Tridecahedron

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 convex polyhedra.

Convex

A polyhedron with 13 faces fills space with its chiral copy.

There are 96,262,938 topologically distinct convex tridecahedra, excluding mirror images, having at least 9 vertices. 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.

Common tridecahedrons

Examples

Name (vertex layout)	Symbol	Stereogram	Expanded view	Faces		Edges	Apexes
Hendecagonal prism	t{2,11} {11}x{}			13	square × 11 hendecagon × 2	33	22
Dodecagonal pyramid	( )∨{12}			13	triangle × 12 dodecagon × 1	24	13
Elongated hexagonal pyramid				13	triangle × 6 square × 6 hexagon × 1	24	13
Space-filling tridecahedron		80px	80px	13	quadrilateral × 6 pentagon × 6 hexagon × 1	30	19
Gyroelongated square pyramid		80px	80px	13	triangle × 12 square × 1	20	9
truncated hexagonal trapohedron		100px		13	1 hexagon base 6 pentagon sides 6 kite sides	30	19
Biaugmented pentagonal prism		80px	80px	13	triangle × 8 square × 3 pentagon × 2	23	12

Hendecagonal prism

Regular hendecagonal prism

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. The vertex configuration of a hendecagonal prism is represented by $4.4.11$; in Schläfli notation it can be represented by {11}×{} or t{2, 11}; can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 | 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:

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

$S=11s(h+\frac{1}{2}s\cot \frac{\pi }{11})\approx 11s(h+1.70284s)$

Dodecagonal pyramid

See also: Pyramid (geometry)

Dodecagonal pyramid

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. 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:

$V=(2+\sqrt{3})h{s}^2\approx 3.73205h{s}^2$

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

Space-filling tridecahedron

Space-filling tridecahedron

Space-filling tridecahedron tiling

A space-filling tridecahedron is a tridecahedron that can, along with its mirror image, completely fill three-dimensional space without leaving gaps. It has 13 faces, 30 edges, and 19 vertices. Among the thirteen faces, there are six quadrilaterals, six pentagons and one regular hexagon. Since it must tile with its mirror image, the space-filling tridecahedron is pseudo-space-filling rather than truly space-filling. The space-filling tridecahedron tiles with the cubic symmetry group Pa3.

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.

	Image	Rotation animation	Expanded view
Original polyhedron tridecahedron	100px	100px	150px
Dual polyhedron enneadecahedron	100px	100px	120px

External links

• Self-dual tridecahedra
• What Are Polyhedra?, with Greek Numerical Prefixes