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Penrose tilings have been seen in nature as well. Notably, some three-dimensional solids have atomic structures that are aperiodic and contain 10-fold symmetry. This was previously thought to be strictly forbidden, as ordinary crystals are strictly periodic. These were discovered in the late 20th century by Dan Shechtman and are called “quasicrystals” due to their unique nature. It was previously believed that only 2, 3, 4, or 6-fold symmetries were found in crystals. An examination of the patterns in these structures showed that atoms form clusters that resemble 3D Penrose tilings and follow these rules.

The discovery of these quasicrystal 3D formations went on to win the Nobel prize in chemistry in 2011. At first, it was thought that the quasicrystals were not stable enough to be naturally occurring and could only be made in a lab, but then two naturally occurring quasicrystals of extraterrestrial origin, dating back billions of years to about the time the solar system formed, were discovered in a meteorite recovered from Russia, in 2009 and 2015 respectively. Now, the discovery of 10-fold symmetry in quasicrystals is being used for non-stick coatings in cooking appliances, as well as in developing stronger steel for armor and medical purposes. We will further discuss the quasicrystal phenomenon in the next section.

Another example of Penrose tilings occurring in the real world, although this is not necessarily naturally occurring, is in Kleenex tissues. In 1997, Roger Penrose filed a lawsuit against the company in 1997, claiming that they had used Penrose tiling patterns to develop a thicker, softer type of toilet paper. Due to its aperiodicity, it did not bunch up. His lawsuit claimed that the product was a form of copyright infringement, as he had obtained a patent for the Penrose tilings decades prior, following his discovery. Ultimately, Kimberly-Clark withdrew the product. The image below is a high-contrast photo of a sample of the toilet paper; a pattern resembling the rhombi Penrose tilings is clearly visible (Hayes, 2017).

Penrose tilings can also be found in old decorative patterns found in North Africa and the Middle East. Academics have noted that several of these medieval patterns, found for example in “girih” patterns in Islamic shrines, showed Penrose tiling patterns, centuries before the concept was mathematically developed and recorded. The image below shows these patterns found on a girih tile pattern in the Darb-i-Imam shrine in Iran (Lu and Steinhardt, 2007).

Penrose Tiling Home

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

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