Quasicrystals

The existence of quasicrystals in matter was firmly believed by the scientific community to be absolutely impossible.

Then Paul Steinhardt predicted that they must exist.

Then Dan Schechtman discovered them in matter. Synthetic matter, but matter.

And then they were discovered in nature. In meteor fragments – but nature.

A quasicrystal is an aperiodic, but not random, pattern. A quasicrystal in any given dimension is created by projecting a crystal – a periodic pattern – from a higher dimension to a lower one. For example, imagine projecting a 3-dimensional checkerboard – or cubic lattice made of equally sized and equally spaced cubes – onto a 2D plane at a certain angle. The 3D cubic lattice is a periodic pattern that stretches out infinitely in all directions. The 2-dimensional, projected object is not a periodic pattern. Rather, it is distorted due to the angle of projection, and instead of containing only one shape that repeats infinitely like the 3D crystal does, it contains a finite number of different shapes (called proto-tiles) that combine with one another in specific ways governed by a set of mathematical/geometrical rules to fill the 2D plane in all directions. It is possible, with the correct mathematical and trigonometrical toolkit to actually recover the mother object in 3D (the cubic lattice in this example) by analyzing the 2D projection. A famous example of a 2D quasicrystal is the Penrose tiling conceived by Roger Penrose in the 1970’s, in which a 2D quasicrystal is created by projecting a 5-dimensional cubic lattice to a 2D plane.

Emergence theory focuses on projecting the 8-dimensional E8 crystal to 4D and 3D. When the fundamental 8D cell of the E8 lattice (a shape with 240 vertices known as the “Gosset polytope”) is projected to 4D, two identical, 4D shapes of different sizes are created. The ratio of their sizes is the golden ratio. Each of these shapes are constructed of 600 3-dimensional tetrahedra rotated from one another by a golden ratio based angle. We refer to this 4D shape as the “600-Cell”. The 600-cells interact in specific ways (they intersect in 7 golden-ratio related ways and “kiss” in one particular way) to form a 4D quasicrystal. We then project this 4D quasicrystal to 3D to form a 3D quasicrystal that has one type of proto-tile: a 3D tetrahedron.

To learn more on the fascinating new world of quasicrystals, here are some academic papers on the subject:

Dan Shechtman, Ilan Blech (1984). “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.”

Alan Mackay (1982). “Crystallography and the Penrose Pattern.”

Veit Elser, N.J.A Sloane. “A Highly Symmetric Four Dimensional Quasicrystal”.

Michael Baake, Franz Gähler. “Symmetry Structure of the Elser-Sloane Quasicrystal”.

Justus A. Kromer, et al. “What Phasons Look Like: Particle Trajectories in a Quasicrystalline Potential.”

Marjolein N. van der Linden, Jonathan P.K. Doye, Ard A. Louis (2012) “Formation of dodecahedral quasicrystals in two-dimensional systems of patchy particles”

Pablo F. Damasceno, et al. (2012) “Predictive Self-Assembly of Polyhedra into Complex Structures.”

John Gardiner (2012) “Fibonacci, quasicrystals and the beauty of flowers.”

Kleman, Maurice (2011). Cosmic Forms.

Amir Haji-Akbari, et al. (2009) “Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra.”

Kleman, Maurice (2002). Phasons and the Plastic Deformation of Quasicrystals.