Pentakis dodecahedron

In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name.[1] There are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include:

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).
Pentakis dodecahedron

(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationkD
Face typeV5.6.6
isosceles triangle
Faces60
Edges90
Vertices32
Vertices by type20{6}+12{5}
Symmetry groupIh, H3, [5,3], (*532)
Rotation groupI, [5,3]+, (532)
Dihedral angle156°43′07″
arccos(−80 + 95/109)
Propertiesconvex, face-transitive

Truncated icosahedron
(dual polyhedron)

Net

Other more non-convex geometric variants include:

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.

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.

Biology

The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron.

Orthogonal projections

The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge:

Orthogonal projections
Projective
symmetry
[2] [6] [10]
Image
Dual
image

Cultural references

References

  1. Conway, Symmetries of things, p.284
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
  • Sellars, Peter (2005). "Doctor Atomic Libretto". Boosey & Hawkes. We surround the plutonium core from thirty two points spaced equally around its surface, the thirty-two points are the centers of the twenty triangular faces of an icosahedron interwoven with the twelve pentagonal faces of a dodecahedron.
  • Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 0730208. (The thirteen semiregular convex polyhedra and their duals, Page 18, Pentakisdodecahedron)
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )
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