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Concept
Kleetope
Summary
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a shallow pyramid. Kleetopes are named after Victor Klee.
The triakis tetrahedron is the Kleetope of a tetrahedron, the triakis octahedron is the Kleetope of an octahedron, and the triakis icosahedron is the Kleetope of an icosahedron. In each of these cases the Kleetope is formed by adding a triangular pyramid to each face of the original polyhedron.
The tetrakis hexahedron is the Kleetope of the cube, formed by adding a square pyramid to each of its faces, and the pentakis dodecahedron is the Kleetope of the dodecahedron, formed by adding a pentagonal pyramid to each face of the dodecahedron.
The base polyhedron of a Kleetope does not need to be a Platonic solid. For instance, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron, formed by replacing each rhombus face of the dodecahedron by a rhombic pyramid, and the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. In fact, the base polyhedron of a Kleetope does not need to be Face-transitive, as can be seen from the tripentakis icosidodecahedron above.
The Goldner–Harary graph may be represented as the graph of vertices and edges of the Kleetope of the triangular bipyramid.
One method of forming the Kleetope of a polytope P is to place a new vertex outside P, near the centroid of each facet. If all of these new vertices are placed close enough to the corresponding centroids, then the only other vertices visible to them will be the vertices of the facets from which they are defined. In this case, the Kleetope of P is the convex hull of the union of the vertices of P and the set of new vertices.
Alternatively, the Kleetope may be defined by duality and its dual operation, truncation: the Kleetope of P is the dual polyhedron of the truncation of the dual of P.
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