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In geometry, a pentakis dodecahedron or kisdodecahedron a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name. 〔Conway, Symmetries of things, p.284〕 There are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include: * The usual Catalan pentakis dodecahedron, a convex hexecontahedron with sixty isosceles triangular faces illustrated in the sidebar figure. It is a Catalan solid, dual to the truncated icosahedron, an Archimedean solid. * As the heights of the pentagonal pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi, and the shape becomes a rhombic triacontahedron. * As the height is raised further, the shape becomes non-convex. In particular, an equilateral or deltahedron version of the pentakis dodecahedron, which has sixty equilateral triangular faces as shown in the adjoining figure, is slightly non-convex due to its taller pyramids (note, for example, the negative dihedral angle at the upper left of the figure). : Other more non-convex geometric variants include: * 50px The small stellated dodecahedron (with very tall pyramids). * 50px Great pentakis dodecahedron (with extremely tall pyramids) * 50px Wenninger's third stellation of icosahedron (with inverted pyramids). If one affixes pentagrammic pyramids into Wenninger's third stellation of icosahedron one obtains the great icosahedron. ==Chemistry== The ''pentakis dodecahedron'' in a model of buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「pentakis dodecahedron」の詳細全文を読む スポンサード リンク
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