In addition to the regular "Platonic" solids already known to the Pythagoreans and described by Plato — the semiregular "archimedean" polyhedrons ascribed to Archimedes — and the non-convex star polyhedrons Johann Kepler discovered and described, he also found two rhombohedrons: the twelve-sided one with its periodic space-filling property, the so-called rhombic dodecahedron, and the thirty-sided triacontahedron, non-packable solid with its five fold symmetry in d, which according to then accepted opinion, does not make a periodic filling of space possible. Both are contained in the Metaeder.

The first is generated via quadratic pyramids on the sides of a cube, the second via three-cornered pyramids on the sides of an icosahedron. The d-position of the rhombic triacontahedron produces an aperiodic surface tessalation, a Penrose pattern - its corresponding forms in space are the quasicrystals discovered by Shechtman in 1984.

Quasicrystals are formed either by the overlapping arrangement of rhombic triacontahedrons or by the aperiodic / complementary packing of rhombohedral solids, all of whose faces are formed with golden rhombuses: two different rhombohedrons, the rhombic dodecahedron, the rhombic icosahedron and/or the rhombic triacontahedron.

This is in fact a classic quasicrystal which, in the d projection, generates the classic Penrose pattern.

A two-dimensional projection in d of the rhombic triacontahedron results in the so-called Penrose pattern.

Another frequently published Penrose pattern consisting of three pentagons, two rhombuses and a crown produces, as a corresponding spatial form, the figure shown below.


The unit cells of a quasicrystal are two different complementary packable golden rhombohedrons, an acute-angled (brown) and an obtuse-angled (green) one, whose sides form golden rhombic figures. These rhombohedrons can be combined in a space-filling manner to form rhombic dodecahedrons, icosahedrons, triacontahedrons and, over and beyond this, quasicrystals. All figures are contained in the Metaeder.


Metaeder at van de Loo in Essen 1960

Two rhombohedrons prolate and obtuse
two prolate + two obtuse
two prolate + two obtuse - rhombic dodecahedron
three prolate+ tree obtuse
four prolate + four obtuse
five prolate + five obtuse
six prolate + six obtuse
seven prolate + seven obtuse
eight prolate + eight obtuse
ten prolate + ten obtuse - rhombic triacontahedron
four rhombic triacontahedrons
space-filling quasicrystal