Rhombic enneacontahedron

Rhombic enneacontahedron
Type	zonohedron
Face polygon	rhombus
Faces	90 rhombi: (60 wide and 30 narrow)
Edges	180
Vertices	92
Faces per vertex	3, 5, and 6
Symmetry group	Ih, [5,3], *532
Properties	convex, zonohedron
Net	(click to enlarge)

A rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.

The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2. The face angles of these rhombi are approximately 70.528° and 109.471°. The thirty slim rhombic faces have face vertex angles of 41.810° and 138.189°; the diagonals are in ratio of 1 to φ².

The rhombic enneacontahedron is called a rhombic enenicontahedron in Lloyd Kahn's Domebook 2.

Close-packing density

The packing-fraction of the close-packed crystal formed by rhombic enneacontrahedra is given by:

$\eta = 16 - \frac{34}{\sqrt{5}} \approx 0.7947377530014315$

It was proven that this close-packed value is assumed in a Bravais-type lattice by de Graaf (2011), who also described the lattice. The proof is conditionally dependent on Hales (2005) proof of the Kepler conjecture and the proof of the inscribed-sphere upper bound for the packing of particles by Torquato (2009).

References

• Weisstein, Eric W., "Rhombic enneacontahedron", MathWorld.
• VRML model: George Hart, [1]
• George Hart's Conway Generator Try dakD
• Domebook2 by Kahn, Lloyd (Editor); Easton, Bob; Calthorpe, Peter; et al., Pacific Domes, Los Gatos, CA (1971), page 102
• de Graaf, J.; van Roij, R.; Dijkstra, M. (2011), "Dense Regular Packings of Irregular Nonconvex Particles", Phys. Rev. Lett. 107: 155501, Bibcode:2011PhRvL.107o5501D, doi:10.1103/PhysRevLett.107.155501 
• Torquato, S.; Jiao, Y. (2009), "Dense packings of the Platonic and Archimedean solids", Nature 460: 876, doi:10.1038/nature08239, PMID 19675649 
• Hales, Thomas C. (2005), "A proof of the Kepler conjecture", Annals of Mathematics 162: 1065, doi:10.4007/annals.2005.162.1065 

Further reading

• Rhombic Enneacontahedron Includes a great deal of metric data