“A small molecule might be called ‘the germ of a solid’. Starting from such a small solid germ, there seem to be two different ways of building up larger and larger associations. One is the comparatively dull way of repeating the same structure in three directions again and again. That is the way followed in a growing crystal. Once the periodicity is established, there is no definite limit to the size of the aggregate. The other way is that of building up a more and more extended aggregate without the dull device of repetition. That is the case of the more and more complicated organic molecule in which every atom, and every group of atoms, plays an individual role, not entirely equivalent to that of many others (as is the case in a periodic structure). We might quite properly call that an aperiodic crystal or solid and express our hypothesis by saying: ‘We believe a gene – or perhaps the whole chromosome fibre’ – to be an aperiodic solid.” – Erwin Schrödinger, What is Life? (1944) chapter entitled ‘The Aperiodic Solid’

Crystals are structures that derive their unique properties (optical transparency, strength, etc.) from the tight packing, symmetric structure of the atoms that comprise them – like quartz, ice, or diamonds. There are only so many ways atoms can be packed together in a periodic pattern to form a two-dimensional crystal: rectangles and parallelograms (i.e. 2-fold symmetry), triangles (3-fold), squares (4-fold), or hexagons like snowflakes or honeycombs (6-fold). These shapes can be connected tightly to one another leaving no gaps in between. Moreover, there is no limit on how extensive crystals can be since attaching more atoms is just a matter of repeating the pattern. Mathematically, we can tessellate an infinite plane with these shapes. Other shapes, like pentagons, don’t work. There are always gaps. In fact, mathematicians have proven no other symmetries are allowed in crystals! These symmetries were “forbidden” in nature and crystallographers never expected to see them. But, in 1982, Dan Shechtman did! When studying the structure of a lab-created alloy of aluminum and manganese () using an electron microscope, he saw a 5-fold symmetric diffraction pattern (Bragg Diffraction) [see Figure 20]. Most crystallographers were skeptical. Shechtman spent two years scrutinizing his work, and, after ruling out all other possible explanations, published his findings in 1984. Turns out, what he discovered was a quasicrystal. In 2011 he was awarded the Nobel Prize in chemistry for his discovery.

Figure 20: Electron diffraction pattern of an icosahedral Zn-Mg-Ho quasicrystal by Materialscientist (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons

Quasicrystals were not supposed to exist in nature because they were thought to require long-range forces to develop. The forces that were thought to guide atomic assembly of crystals, electromagnetic Coulomb forces, are dominated by local (nearest neighbor) interactions. Still, today, we can make dozens of different quasicrystals in the lab, and, they have been found a handful of times in nature. Physicists have postulated that the non-local effects of quantum mechanics are involved and this is what enables quasicrystals to exist.

Figure 21: Example of 5-fold symmetry may be indicative of biological quasicrystals. (First) flower depicting 5-fold symmetry from “Lotsa Splainin’ 2 Do”, (second) plant with 5-fold symmetric spiral from www.digitalsynopsis.com, (third) starfish from www.quora.com, (last) Leonardo Da Vinci’s “The Vitruvian Man” (1485) via Wikipeida

There is evidence of quasicrystals in biological systems as well: protein locations in the bovine papilloma virus appear to show dodecahedral symmetry [see figure 22], the Boerdijk-Coxeter helix (which forms the core of collagen) packs extremely densely and is proposed to have a quasicrystalline structure, pentameric symmetry of neurotransmitters may be indicative of quasicrystals, and general five-fold symmetries in nature [see figure 21] may also be indicative of their presence. Also, the golden ratio which appears frequently in biological systems is implicit in quasicrystal geometry.

Figure 22: Protein locations in a capsid of bovine papilloma virus. (a) Experimental protein density map. (b) Superimposition of the protein density map with a dodecahedral tessellation of the sphere. (c) The idealized quasilattice of protein density maxima. Kosnetsova, O.V.  Rochal, S.B.  Lorman, V.L. “Quasicrystalline Order and Dodecahedron Geometry in Exceptional Family of Viruses“, Phys. Rev. Lett., Jan. 2012, Hat tip to Prescribed Evolution.

Aperiodic tilings give a mathematical description of quasicrystals. We can trace the history of such tilings back to Johannes Kepler in the 1600’s. The most well-known examples are Penrose tilings [see figure 23], discovered by Roger Penrose in 1974. Penrose worked out that a 2-D infinite plane could, indeed, be perfectly tessellated in a non-periodic way -first, using six different shapes, and later with only two. Even knowing what two shapes to use, it is not easy to construct a tiling that will cover the entire plane (a perfect Penrose tile). More likely is that an arrangement will be chosen that will lead to an incomplete tiling with gaps [see figure 23]. For example, in certain two-tile systems, only 7 of 54 combinations at each vertex will lead to a successful quasicrystal. Selected randomly, the chance of successfully building a quasicrystal quickly goes to zero as the number of vertices grows. Still, it has been shown that in certain cases it is possible to construct Penrose tiles with only local rules (e.g. see “Growing Perfect Quasicrystals“, Onoda et al., 1988). However, this is not possible in all cases, e.g. quasicrystals that implement a one-dimensional Fibonacci sequence.

