Pentakis dodecahedron

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 + 9√5/109)
Propertiesconvex, face-transitive

Truncated icosahedron
(dual polyhedron)
Net

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.[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).

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
Spherical pentakis dodecahedron

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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