}$ 

${\displaystyle S=4s\left({\sqrt {4h^{2}+s^{2}\cot ^{2}{\frac {\pi }{16}}}}+s\cot {\frac {\pi }{16}}\right)\approx 4s\left({\sqrt {4h^{2}+25.2741s^{2}}}+5.02734s\right)}$ 

Elongated octagonal pyramid

Main article: Elongated octagonal pyramid

 

Elongated octagonal pyramid

An elongated octagonal pyramid is formed by attaching an octagonal prism to an octagonal pyramid. It consists of 17 faces, 32 edges, and 17 vertices.

Pentagonal rotunda

Main article: Pentagonal rotunda

 

Pentagonal rotunda

A pentagonal rotunda is a rotunda with a pentagon as its base. It consists of 15 faces, 35 edges and 20 vertices. Its 17 faces consist of 1 pentagon top, 1 decagon base, 5 pentagon sides, and 10 triangle sides.

A regular pentagon rotunda is one whose base is a regular pentagon. It has regular pentagons on both its top and side faces. All faces of this polyhedron are regular polygons. It is therefore a Johnson solid, and is the only rotunda that belongs in the Johnson group.^[10]

The symmetry group of the pentagonal rotunda is C_5v, and its order is 10.

Triangular cupolarotunda

Main article: Triangular cupolarotunda

A triangular cupolarotunda is formed by joining a triangular cupola and a triangular rotunda with the base having more edges. It consists of 17 faces, 30 edges, and 15 vertices. Its 17 faces consist of 2 triangle bases, 9 triangle sides, 3 rectangle sides, and 3 pentagon sides.

The triangular cupolarotunda can be either ortho- or gyro- depending on how the cupola and rotunda are joined.

Triangular orthocupolarotunda

Triangular gyrocupolarotunda

List of heptadecahedra

Name	Image	Symbol	V	E	F	χ	Faces	Symmetry
Pentadecagonal prism		t{2,15} {15}x{}	30	45	17	2	2 pentadecagons 15 rectangles	D15h, [15,2], (*15 2 2), order 60
Hexadecagonal pyramid		( )∨{16}	17	32	17	2	1 hexadecagon 16 triangles	C16v, [16], (*16 16)
Elongated octagonal pyramid		P8+Y8	17	32	17	2	8 triangles 8 squares 1 octagon	C8v, [8], (*88)
Truncated octagonal bipyramid			17	32	17	2	1 octagon 8 trapezoids 8 triangles	C8v, [8], (*88)
Pentadecagonal frustum			30	45	17	2	2 pentadecagons 15 trapezoids	D15h, [15,2], (*15 2 2), order 60
Pentagonal rotunda			20	35	17	2	1 pentagon top 1 decagon base 5 pentagon sides 10 triangle sides	C5v, [5], (*55), order 10
Triangular orthocupolarotunda			15	30	17	2	2 triangle bases 9 triangle sides 3 rectangle sides 3 pentagon sides	C3v
Triangular gyrocupolarotunda

References

1. V. Gold (1977). Advances in Physical Organic Chemistry APL. Vol. 4. Academic Press. p. 227. ISBN 9780080581538.
2. Commonwealth Scientific and Industrial Research Organization (Australia), Institute of Physics (Great Britain). Australian Branch, Australian National Research Council, Australian Academy of Science (1999). Australian Journal of Chemistry. Vol. 52. Commonwealth Scientific and Industrial Research Organization. p. 343-822, 497.
3. Pedrita, alongside Água de Luso. "The bottle's 17 faces". Archived from the original on 2016-08-20. Retrieved 2016-08-20.
4. "以節省空間與堆疊的17面體瓶身設計". poloves. Archived from the original on 2014-08-04. Retrieved 2016-08-20.
5. Counting polyhedra
6. Schoen, Alan H. (June–July 2008), "On the graph (10,3)-a" (PDF), Notices of the American Mathematical Society, 55 (6): 663.
7. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
8. "pentadecagonal prism". Wolfram Alpha Site.
9. "hexadecagonal pyramid". Wolfram Alpha Site.
10. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.

• What Are Polyhedra?, with Greek Numerical Prefixes

External links