In 1973, mathematician and physicist Roger Penrose “discovered” his famous aperiodic patterns, which became known as Penrose Tiles. Penrose Tiles are formed by two or more types of geometric primitives, which are fit together to build a closed pattern. The interesting property of such a pattern is, that it never repeats, when shifted horizontally or vertically. Using Penrose Tiles, you can create tilings that have symmetries based on a pentagon, for example, even though a pentagon usually doesn’t tile (a hexagon does, hence the use of hexagons in many industrial use cases, and of course also in bee hives).
The story goes, that Roger Penrose, still proud of his mathematical discovery, once entered a centuries-old mosque in the South of Spain, and to his surprise, the floor was covered with exactly the kinds of patterns that he thought he had invented.
We should not really be surprised. During the Golden Age of Islam, mathematical knowledge grew by leaps and bounds. Our numeric system is based on Hindu-Arabic numerals. The translations of the writings of Al Khwarizmi (الخوارزمي), whose latinized name became famous in the word Algorithm, introduced the decimal system to Europe at the end of the Middle Ages, and helped spark the Renaissance. So it is no surprise that the patterns in the mosque were based on mathematical knowledge, that had to be rediscovered half a millennium later.
In fact, these tiles, called Girih (Persian for “knot”), can be found everywhere in Islamic buildings here in the Middle East.
Source Wikipedia / User Pentocelo — See Wikimedia page for licensing information
For a long time, Penrose Tiles seemed a mathematical curiosity, useful only for ornamentation. Until the discovery of Quasicrystals in the molecular makeup of advanced materials, which won Dan Shechtman the Nobel Prize for Chemistry in 2011. Many Quasi-crystalline alloys have very advantageous properties, specifically, because of their aperiodic nature.
To the best of my knowledge, Penrose Tiles, let alone 3D quasi-crystalline structures, have never been used in engineering. This may sound surprising, given the fact that the mathematical foundation has been around for decades, if not centuries. Then, on the other hand, until Buckminster Fuller made the Buckyball famous, nobody had ever thought of building geodesic domes.
Aerospace hardware usually uses honeycomb or Isogrid structures to reinforce material, which are essentially based on Buckminster Fuller’s work. These regular patterns have the advantage of being easily created on today’s Computer Aided Design systems. They bring with them the many disadvantages of being periodic. Periodic patterns are, for example, not ideal when it comes to vibration and acoustical behavior. If external oscillations have a frequency close to the Eigenfrequency of an object, the structure starts to resonate, which can result in catastrophic destruction, due to fatigue. Aperiodic patterns, very likely, will have much better properties, since they have an ordered (and therefore predictable), but aperiodic structure.
Fatigue due to acoustic oscillations is a huge problem in many applications, but it is particularly relevant in Hypersonic air- and spacecraft. Traditionally, engineers have been able to address this issue, by fine tuning their systems through much trial and error. For Hypersonics, this is not easy, as it usually requires a full test run under realistic conditions. Iteration is costly and takes a lot of time — especially, because the test result is often a rather violent destruction of the test vehicle.
What if we used an aperiodic pattern to reinforce aircraft and spacecraft components? Would those not have a huge advantage over the traditional Isogrids? Furthermore, could we create quasi-crystal-like three-dimensional lattices, that bring the microscopic advantages of quasicrystalline materials to the macroscopic world of airframes?
One challenge of course is to create the design for these objects. A traditional engineer working in a graphical environment such as CAD will never be able to, except in very simple use cases. In the world of Computational Engineering, things are different. Even though Penrose Tiles are far from trivial, let alone Quasi Crystals, these algorithms are readily available and can be applied at scale.
Last year, on one of our long walks, Josefine and I had a deep conversation about Penrose Patterns, and whether they could be used in engineering. While still speculative, she decided to write a public forum post in the Hyperganic Forum in June 2022, explaining how to do this, and elaborating on possible use cases.
She just recreated it from scratch for a side project she is working on with her new company, LEAP 71. It would be extremely interesting to see how these objects perform in the real world.
One fun fact about Computational Engineering — if you have an algorithm for Penrose Tiling, why would you ever use an Isogrid again, unless it’s better for the use case? Just like you would never use an inferior block of computer code in software engineering, if something better exists, you will never utilize a less sophisticated engineering solution, if you have encoded a better one algorithmically.
Let’s discuss.
