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.