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Truncated triakis tetrahedron

Near-miss Johnson solid with 16 faces

Field	Value
image	Truncated triakis tetrahedron.svg
caption	(compare animation)
type	Fullerene,
Near-miss Johnson solid
faces	4 hexagons
12 irregular pentagons
edges	42
vertices	28
symmetry	tetrahedral symmetry Td
properties	convex
net	Conway_dk6tT_net.png

Near-miss Johnson solid 12 irregular pentagons

In geometry, the truncated triakis tetrahedron is a convex polyhedron with 16 faces: four sets of three pentagons with a shared vertex, arranged in a tetrahedral arrangement, with four hexagons in the remaining gaps. The faces cannot all be regular polygons, so it is a near-miss Johnson solid. As a fullerene, it is called tetrahedral fullerene or C28 fullerene, and has been suggested as the smallest stable carbon fullerene.

Chemistry

This structure is a fullerene, one of two 28-vertex fullerenes. In this context, it is called tetrahedral fullerene{{cite journal

Geometry

This polyhedron is one of the polyhedra studied in 1935 by Michael Goldberg as a possible solution to the isoperimetric problem of maximizing the volume for a given number of faces (16 in this case) and a given surface area.{{cite journal

The pentagons of this polyhedron cannot be regular pentagons, because if they were then the hexagonal faces would not be flat. As a polyhedron that has near-regular but not regular faces, it is an example of a near-miss Johnson solid, "the closest near miss with higher than prismatic symmetry".

A combinatorially equivalent (but not necessarily geometrically congruent) polyhedron can be constructed from a triakis tetrahedron by truncating the order-6 vertices. The truncated vertices become four hexagon faces, and the obtuse triangles of the triakis tetrahedron are truncated at their acute vertices into pentagons.

References

{{cite OEIS. A007894. Number of fullerenes with 2n vertices (or carbon atoms)
[http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm Johnson Solid Near Misses: Number 22]

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

fullerenes polyhedra truncated-tilings

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