tridecahedron

There are 96,262,938 topologically distinct convex tridecahedra, excluding mirror images, having at least 9 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.) There is a pseudo-space-filling tridecahedron that can fill all of 3-space together with its mirror-image.{{cite web |last1=Ludacer |first1=Randy |title=Honeycombs and Structural Package Design: More Ways of Taking Up Space |url=http://www.beachpackagingdesign.com/wp/2011/09/honeycombs-and-structural-package-design-more-ways-of-taking-up-space.html |website=Beach Branding & Packaging Design |archiveurl=https://web.archive.org/web/20160307203424/http://beachpackagingdesign.com/boxvox/honeycombs-and-structural-package-design-more-ways-of-taking-up-space |archivedate=2016-03-07}}

{{Main article|Hendecagonal prism}}

File:Hendecagonal prism.png

A hendecagonal prism is a prism with a hendecagon base. It is a type of tridecahedron, which consists of 13 faces, 22 vertices, and 33 sides. A regular hendecagonal prism is a hendecagonal prism whose faces are regular hendecagons, and each of its vertices is a common vertex of 2 squares and 1 hendecagon. In a vertex figure a hendecagonal prism is represented by $4\{.\}4\{.\}11$; in Schläfly notation it can be represented by {11}×{} or t{2, 11}; {{CDD|node_1|1x|1x|node|2|node_1}} can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 {{pipe}} 2; in Conway polyhedron notation it can be represented by P11. If the side length of the base of a regular hendecagonal prism is $s$ and the height is $h$, then its volume $V$ and surface area $S$ are:{{WolframAlpha| title=Hendecagon prism}}

:$V=\backslash frac\{11\; h\; s^2\; \backslash cot\{\backslash frac\{\backslash pi\}\{11\}\}\}\{4\}\backslash approx\; 9.36564\; h\; s^2$

:$S=11s\backslash left(h+\backslash frac\{1\}\{2\}s\backslash cot\{\backslash frac\{\backslash pi\}\{11\}\}\backslash right)\backslash approx\; 11s\backslash left(h+1.70284s\backslash right)$

{{See also|Pyramid (geometry)}}

File:Dodecagonal pyramid.svg

A dodecagonal pyramid is a pyramid with a dodecagonal base. It is a type of tridecahedron, which has 13 faces, 24 edges, and 13 vertices, and its dual polyhedron is itself.{{WolframAlpha| title=Dodecagon pyramid}} A regular dodecagonal pyramid is a dodecagonal pyramid whose base is a regular dodecagon. If the side length of the base of a regular twelve-sided pyramid is $s$ and the height is $h$, then its volume $V$ and surface area $S$ are:

:$V=\backslash left(2+\backslash sqrt\{3\}\backslash right)hs^2\backslash approx\; 3.73205\; h\; s^2$

:$S=3s\backslash left(\backslash sqrt\{4h^2+\backslash left(7+4\backslash sqrt\{3\}\backslash right)s^2\}+\backslash left(2+\backslash sqrt\{3\}\backslash right)s\backslash right)\backslash approx\; 3s\backslash left(\backslash sqrt\{4h^2+13.9282s^2\}+3.73205s\backslash right)$

{{Infobox polyhedron

| name = Space-filling tridecahedron

| image = Space-Filling_Triskaidecahedron.svg

| Type = Space-filling polyhedra

| faces = 13

| edges = 30

| vertices = 19

| type = 6 trapezoids 20px
6 pentagons 20px
1 regular hexagons 20px

}}

A space-filling tridecahedron[http://science.unitn.it/~gabbrielli/javaview/scp8f.html Oblate Rhombohedra] {{Webarchive|url=https://web.archive.org/web/20160921131728/http://science.unitn.it/~gabbrielli/javaview/scp8f.html |date=2016-09-21 }} science.unitn.it [2016-1-10]Virtual Polyhedra, [http://www.georgehart.com/virtual-polyhedra/greek-prefixes.html Greek Numerical Prefixes] {{Webarchive|url=https://web.archive.org/web/20160115214334/http://www.georgehart.com/virtual-polyhedra/greek-prefixes.html |date=2016-01-15 }}, 1996, George W. Hart, georgehart.com [2016-1-10] is a tridecahedron that can completely fill three-dimensional space without leaving gaps. It has 13 faces, 30 edges, and 19 vertices. Among the thirteen faces, there are six trapezoids, six pentagons and one regular hexagon.[http://science.unitn.it/~gabbrielli/javaview/p8.html A space-filling polyhedron with 13 faces] {{webarchive|url=https://web.archive.org/web/20170701081626/http://science.unitn.it/~gabbrielli/javaview/p8.html}} science.unitn.it [2016-1-10]

;Dual polyhedron

The polyhedron's dual polyhedron is an enneadecahedron. It is similar to a twisted half-cube, but one of its vertices is treated as a face before twisting.

{{reflist|group="notes"}}

{{reflist}}

• [http://dmccooey.com/polyhedra/SelfDualTridecahedron1.html Self-dual tridecahedra]
• [https://web.archive.org/web/20050212114016/http://members.aol.com/Polycell/what.html What Are Polyhedra?], with Greek Numerical Prefixes

{{Polyhedra}}

Category:Polyhedra