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.
A very few specimens of grossular were observed to contain metallic dodecahedrons
believed to be bornite.
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
. Creating a definition that includes these five solids and only these five solids requires careful consideration of properties and their relationships.
As an example, we reconsider our previous example of the dodecahedron
, although we flatten the dodecahedron
to illustrate the difficulty of finding integer shadows.
(1) In 1984 it was confirmed that nature, too, uses pentagons and dodecahedrons
. In a rapidly cooled alloy of aluminium and manganese a crystalline structure with fivefold symmetry was clearly formed.
of platonic solids: dodecahedrons
and icosahedrons that have epitomized human conception on an abstract plane since the first millennium BCE.
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.
in Albuquerque, N.M., 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.
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.
Sharp, honey-brown to yellow-green dodecahedrons
of andradite reaching 1.5 cm individually form clusters to 12 cm across, some on matrix of massive black hematite.
If one now takes a look at Caris's combinations of icosahedrons or dodecahedrons
, then they seem to mock the laws of crystallography (Figs.