Triaugmented triangular prism | |
---|---|
Type | Deltahedron, Johnson J50 – J51 – J52 |
Faces | 14 triangles |
Edges | 21 |
Vertices | 9 |
Vertex configuration | |
Symmetry group | |
Dual polyhedron | Associahedron |
Properties | convex |
Net | |
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, [1] tricapped trigonal prism, [2] tetracaidecadeltahedron, [3] [4] or tetrakaidecadeltahedron; [1] 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. [5] 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. [6] [7] More generally, the convex polyhedra in which all faces are regular polygons are called the Johnson solids, and every convex deltahedron is a Johnson solid. The triaugmented triangular prism is numbered among the Johnson solids as . [8]
One possible system of Cartesian coordinates for the vertices of a triaugmented triangular prism, giving it edge length 2, is: [1]
A triaugmented triangular prism with edge length has surface area [9]
the area of 14 equilateral triangles. Its volume, [9]
can be derived by slicing it into a central prism and three square pyramids, and adding their volumes. [9]
It has the same three-dimensional symmetry group as the triangular prism, the dihedral group of order twelve. Its dihedral angles can be calculated by adding the angles of the component pyramids and prism. The prism itself has square-triangle dihedral angles and square-square angles . The triangle-triangle angles on the pyramid are the same as in the regular octahedron, and the square-triangle angles are half that. Therefore, for the triaugmented triangular prism, the dihedral angles incident to the degree-four vertices, on the edges of the prism triangles, and on the square-to-square prism edges are, respectively, [10]
The graph of the triaugmented triangular prism has 9 vertices and 21 edges. It was used by Fritsch & Fritsch (1998) as a small counterexample to Alfred Kempe's false proof of the four color theorem using Kempe chains, and its dual map was used as their book's cover illustration. [11] Therefore, this graph has subsequently been named the Fritsch graph. [12] An even smaller counterexample, called the Soifer graph, is obtained by removing one edge from the Fritsch graph (the bottom edge in the illustration here). [12] [13]
The Fritsch graph is one of only six connected graphs in which the 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 topological surface called a 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—excluding the two 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. [14]
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) [15] that can be realized with three non-adjacent square faces, and six more faces that are congruent irregular pentagons. [16] It is also known as an order-5 associahedron, a polyhedron whose vertices represent the 14 triangulations of a regular hexagon. [15] 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. [17]
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 , associated with the Dynkin diagram , the root system, and the cluster algebra. [18] 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. [15]
In the geometry of chemical compounds, it is common to visualize an atom cluster surrounding a central atom as a polyhedron—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. [2] For example, the lanthanides from lanthanum to dysprosium dissolve in water to form cations surrounded by nine water molecules arranged as a triaugmented triangular prism. [19]
In the Thomson problem, concerning the minimum-energy configuration of charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the minimum solution known for places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, inscribed in a sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known. [20]
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra.
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.
In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.
In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid. Each bipyramid is the dual of a uniform prism.
In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral, and it is called an equilateral square pyramid.
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.
In geometry, the triangular hebesphenorotunda is one of the Johnson solids.
In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.
In mathematics, an associahedronKn is an (n – 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari.
In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.