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pentak
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There are 6 letters in PENTAK ( A1E1K5N1P3T1 )
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Definitions of pentak in various dictionaries:
PENTAK - In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope o...
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In geometry, a pentakis dodecahedron or kisdodecahedron is 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. * 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. The critical height of each of the pyramids above the faces of the original unit dodecahedron is * * * * h * = * * * * 65 * + * 22 * * * 5 * * * * * 19 * * * 5 * * * * * * ≈ * 0.2515 * * * {\displaystyle h={\frac {\sqrt {65+22{\sqrt {5}}}}{19{\sqrt {5}}}}\approx 0.2515} * * At this size, the dihedral angle between all neighbouring triangular faces is equal to the value in the table above. Flatter pyramids have higher intra-pyramid dihedrals and taller pyramids have higher inter-pyramid dihedrals.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: * * The small stellated dodecahedron (with very tall pyramids). * Great pentakis dodecahedron (with extremely tall pyramids) * Wenninger's third stellation of icosahedron (with inverted pyramids).If one affixes pentagrammic pyramids into an excavated dodecahedron one obtains the great icosahedron. * If one keeps the center dodecahedron, one get the net of a Dodecahedral pyramid. |