# 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

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).