Heptadecahedron

{{short description|Polyhedron with 17 faces}}

{{Expand Chinese|topic=sci|date=June 2023}}

A heptadecahedron (or heptakaidecahedron) is a polyhedron with 17 faces. No heptadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a heptadecahedron; for example, the hexadecagonal pyramid and pentadecagonal prism.

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.{{citation

| last = Schoen | first = Alan H.

| date = June–July 2008

| issue = 6

| journal = Notices of the American Mathematical Society

| page = 663

| title = On the graph (10,3)-a

| url = https://www.ams.org/notices/200806/tx080600663p.pdf

| volume = 55}}. 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.{{citation

| last = Berman | first = Martin

| year = 1971

| title = Regular-faced convex polyhedra

| journal = Journal of the Franklin Institute

| volume = 291

| issue = 5

| pages = 329–352

| doi = 10.1016/0016-0032(71)90071-8

| mr = 290245

}}.

12-3 plesiohedron.png|The heptadecahedron that tiles space in the Voronoi diagram of the Laves graph.

Pentagonal rotunda.png|Pentagonal rotunda, the sixth Johnson solid $J\_6$

Triaugmented hexagonal prism.png|Triaugmented hexagonal prism, the fifty-seventh Johnson solid $J\_\{57\}$

Augmented sphenocorona.png|Augmented sphenocorona, the eighty-seventh Johnson solid $J\_\{87\}$

There are 6,415,851,530,241 topologically distinct convex heptadecahedra, excluding mirror images, having at least 11 vertices.[http://www.numericana.com/data/polycount.htm Counting polyhedra] (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.)