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

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. Conway generalizes Kepler's *kis* prefix as this same kis operator.

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.

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.

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. |

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*.

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 *d*^{2}/2, then *P ^{K}* is dimensionally unambiguous.

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 log_{3} 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] }

- ↑ Grünbaum ( 1963 , 1967 ).
- ↑ Malkevitch, Joseph,
*People Making a Difference*, American Mathematical Society . - ↑ Grünbaum (1967), p. 217.
- ↑ Grünbaum (1963); Grünbaum (1967), p. 227.
- ↑ Grünbaum (1967), p. 357; Goldner & Harary (1975).
- ↑ Moon & Moser (1963).
- ↑ Jendro'l & Madaras (2005).

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, any polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as *poly-* + *-hedron*.

In geometry, **stellation** is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in *n* dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word *stellation* comes from the Latin *stellātus*, "starred", which in turn comes from Latin *stella*, "star".

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or *j*-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, the **rhombic dodecahedron** is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a *d*-dimensional **simple polytope** is a *d*-dimensional polytope each of whose vertices are adjacent to exactly *d* edges. The vertex figure of a simple *d*-polytope is a (*d* − 1)-simplex.

In geometry, a **simplicial polytope** is a polytope whose facets are all simplices. For example, a *simplicial polyhedron* in three dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph.

In geometric graph theory, a branch of mathematics, a **polyhedral graph** is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.

In graph theory, a branch of mathematics, the **Herschel graph** is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.

In the mathematical field of graph theory, the **Goldner–Harary graph** is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967.

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.

In geometry, **chamfering** or **edge-truncation** is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

In polyhedral combinatorics, a **stacked polytope** is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.

In three-dimensional hyperbolic geometry, an **ideal polyhedron** is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

- 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 .

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.