It is also possible to construct sets of nontransitive dodecahedra such that there are no repeated numbers and all numbers are primes.
12.
During the hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe.
13.
The three dodecahedra are all stellations of the regular convex dodecahedron, and the great icosahedron is a stellation of the regular convex icosahedron.
14.
The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.
15.
It is a somewhat rare mineral sought after by collectors as it typically forms euhedral isometric crystals exhibiting various cubes, octahedra, and dodecahedra.
16.
There are 4800 edges of two types : 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron.
17.
There are 4800 edges of two types : 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron.
18.
The small stellated dodecahedron and the great icosahedron are facettings of the convex dodecahedron, while the two great dodecahedra are facettings of the regular convex icosahedron.
19.
So do regular geometric figures and pattens with 5-fold symmetry ( e . g ., pentagons, pentagrams, dodecahedra, Penrose tilings ).
20.
Simulations combining local improvement methods with random packings suggest that the lattice packings for icosahedra, dodecahedra, and octahedra are optimal in the broader class of all packings.
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