In the three dimensional Euclidean space, there are just five types of regular polyhedrons: a regular tetrahedron, cube, regular octahedron, regular dodecahedron
and a regular icosahedron.
The universe may have a particular finite shape, modeled on a 12-sided geometric object known as a dodecahedron
, they propose.
The faces can be bounded by equilateral triangles, as are the tetrahedron, octahedron, and icosahedron; by squares, resulting in the cube; or by regular pentagons, resulting in the dodecahedron
As an example, we reconsider our previous example of the dodecahedron
, although we flatten the dodecahedron
to illustrate the difficulty of finding integer shadows.
This, too, is a difficult figure, which does not comply with the laws of regular space filling and crystallography The dodecahedron
is one of the five Platonic bodies (regular polyhedrons, cf.
A few crude dodecahedrons
visually identified as bornite were found with grossular crystals, diopside crystals and chalcopyrite in 2000-2001.
discovered that rods sprouting from nodes in the shape of dodecahedrons
-- three-dimensional forms having 12 pentagonal faces--could be assembled into polyhedral building units, which in turn could be combined to create an array of larger structures displaying a fivefold symmetry.
The diamonds are typical of the stones recovered to date from the Zapadnya Area, being mainly fragments of dodecahedrons
Gerard Caris succeeds in combining his basic visual elements: regular pentagons (two-dimensional) and pentagonal dodecahedrons
(three-dimensional) into complex structures with apparent ease, but that ease is illusory.
Lustrous, partially gemmy, deep blue (some are purplish blue) spinel crystals are here found as loose singles and clusters, and also in a scapolite-diopside-mica matrix, and, most surprisingly, the crystals are not octahedrons but dodecahedrons
, a few of them ideally formed, most of them somewhat distorted.
If one now takes a look at Caris's combinations of icosahedrons or dodecahedrons
, then they seem to mock the laws of crystallography (Figs.
Sharp, honey-brown to yellow-green dodecahedrons
of andradite reaching 1.