Figure 23: (Left) A failed Penrose tiling. (Right) A successful Penrose tiling. Both are from Paul Steinhardt’s Introduction to Quasicrystals here.

Phasons are a kind of dynamic structural macro-rearrangement of particles. Like phonons they are a quasiparticle. Several particles in the quasicrystal can simultaneously restructure themselves to phase out of one arrangement and into another [see Figure 24-right]. This paper from 2009 entitled “A phason disordered two-dimensional quantum antiferromagnet” studied a theoretical quasicrystal of ultracold atomic gases in optical lattices after undergoing phason distortions. The authors show that delocalized quantum effects grow stronger with the level of disorder in the quasicrystal. One can see how phason-flips disorder the perfect quasicrystaline pattern [see Figure 24-left].

Figure 24: (Left) The difference between an ordered and disordered quasicrystal after several phason-flips from “A phason disorder two-dimension quantum antiferromagnet” by A. Szallas and A. Jagannathan. (Right) HBS tilings of d-AlCoNi (a) boat upright (b) boat flipped. Atomic positions are indicated as Al¼white, Co¼blue, Ni¼black. Large/small circles indicate vertical position. Tile edge length is 6.5A˚. Caption and image from “Discussion of phasons in quasicrystals and their dynamics” by M. Widom.

Figure 25: Physical examples of quasicrystals created in the lab. Both are from Paul Steinhardt’s “Introduction to Quasicrystals“.

In 2015 K. Edagawa et al. captured video via electron microscopy of a quasicrystal, , growing. They published their observations here: “Experimental Observation of Quasicrystal Growth“. This write-up, “Viewpoint: Watching Quasicrystals Grow” by J. Jaszczak, provides an excellent summary of Edagawa’s findings and we will follow it here: certain quasicrystals, like this one, produce one-dimensional Fibonacci chains. A Fibonacci chain can be generated by starting with the sequence “WN” (W for wide, N for narrow referring to layers of the quasicrystal) and then use the following substitution rules: replace “W” with “WN” and replace “N” with “‘W”. Applying the substitutions one time transforms “WN” into “WNW”. Subsequent application expands the Fibonacci sequence: “WNWWN”, “WNWWNWNW”, “WNWWNWNWWNWWN”, and so on. The continued expansion of the sequence cannot be done without knowledge of the whole one-dimensional chain. Turns out that when new layers of atoms are added to the quasicrystal, they are usually added incorrectly leaving numerous gaps [see Figure 26]. This creates “phason-strain” in the quasicrystal. There may be, in fact, several erroneous layers added before the atoms undergo a “phason-flip” into a correct arrangement with no gaps.

Figure 26:
Portion of an ideal Penrose tiling illustrating part of a Fibonacci sequence of wide (W) and narrow (N) rows of tiles (green). The W and N layers are separated by rows of other tiles (light blue) that have edges perpendicular to the average orientation of the tiling’s growth front. The N layers have pinch points (red dots) where the separator layers touch, whereas the W layers keep the divider layers fully separated. An ideal tiling would require the next layer to be W as the growth front advances. However, Edagawa and colleagues observed a system in which the newly grown layer would sometimes start as an N layer, until a temperature-dependent time later upon which it would transition through tile flipping to become a W layer. (graph and caption are from Jaszczak, J.A. APS Physics)

How does nature do this? Non-local quantum mechanical effects may be the answer. Is the quasicrystal momentarily entangled together so that it not only may be determined what sort of layer, N or W, goes next, but also, so that the action of several atoms may be coordinated together in one coordinated phason-flip?

One cannot help but wonder, does quantum mechanics understand the Fibonacci sequence? In other words, has it figured out that it could start with “WN” and then follow the two simple substitution rules outlined above? This would represent a rather simple description (MDL) of the quasicrystal. And, if so, where does this understanding reside, i.e. where is the quasicrystal’s DNA? Suffice it to say, it has, at the very least, figured out something equivalent. In other words, whether it has understood the Fibonacci sequence or not, whether it has understood the substitution rules or not, it has developed the equivalent to an understanding as it can extend the sequence! So, even if quantum mechanics did not keep some sort of log, or blueprint of how to construct the Fibonacci quasicrystal, it certainly has the information to do so!