Pentakis dodecahedron

Pentakis dodecahedron
Pentakisdodecahedron.jpg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png
Conway notation kD
Face type V5.6.6
DU25 facets.png

isosceles triangle
Faces 60
Edges 90
Vertices 32
Vertices by type 20{6}+12{5}
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 156°43′07″
arccos(−80 + 95/109)
Properties convex, face-transitive
Truncated icosahedron.png
Truncated icosahedron
(dual polyhedron)
Pentakis dodecahedron Net
Net

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. It is a Catalan Solid, meaning that it is a dual of an Archimedean Solid, in this case, the Truncated Icosahedron.

3d model of a pentakis dodecahedron

Cartesian coordinatesEdit

Let   be the golden ratio. The 12 points given by   and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points   together with the points   and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of   gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin. The length of its long edges equals  . Its faces are acute isosceles triangles with one angle of   and two of  . The length ratio between the long and short edges of these triangles equals  .

ChemistryEdit

 
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.

BiologyEdit

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 projectionsEdit

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
     

Related polyhedraEdit

 
Spherical pentakis dodecahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
               
                                               
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
               
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
                     
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
               
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6

Cultural referencesEdit

ReferencesEdit

  • 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 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )

External linksEdit