Triaugmented triangular prism: Difference between revisions

Revision as of 07:46, 28 November 2022

Revision as of 02:25, 9 December 2022

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In [[geometry]], theThe '''triaugmented triangular prism'''{{r|trigg}}, in geometry, is a [[convex polyhedron]] with 14 [[equilateral triangle]]s as its faces. It can be constructed from a [[triangular prism]] by attaching [[equilateral square pyramid]]s to each of its three square faces, a process called [[Augmentation (geometry)|augmentation]].{{r|pughtrigg}}

The same shape is also called the '''tetrakis triangular prism''',{{r|shdc}} '''tricapped trigonal prism''',{{r|kepert}} '''tetracaidecadeltahedron''',{{r|burgiel|pugh}} or '''tetrakaidecadeltahedron''';{{r|shdc}} these last names mean a polyhedron with 14 triangular faces. It is an example of a [[deltahedron]] and of a [[Johnson solid]].

The edges and vertices of the triaugmented triangular prism form a [[maximal planar graph]] with 9 vertices and 21 edges, called the '''Fritsch graph'''. It was used by Rudolf and Gerda Fritsch to show that [[Alfred Kempe]]'s attempted proof of the [[four color theorem]] was incorrect. The Fritsch graph is one of only six graphs in which every [[Neighbourhood (graph theory)|neighborhood]] is a 4- or 5-vertex cycle.

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==Construction==

[[File:J51 triaugmented triangular prism.stl|thumb|3D model of the triaugmented triangular prism]]

The triaugmented triangular prism can be constructed by attaching [[equilateral square pyramid]]s to each of the three square faces of a [[triangular prism]], a process called [[Augmentation (geometry)|augmentation]].{{r|pughtrigg}} These pyramids cover each square, replacing it with four [[equilateral triangle]]s, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces, like this one, is called a [[deltahedron]]. There are only eight different [[Convex set|convex]] deltahedra, one of which is the triaugmented triangular prism.{{r|fw47|cundy}} More generally, the convex polyhedra in which all faces are [[regular polygon]]s are called the [[Johnson solid]]s, and every convex deltahedron is a Johnson solid. The triaugmented triangular prism is numbered among the Johnson solids {{nowrap|as <math>J_{51}</math>.{{r|francis}}}}

One possible system of [[Cartesian coordinates]] for the vertices of a triaugmented triangular prism, giving it edge length 2, is:{{r|shdc}}