Some further reading is given at the end.

Some apparently crystalline structures have no regular periodic
translation unit. They are called quasicrystals and until recently their
structure has been a mystery. Steinhard and Tsai have solved the puzzle
for a layered 2-dimensional structure found in an Al_{72}Ni_{20}Co_{8}
alloy (New Scientist 27 Feb 1997), showing that it comprises overlapping
decagonal atomic clusters. The overlapping pattern of the decagons is
closely related to the Penrose tiling, which comprises two types of rhomb,
sharp and blunt (the sharp ones tend to group in pairs, as non-convex
trapezia). According to the New Scientist article, no 3-D quasicrystal
structure has yet been resolved (but see below).

There is at least one promising candidate for just such a 3-D solution, which is summarised in the figures. Six root 5 rhombs (1) can form either of two rhombic hexahedra (or rhombic parallepipeds) (2); a flattened, oblate one or a sharp, prolate one.

Two of each type of hexahedron (3) form Bilinski's rhombic dodecahedron (4). This dodecahedron fills space - though its packing is periodic, it demonstrates that the various solid angles of the hexahedra can combine to form closed vertices - a necessary precondition for space-filling.

Three more of each hexahedron pack round the rhombic dodecahedron (5) to form the rhombic icosahedron (6), around which five more of each (not shown) pack to form the rhombic triacontahedron (7).

The two rhombic hexahedra are analogous to the two Penrose tiles, and the rhombic triacontahedron to a decagon formed from Penrose tiles. The decagons formed by a Penrose tiling are larger than the decagons of the underlying quasicrystal, so one could expect a similar relationship in any 3-D analogue. In the 2-D quasicrystal, the intersection of two overlapping decagons can also be formed from Penrose tiles. It is also possible to find an intersection of two rhombic triacontahedra formed from rhombic hexahedra. The analogous geometric principles are that all the 2-D polygons are parallel sided (zonogons), and all the 3-D polyhedra are parallel edged (zonohedra). The analogies are so close that the 3-D tiling is sometimes called the 3-D Penrose tiling.

The key to unlocking Steinhard and Tsai's structure is that many decagon tilings are possible but not all can be aligned with (or superimposed on) the Penrose tiling: theirs is just such a non-aligned tiling. This principle has not yet been extended to three dimensions, so we do not know whether tilings of overlapping rhombic triacontahedra exist, whether 'non-Penrose' variants exist, and ultimately whether there exist real quasicrystals with such a structure - or even whether quasicrystals are necessarily non-Penrose.

Some further reading is given below. I do not have the resources to follow it up, but woul be glad to hear from anyone who does.

Guy Inchbald guy@steelpillow.com

**Further reading on dissection of the rhombic
triacontahedron and on quasicrystals:** (Thanks to Alex Day and Kate
Crennell)

British Crystallographic Association.

Hargittai, I (1997) The Story of Quasicrystals, *Chemical
Intelligencer* 3(4), p 25-29.

Hart, George W, *encyclopaedia of polyhedra*
http://www.georgehart.com/virtual-polyhedra/dissection-rt.html.
Dissection of the rhombic triacontahedron.

Kowalewski, Gerhard, *Der Keplersche Korper und andere Bauspiele*,
Koehlers, Leipzig (1938). Shows how to dissect a five-coloured rhombic
triacontahedron into twenty three-coloured rhombic hexahedra (rhombic
parallelepipeds). (In German)

Mackay, A *Physica* 114A p 609-613 (1982). Extends the Penrose
pattern into 3 dimensions.

Schectman, D et al *Phys Rev Letters* 53 p 1951-1953 (1984).
Discovery of a solid which extends the Penrose pattern into 3 dimensions.

US patent No 5603188, "Architectural Body Having a Quasicrystal Structure" references cited include:

- "Quasicrystals," David Nelson,
*Scientific American*, pp 43-57, Aug 1986. - "Quasicrystals," Paul Joseph Steinhardt,
*American Scientist*, vol 74, pp 586-597, Nov-Dec 1986. - "Quasicrystals with arbitrary orientational symmetry,"
Joshua ES Socolar, Paul J Steinhardt and Dov Levine (Socolar),
*Physical Review B*, vol. 32 No. 8, pp 5547-5550, 15 Oct 1985.