The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its 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.

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 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 is a composite polyhedron, meaning it can be constructed by attaching equilateral square pyramids to each of the three square faces of a triangular prism, a process called augmentation.

A polyhedron with only equilateral triangles as faces is called a deltahedron.

[7][8] 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

[9] One possible system of Cartesian coordinates for the vertices of a triaugmented triangular prism, giving it edge length 2, is:[1]

can be derived by slicing it into a central prism and three square pyramids, and adding their volumes.

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.

[13] 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).

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.

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.

[15] 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.

[17] It is also known as an order-5 associahedron, a polyhedron whose vertices represent the 14 triangulations of a regular hexagon.

[16] 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.

[18] 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

[19] 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 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.

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.