Polyhedra

cubic graph with 12 vertices and 18 edges

polyhedron formed from two hexagonal pyramids joined at their bases

polyhedron

thumb|The bisymmetric hendecahedron contains 11 faces and can be arranged in 3D without gaps. A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are many topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism.

{|class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2|Regular-based right pyramids |- |align=center colspan=2|240pxSix tetrahedra whose vertices meet at the center. Blue edges are doubled with pairs of faces hidden. |- |bgcolor=#e7dcc3|Faces||24 isosceles triangles |- |bgcolor=#e7dcc3|Edges||36 (6 as degenerate pairs) |- |bgcolor=#e7dcc3|Vertices||12 |- |bgcolor=#e7dcc3|Symmetry group||C3v, [3], (*33), order 6 |- |bgcolor=#e7dcc3|Properties||torus |- align=center |colspan=2|240pxNet |} 120px|thumb|A kaleidocycle befo

thumb|right|[[Ball-and-stick model of the octadecahedral closo-undecaborate ion, [B11H11]2−, as found in the crystal structure of the benzyltriethylammonium salt.]]

{| class=wikitable align=right |- align=center |200pxTriakis octahedron |200pxTetrakis hexahedron |- align=center |200pxDeltoidal icositetrahedron |200pxPentagonal icositetrahedron |} In geometry, an icositetrahedron refers to a polyhedron with 24 faces, none of which are regular polyhedra. However, many are composed of regular polygons, such as the triaugmented dodecahedron and the disphenocingulum. Some icositetrahedra are near-spherical, but are not composed of regular polygons. A minimum of 14 vertices is required to form a icositetahedron.

15th century essay by Piero della Francesca

sequence of faces of a polytope

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a pyramid. In some cases, the pyramid is chosen to have regular sides, often producing a non-convex polytope; alternatively, by using sufficiently shallow pyramids, the results may remain convex. Kleetopes are named after Victor Klee, although the same concept was known under other names long before the work of Klee.

property of a uniform tiling or polyhedron that is colored to be vertex-transitive

Wikimedia list article

polyhedron

{|style="float: right; margin: 0 0 1em 1em; width: 22.5em; font-size: 0.86em; line-height: normal; border: 1px solid #CCD2D9; background: #F0F6FA" ! colspan=2 style="padding-top:1.0em; padding-bottom:1.0em;"|Common tridecahedra |- |align=center|120pxSpace-filling tridecahedron |align=center|120pxElongated hexagonal pyramid |- |align=center|120pxHendecagonal prism |align=center|120pxGyroelongated square pyramid |}

thumb|right|An example of an Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism.

redirect Dodecahedron#Pyritohedron

polyhedral approximation to a cylinder obtained by stacking antiprisms, used as a pathological example to demonstrate that limits of surface areas of polyhedral approximations do not always equal the true surface area

polyhedron

polyhedron obtained by alternating a corresponding omnitruncated polyhedron

polyhedron with twelve kite-shaped faces and face-transitive dihedral symmetry

property of polyhedra invariant under scissors congruence

A heptadecahedron (or heptakaidecahedron) is a polyhedron with 17 faces. No heptadecahedron is regular; hence, the name is ambiguous. There are heptadecahedra which are nearly spherical, like those seen in some chemical structures, however their faces are not composed of regular polygons. There also exist heptadecahedra made up of regular polygons, such as the pentagonal rotunda and augmented sphenocorona, but their symmetry is low. In addition, there are numerous topologically distinct forms of a heptadecahedron; for example, the hexadecagonal pyramid and pentadecagonal prism.

thumb|Octadecagonal (18-sided) Pyramid (geometry)|pyramid right|thumb|169px|3D model of 18-sided pyramid thumb|a 19-sided 3d object composed of 1 hexagonal face, 12 rectangular faces, 6 triangular facesthumb|Heptadecagonal (17-sided) Prism (geometry)|prism A enneadecahedron (or enneakaidecahedron) is a polyhedron with 19 faces. No enneadecahedron is regular; hence, the name is ambiguous.

goldberg polyhedron with 42 faces

polyhedron

{| class=wikitable align=right |- align=center |200pxPentakis dodecahedron |200pxDeltoidal hexecontahedron |- align=center |200pxPentagonal hexecontahedron |200pxTriakis icosahedron |- align=center |200pxRhombic hexecontahedron |200pxSmall hexagonal hexecontahedron |}

polytope which can stand on only one face

near-miss Johnson solid with 16 faces

{|style="float: right; margin: 0 0 1em 1em; width: 22.5em; font-size: 0.86em; line-height: normal; border: 1px solid #CCD2D9; background: #F0F6FA" ! colspan=2 style="padding-top:1.0em; padding-bottom:1.0em;" | Examples of pentadecahedra |- |align="center"|120pxDual elongated triangular cupola |align=center|120pxElongated pentagonal bipyramid |- |align=center|120pxTridecagonal prism |align=center|120pxElongated heptagonal pyramid |}

extension of the idea of a polyhedron

operation that modifies one polytope into another