Toilet Paper Lawsuit

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.

Enhanced image of Kimberly-Clark toilet paper (Hayes, 2017)

Medieval Islamic Architecture

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.

Darm-i-Imam shrine, Iran (Lu and Steinhardt, 2007)

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