Triaugmented triangular prism: Difference between revisions

Revision as of 21:36, 25 November 2022

Revision as of 21:44, 25 November 2022

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== Fritsch graph ==

[[File:Fritsch map.svg|thumb|left|The Fritsch graph and its dual map. For the partial 4-coloring shown, the red–green and blue–green [[Kempe chain]]s cross. It is not possible to free a color for the uncolored center region by swapping colors in a single chain, contradicting [[Alfred Kempe]]'s false proof of the four- color theorem.]]

The graph of the triaugmented triangular prism has 9 vertices and 21 edges. It was used by {{harvtxt|Fritsch|Fritsch|1998}} as a small counterexample to [[Alfred Kempe]]'s false proof of the [[four color theorem]] using [[Kempe chain]]s, and its dual map was used as their book's cover illustration.{{r|ff98}} Because of this usageTherefore, this graph has subsequently been named the '''Fritsch graph'''.{{r|involve}} An even smaller counterexample, called the Soifer graph, can beis obtained by removing one edge from the Fritsch graph (the bottom edge in the illustration).{{r|involve|soifer}}

The Fritsch graph is one of only six connected graphs within the property thatwhich the [[Neighbourhood (graph theory)|neighborhood]] of every vertex is a cycle of length four or five. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a [[manifold|topological surface]] called a [[Triangulation (topology)|Whitney triangulation]]. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive [[angular defect]] at every vertex. This makes them a combinatorial analogue of the positively- curved smooth surfaces. They come from six of the eight deltahedra, omittingdeltahedra—excluding the onestwo that have a vertex with a triangular neighborhood. As well as the Fritsch graph, the other five are the graphs of the [[regular octahedron]], [[regular icosahedron]], [[pentagonal bipyramid]], [[snub disphenoid]], and [[gyroelongated square bipyramid]].{{r|knill}}

==Dual associahedron==

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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> [[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 each other, 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 a dimension equal to the number of sides of the polygon minus three.{{r|fr07}}

==Applications==

In the geometry of [[chemical compound]]s, it is common to visualize an [[atom cluster]] surrounding a central atom as a polyhedron, thepolyhedron—the [[convex hull]] of the surrounding atoms' locations. The [[tricapped trigonal prismatic molecular geometry]] describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.{{r|kepert}}

In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on the surface of a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n=9</math> places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, [[Circumscribed sphere|inscribed in a sphere]]. This hasconfiguration beenis proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.{{r|whyte}}

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