Pentakis dodecahedron

Let ${\displaystyle \phi }$ be the golden ratio. The 12 points given by ${\displaystyle (0,\pm 1,\pm \phi )}$ 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 ${\displaystyle (\pm 1,\pm 1,\pm 1)}$ together with the points ${\displaystyle (\pm \phi ,\pm 1/\phi ,0)}$ and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of ${\displaystyle (3\phi +12)/19\approx 0.887\,057\,998\,22}$ 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 ${\displaystyle 2/\phi }$. Its faces are acute isosceles triangles with one angle of ${\displaystyle \arccos((-8+9\phi )/18)\approx 68.618\,720\,931\,19^{\circ }}$ and two of ${\displaystyle \arccos((5-\phi )/6)\approx 55.690\,639\,534\,41^{\circ }}$. The length ratio between the long and short edges of these triangles equals ${\displaystyle (5-\phi )/3\approx 1.127\,322\,003\,75}$.

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.

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.

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

A concave pentakis dodecahedron adds inverted pyramids on the pentagonal faces of a dodecahedron.

A pentakis dodecahedron (left) with inverted pyramids (right) has the same surface area. Related polyhedra 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 See also Excavated dodecahedron Cultural references The Spaceship Earth structure at Walt Disney World's Epcot is a derivative of a pentakis dodecahedron. The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https://books.google.com/books?id=JD8EAAAAMBAJ&dq=jeffrey+lindsay&pg=PA92 The shape of the "Crystal Dome" used in the popular TV game show The Crystal Maze was based on a pentakis dodecahedron. In Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron. These Domes also appear whenever the player transforms on a dome in the Hypno Ray level. Some Geodomes in which people play on are Pentakis Dodecahedra. References 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 ) External links Eric W. Weisstein, Pentakis dodecahedron (Catalan solid) at MathWorld. Pentakis Dodecahedron – Interactive Polyhedron Model