The beautiful mathematical structure of Penrose patterns have advanced our understanding of quasicrystals, a new breed of high-tech physical materials discovered in meteorites. Like all physical materials, these are collections of one or a few types of “particles” – atoms, molecules, or larger units – that are arranged in some pattern. The most familiar patterns are crystalline arrangements in which a simple unit is repeated in a regular way.

Periodic pattern of the honeycomb

During last Friday’s Visualization Forum, Josh Socolar, a Duke physics professor, conveyed his enthusiasm for the exotic patterns generated by non-periodic crystalline structures to a large audience munching on barbecue chicken and Alfredo pasta in the LSRC (Levine Science Research Center). Unlike many of the previous talks on visualizing data, Professor Socolar is not trying to find a new technique of visualization, but rather aims to emphasize the importance of visualizing certain structures.

Equations in chemistry for calculating vibrations when a material is heated are often based on the assumption that the material has a uniform structure such as the honeycomb pattern above. However, the atoms of a non-periodic crystalline object will behave differently when heated, making it necessary to revise the simplified mathematical models – since they can no longer be applied to all physical materials.

Quasicrystals, one type of non-periodic structured material, can be represented by the picture below. The pattern contains features with 5-fold symmetry of various sizes (highlighted in red, magenta, yellow, and green).

Quasicrystal structure with 5-fold symmetry

Quasicrystal structure with 5-fold symmetry

Drawing straight lines within each tile – as shown on the bottom half of the diagram below – produces lines running straight through the material with various lengths. Professor Socolar computed the lengths of these line segments and was amazed to discover that they follow the Fibonacci sequence. This phenomenon was recently discovered to occur naturally in icosahedrite, a rare and exotic mineral found in outer space.

Lines drawn through a quasicrystal structure

Lines drawn through a quasicrystal structure

By using software programs like Mathematica, we can create 3D images and animations for the expansion of such quasicrystal structures (a) as well as computing Sierpinski patterns formed when designing other types of non-periodic tile shapes (b).

Still of animation of expanding quasicrystal tiles - that looks like a cup of coffee.

(a) Still of animation of expanding quasicrystal tiles – that looks like a cup of coffee.


(b) Sierpinski triangle pattern drawn for other non-periodic tile shapes

(b) Recolored diagram of

(b) Recolored diagram of Sierpinski triangle pattern

Most importantly, Professor Socolar concludes, neither the Fibonacci nor non-periodic Penrose patterns would have been identified in quasicrystal structures without the visualization tools we have today. With Fibonacci sequence patterns discovered in the sunflower seed spiral as well as in the structure of the icosahedrite meteorite, we have found yet another mathematical point of unity between our world and the rest of the cosmos.

Professor Socolar taking questions from the audience.

Professor Socolar taking questions from the audience.

Post by Anika Radiya-Dixit