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Polytope made by turning a polytope's facets into pyramids
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed
Kleetope
Catalan solid with 60 faces
an instance of a general construction called the Kleetope; the triakis icosahedron is the Kleetope of the icosahedron. This interpretation is also expressed
Triakis_icosahedron
Catalan solid with 12 faces
triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. This replaces the equilateral triangular faces of the
Triakis_tetrahedron
Catalan solid with 24 faces
barycentric subdivision of a tetrahedron. The name "tetrakis" is used for the Kleetopes of polyhedra with square faces. Hence, the tetrakis hexahedron can be
Tetrakis_hexahedron
Catalan solid with 60 faces
pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. Specifically, the term typically refers to a particular
Pentakis_dodecahedron
Shape with three sides
bipyramids. The Kleetope of a polyhedron is a new polyhedron made by replacing each face of the original with a pyramid, and so the faces of a Kleetope will be
Triangle
Two tetrahedra joined by one face
triangular bipyramids may be derived in different ways. The Kleetope of a triangular bipyramid, its Kleetope can be constructed from a triangular bipyramid by attaching
Triangular_bipyramid
Catalan solid with 48 faces
each face of the rhombic dodecahedron with a flat pyramid results in the Kleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron
Disdyakis_dodecahedron
Polytope whose facets are all simplices
bipyramids, deltahedra (wherein the faces are equilateral triangles, and Kleetope of polyhedra. The simplicial polyhedron corresponds via Steinitz's theorem
Simplicial_polytope
Catalan solid with 120 faces
disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision
Disdyakis_triacontahedron
Polyhedron with 12 faces
D6d symmetry of order 24. Triakis tetrahedron: a Catalan solid and the Kleetope of a regular tetrahedron. Obtained by affixing four triangular pyramids
Dodecahedron
Christmas decoration
is missing and used for mounting. This shape is technically known as a Kleetope of a rhombicuboctahedron. Each face of the geometric solid in the middle
Moravian_star
Catalan solid with 24 faces
octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more
Triakis_octahedron
Solid with six equal square faces
opposite faces. Attaching a low pyramid to each face of a cube produces its Kleetope, the tetrakis hexahedron, dual to the truncated octahedron. The barycentric
Cube
Polyhedron with 60 faces
hexecontahedron can be constructed as a Kleetope of a pentakis dodecahedron. It is therefore a second order Kleetope of the regular dodecahedron. In other
Small hexagonal hexecontahedron
Small_hexagonal_hexecontahedron
Solid with four equal triangular faces
with four triangular pyramids attached to each of its faces. i.e., its Kleetope. Some Johnson solid such as elongated triangular pyramid and elongated
Regular_tetrahedron
13 polyhedra; duals of the Archimedean solids
by adding pyramids to the faces of Platonic solids. These examples are Kleetopes of Platonic solids: triakis tetrahedron, tetrakis hexahedron, triakis
Catalan_solid
Method of describing higher-order polyhedra
Truncate: t Kis raises a pyramid on each face, and is also called akisation, Kleetope, cumulation, accretion, or pyramid-augmentation. Truncate cuts off the
Conway_polyhedron_notation
American mathematician (1925–2007)
papers. He proposed Klee's measure problem and the art gallery problem. Kleetopes are also named after him, as is the Klee–Minty cube, which shows that
Victor_Klee
Solid with 12 equal pentagonal faces
filling the gap with equilateral triangles. Pentakis dodecahedron is the Kleetope of a regular dodecahedron, obtained by affixing pentagonal pyramids to
Regular_dodecahedron
Solid with eight equal triangular faces
out six square pyramids. The triakis octahedron is a Catalan solid, the Kleetope of a regular octahedron, by attaching triangular pyramids onto its faces
Regular_octahedron
Solid with twenty equal triangular faces
base of triangular pyramids onto each face of a regular icosahedron, the Kleetope of an icosahedron. The truncated icosahedron is an Archimedean solid constructed
Regular_icosahedron
Two tetrahedra crossing each other
of these pyramid attachments, convex or non-convex, may be known as the Kleetope of an octahedron. For another choice of pyramid height, intermediate between
Stellated_octahedron
(geometry) Diminishment (geometry) Greatening (geometry) Aggrandizement (geometry) Stellation Kleetope Conway polyhedron notation List of geometry topics
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polyhedra are determined by surface distance
creases into a non-convex polyhedron with 24 equilateral triangle faces, the Kleetope obtained by gluing square pyramids onto the squares of a cube. Six triangles
Alexandrov's theorem on polyhedra
Alexandrov's_theorem_on_polyhedra
Undirected graph with 11 nodes and 27 edges
constructed by gluing tetrahedra onto each face of a triangular dipyramid, the Kleetope of the triangular dipyramid. If the tetrahedra are regular tetrahedron
Goldner–Harary_graph
bases are systematically attached to faces of the polyhedra (akin to kleetopes, augmenting them into a "star-like" polyhedron). In this same work, da
List of polyhedral stellations
List_of_polyhedral_stellations
631 {\displaystyle \log _{3}2\approx 0.631} . A construction based on kleetopes shows that some polyhedral graphs have longest cycle length O ( n log
Shortness_exponent
Graph formed by subdivision of triangles
networks correspond geometrically to a type of stacked polyhedron called a Kleetope. Other authors applied the same name more broadly to planar 3-trees in
Apollonian_network
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