Harman Patil (Editor)

Pentakis dodecahedron

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Pentakis dodecahedron

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:

Contents

  • 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:

  • 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 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.

    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:

    References

    Pentakis dodecahedron Wikipedia
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