Kleetope

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope P^K formed by replacing each facet of P with a shallow pyramid.^[1] Kleetopes are named after Victor Klee.^[2]

Examples

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.

Kleetopes of the Platonic solids

triakis tetrahedron Kleetope of tetrahedron.

tetrakis hexahedron Kleetope of cube.

triakis octahedron Kleetope of octahedron.

pentakis dodecahedron Kleetope of dodecahedron.

triakis icosahedron Kleetope of icosahedron.

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.

Some other convex Kleetopes

disdyakis dodecahedron Kleetope of rhombic dodecahedron.

disdyakis triacontahedron Kleetope of rhombic triacontahedron.

tripentakis icosidodecahedron Kleetope of icosidodecahedron.

Bipyramids, such as this pentagonal bipyramid, can be seen as the Kleetope of their respective dihedra.

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.

Some nonconvex Kleetopes, based on the Kepler-Poinsot solids

small stellapentakis dodecahedron Kleetope of small stellated dodecahedron.

great stellapentakis dodecahedron Kleetope of great stellated dodecahedron.

great pentakis dodecahedron Kleetope of great dodecahedron.

great triakis icosahedron Kleetope of great icosahedron.

Definitions

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.^[3]

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.

Properties and applications

If P has enough vertices relative to its dimension, then the Kleetope of P is dimensionally unambiguous: the graph formed by its edges and vertices is not the graph of a different polyhedron or polytope with a different dimension. More specifically, if the number of vertices of a d-dimensional polytope P is at least d²/2, then P^K is dimensionally unambiguous.^[4]

If every i-dimensional face of a d-dimensional polytope P is a simplex, and if i ≤ d − 2, then every (i + 1)-dimensional face of P^K is also a simplex. In particular, the Kleetope of any three-dimensional polyhedron is a simplicial polyhedron, a polyhedron in which all facets are triangles.

Kleetopes may be used to generate polyhedra that do not have any Hamiltonian cycles: any path through one of the vertices added in the Kleetope construction must go into and out of the vertex through its neighbors in the original polyhedron, and if there are more new vertices than original vertices then there are not enough neighbors to go around. In particular, the Goldner–Harary graph, the Kleetope of the triangular bipyramid, has six vertices added in the Kleetope construction and only five in the bipyramid from which it was formed, so it is non-Hamiltonian; it is the simplest possible non-Hamiltonian simplicial polyhedron.^[5] If a polyhedron with n vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length O(n^{log₃ 2}); that is, the shortness exponent of these graphs is log₃ 2, approximately 0.630930. The same technique shows that in any higher dimension d, there exist simplicial polytopes with shortness exponent log_d 2.^[6] Similarly, Plummer (1992) used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching.

Kleetopes also have some extreme properties related to their vertex degrees: if each edge in a planar graph is incident to at least seven other edges, then there must exist a vertex of degree at most five all but one of whose neighbors have degree 20 or more, and the Kleetope of the Kleetope of the icosahedron provides an example in which the high-degree vertices have degree exactly 20.^[7]

Notes

1. Grünbaum (1963, 1967).
2. Malkevitch, Joseph, People Making a Difference, American Mathematical Society.
3. Grünbaum (1967), p. 217.
4. Grünbaum (1963); Grünbaum (1967), p. 227.
5. Grünbaum (1967), p. 357; Goldner & Harary (1975).
6. Moon & Moser (1963).
7. Jendro'l & Madaras (2005).

References

• Jendro'l, Stanislav; Madaras, Tomáš (2005), "Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs", Tatra Mountains Mathematical Publications, 30: 149–153, MR 2190255.
• Goldner, A.; Harary, F. (1975), "Note on a smallest nonhamiltonian maximal planar graph", Bull. Malaysian Math. Soc., 6 (1): 41–42. See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications.
• Grünbaum, Branko (1963), "Unambiguous polyhedral graphs", Israel Journal of Mathematics, 1 (4): 235–238, doi:10.1007/BF02759726, MR 0185506, S2CID 121075042.
• Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.
• Moon, J. W.; Moser, L. (1963), "Simple paths on polyhedra", Pacific Journal of Mathematics, 13 (2): 629–631, doi:10.2140/pjm.1963.13.629, MR 0154276.
• Plummer, Michael D. (1992), "Extending matchings in planar graphs IV", Discrete Mathematics, 109 (1–3): 207–219, doi:10.1016/0012-365X(92)90292-N, MR 1192384.