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Concept
Prisme triangulaire triaugmenté
Résumé
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.
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 neighborhood is a 4- or 5-vertex cycle.
The dual polyhedron of the triaugmented triangular prism is an associahedron, a polyhedron with four quadrilateral faces and six pentagons whose vertices represent the 14 triangulations of a regular hexagon. In the same way, the nine vertices of the triaugmented triangular prism represent the nine diagonals of a hexagon, with two vertices connected by an edge when the corresponding two diagonals do not cross. Other applications of the triaugmented triangular prism appear in chemistry as the basis for the tricapped trigonal prismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem and Tammes problem.
The triaugmented triangular prism can be constructed by attaching equilateral square pyramids to each of the three square faces of a triangular prism, a process called augmentation. These pyramids cover each square, replacing it with four equilateral triangles, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the triaugmented triangular prism.
Source officielle
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