Triaugmented triangular prism: Difference between revisions

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Revision as of 20:50, 27 November 2022

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The [[dual polyhedron]] of the triaugmented triangular prism has a face for each vertex of the triaugmented triangular prism, and a vertex for each face. It is an [[enneahedron]] (that is, a nine-sided polyhedron){{r|fr07}} that can be realized with three non-adjacent [[Square (geometry)|square]] faces, and six more faces that are congruent irregular [[pentagon]]s.{{r|as18}} It is also known as an order-5 [[associahedron]], a polyhedron whose vertices represent the 14 triangulations of a [[regular hexagon]].{{r|fr07}} A less-symmetric form of this dual polyhedron, obtained by slicing a [[truncated octahedron]] into four congruent quarters by two planes that perpendicularly bisect two parallel families of its edges, is a [[space-filling polyhedron]].{{r|goldberg}}

More generally, when a polytope is the dual of an associahedron, its boundary (a [[simplicial complex]]) of triangles, tetrahedra, or higher-dimensional simplices) is called a "cluster complex". In the case of the triaugmented triangular prism, it is a cluster complex of type <math>A_3</math>, associated with the <math>A_3</math> [[Dynkin diagram]] {{Dynkin|node|3|node|3|node}}, the <math>A_3</math> [[root system]], and the <math>A_3</math> [[cluster algebra]].{{r|bsw13}} The connection with the associahedron provides a correspondence between the nine vertices of the triaugmented triangular prism and the nine diagonals of a hexagon. The edges of the triaugmented triangular prism correspond to pairs of diagonals that do not cross, and the triangular faces of the triaugmented triangular prism correspond to the triangulations of the hexagon (consisting of three non-crossing diagonals). The triangulations of other regular polygons correspond to polytopes in the same way, with dimension equal to the number of sides of the polygon minus three.{{r|fr07}}

==Applications==