**This post provides a historical and conceptional perspective for the theoretical discovery of non-periodic 3d space-fillings by Peter Kramer, later experimentally found and now called quasicrystals. See also these previous blog entries for more quasicrystal references and more background material here.
The following post is written by Peter Kramer.
**

When sorting out old texts and figures from 1981 of mine published in *Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells*, Acta Cryst. (1982). A38, 257-264), I came across the figure of a regular pentagon of edge length L, which I denoted as p(L). In the left figure its red-colored edges are star-extending up to their intersections. Straight connection of these intersection points creates a larger blue pentagon. Its edges are scaled up by τ^{2}, with τ the golden section number, so the larger pentagon we call p(τ^{2} L). This blue pentagon is composed of the old red one plus ten isosceles triangles with golden proportion of their edge length. Five of them have edges t_{1}(L): (L, τ L, τ L), five have edges t_{2}(L): (τ L,τ L, τ^{2} L). We find from Fig 1 that these golden triangles may be composed face-to-face into their τ-extended copies as t_{1}(τ L) = t_{1}(L) + t_{2}(L) and t_{2}(τ L) = t_{1}(L) + 2 t_{2}(L).

Moreover we realize from the figure that also the pentagon p(τ^{2} L) can be composed from golden triangles as p(τ^{2} L) = t_{1}(τ L) + 3 t_{2}(τ L) = 4 t_{1}(L) + 7 t_{2}(L).

This suggests that the golden triangles t_{1},t_{2} can serve as elementary cells of a triangle tiling to cover any range of the plane and provide the building blocks of a quasicrystal. Indeed we did prove this long range property of the triangle tiling (see *Planar patterns with fivefold symmetry as sections of periodic structures in 4-space*).

##### An icosahedral tiling from star extension of the dodecahedron.

In the same paper, I generalized the star extension from the 2D pentagon to the 3D dodecahedron d(L) of edge length L in 3D (see next figure) by the following prescription:

- star extend the edges of this dodecahedron to their intersections
- connect these intersections to form an icosahedron

The next star extension produces a larger dodecahedron d(τ^{3}L), with edges scaled by τ^{3}. In the composition of the larger dodecahedron I found four elementary polyhedral shapes shown below. Even more amusing I also resurrected the paper models I constructed in 1981 to actually demonstrate the complete space filling!

These four polyhedra compose their copies by scaling with τ^{3}. As for the 2D case arbitrary regions of 3D can be covered by the four tiles.

The four elementary cells shown in the 1982 paper, Fig. 4. The four shapes are named dodecahedron (d) skene (s), aetos (a) and tristomos (t). |
The paper models from 1981 are still around in 2014 and complete enough to fill the 3D space without gaps. You can spot all shapes (d,s,a,t) in various scalings and they all systematically and gapless fill the large dodecahedron shell on the back of the table. |

##### Quasiperiodicity.

The only feature missing for quasicrystals is aperiodic long-range order which eventually leads to sharp diffraction patterns of 5 or 10 fold point-symmetries forbidden for the old-style crystals. In my construction shown here I strictly preserved central icosahedral symmetry. Non-periodicity then followed because full icosahedral symmetry and periodicity in 3D are incompatible.

In 1983 we found a powerful alternative construction of icosahedral tilings, independent of the assumption of central symmetry: the projection method from 6D hyperspace (*On periodic and non-periodic space fillings of E ^{m} obtained by projection*) This projection establishes the quasiperiodicity of the tilings, analyzed in line with the work

*Zur Theorie der fast periodischen Funktionen (i-iii)*of Harald Bohr from 1925 , as a variant of aperiodicity (more background material here).