Pentadecahedron

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Some pentadecahedrons
Dual elongated triangular cupola.png
Dual elongated triangular cupola
Elongated pentagonal dipyramid.png
Elongated pentagonal dipyramid
Tridecagonal prism.svg
Tridecagonal prism
Elongated heptagonal pyramid.svg
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]

Contents

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

NameTypeImageSymbolVerticesSidesFaces χ Face typeSymmetry
tridecagonal prismprism Tridecagonal prism.svg t{2,13}
{13}x{}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 1x.pngCDel 3x.pngCDel node.png
26391522 tridecagons
13 rectangles
D13h, [13,2], (*13 2 2)
tetradecagonal pyramidpyramid Tetradecagonal pyramid.svg ( )∨{14}15281521 tetradecagon
14 triangles
C14v, [14], (*14 14)
elongated heptagonal pyramid pyramid Elongated heptagonal pyramid.svg 15281527 triangles
7 rectangles
1 heptagon
D7h, [7,2], (*227), order 28
Heptagonal truncated conetruncated cone Qi Jiao Zhui Tai Zhui .svg 15281527 triangles
7 kites
1 heptagon
D7h, [7,2], (*227), order 28
Elongated_pentagonal bipyramid Bipyramid
Johnson solid
Elongated pentagonal dipyramid.png 122515210 triangles
5 squares
D5h, [5,2], (*225)


Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Dual polyhedron</span> Polyhedron associated with another by swapping vertices for faces

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

A hexahedron or sexahedron is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.

<span class="mw-page-title-main">Truncated octahedron</span> Archimedean solid

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.

<span class="mw-page-title-main">Snub disphenoid</span> 84th Johnson solid (12 triangular faces)

In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

<span class="mw-page-title-main">Gyrobifastigium</span> 26th Johnson solid (8 faces)

In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

<span class="mw-page-title-main">Triangular tiling</span> Regular tiling of the plane

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

<span class="mw-page-title-main">Conway polyhedron notation</span> Method of describing higher-order polyhedra

In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

<span class="mw-page-title-main">Uniform polytope</span> Isogonal polytope with uniform facets

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

<span class="mw-page-title-main">Tetradecahedron</span> Polyhedron with 14 faces

A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.

<span class="mw-page-title-main">Enneahedron</span> Polyhedron with 9 faces

In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.

<span class="mw-page-title-main">Octadecahedron</span> Polyhedron with 18 faces

In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.

<span class="mw-page-title-main">Tridecahedron</span> Polyhedron with 13 faces

A tridecahedron, or triskaidecahedron, is a polyhedron with thirteen faces. There are numerous topologically distinct forms of a tridecahedron, for example the dodecagonal pyramid and hendecagonal prism. However, a tridecahedron cannot be a regular polyhedron, because there is no regular polygon that can form a regular tridecahedron, and there are only five known regular polyhedra.

<span class="mw-page-title-main">Ideal polyhedron</span> Shape in hyperbolic geometry

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

References

  1. 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.
  2. 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. Lagunov, VA and Sinani, AB (1998). "Formation of a bistructure of a solid in a computer experiment". Physics of the Solid State. Springer. 40 (10): 1742–1747. doi:10.1134/1.1130648.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. Counting polyhedra