First new geometric pattern discovered in 30 years.
25 Aug 2015

Three mathematicians Casey Mann, Jennifer McLoud and David Von Derau (who is an undergraduate student) from Washington University recently made history by discovering a new class of non-regular pentagon for the first time in thirty years. Just to make clear the importance of this discovery, the team said that for those in the maths world this is the equivalent of discovering a new atomic particle.

The unique thing about this kind of pentagon is that it can “tile the plane” which refers to shapes that can tile an entire surface without leaving any gaps or overlapping itself. Basic squares and triangles can tile the plane whereas tiling the plane with pentagons is normally impossible, making non-regular pentagons that are able to do this all the rarer.

This new discovery is now one of 15 of these shapes that have been discovered but it is thought that there could potentially be many more left to discover.

New PentagonAbove: The new pentagon shape, shown in different colours to exhibit its tiling pattern.

“The problem of classifying convex pentagons that tile the plane is a beautiful mathematical problem that is simple enough to state so that children can understand it, yet the solution to the problem has eluded us for over 100 years,” according to Casey. “The problem also has a rich history, connecting back to the 18th of David Hilbert’s famous 23 problems.”

These kinds of geometric patterns have appeared frequently in nature (e.g. Honeycombs, plant cells, turtle shells as well as in different kinds of viruses) and it's thought that the study of these shapes could have a range of benefits from creating more structurally stable architecture, to being better able to combat viruses like the flu (and yes I suppose there are benefits for floor tiler makers as well!).

The quest to identify all possible non regular pentagons has actually been going on for the past hundred years, starting with the German mathematician Karl Reinhardt who managed to identify five of these shapes. It was thought that Reinhardt had completed the list until in 1968 R. B. Kershner discovered another three. In 1973 Richard James thought he had finally finished this list with a total count of nine, until in 1975 Marjorie Rice an amateur mathematician based in San Diego began spending her free time figuring out more of these shapes eventually going on to discover another four of them by 1977 bringing the total number up to 13, that number was later raised in 1985 to 14 by Rolf Stein. This new shape was the first to be discovered in 30 years.

Fifteen non regular pentagonsAbove: All 15 known tiling pentagon shapes discovered to date.

When asked about the team discovering more of these elusive shapes Casey replied “I am too cautious to make predictions about whether or not more pentagon types will be found, but we have found no evidence preventing more from being found and are hopeful that we will see a few more. As we continue our computerized enumerations, we also hope to gather enough data to start making specific predictions that can be tested.”

So while it seems that we may not have to wait another thirty years it's still probably a bad idea to be holding your breath in anticipation for some new plane tiling shapes to appear any time soon!

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