UW Bothell associate profs, student find rare geometry pattern

BOTHELL — One person’s idle doodling is another’s mathematical breakthrough.

Two mathematics professors and one of their former students at the University of Washington at Bothell have made a discovery in mathematics that could have applications in crystallography or self-assembling nanomachines.

Or it could provide an interesting tiling project for the bathroom floor.

Their discovery is in an esoteric branch of geometry called tessellation, or tiling of identical shapes that can cover a two-dimensional plane with no gaps and no overlaps, out to infinity.

Casey Mann and Jennifer McCloud-Mann are both associate professors of mathematics at UW Bothell. They’re also married to each other.

The Manns and a former undergraduate student, David Von Derau, have discovered a pattern of tiling convex irregular pentagons.

It’s only the 15th tiling pattern for pentagons ever discovered, and they only did it with the help of a computer program that Von Derau wrote.

Pentagons present a unique problem in geometry.

Tiling can be easy to grasp at first, but it gets complicated quickly.

There are only three regular polygons — shapes whose interior angles are the same and the sides are the same lengths — that tile in a plane: triangles, quadrangles and hexagons.

They are convex shapes, meaning all the interior angles are less than 180 degrees.

It’s easy to tile triangles and quadrangles because we’ve all seen graph paper, or been stuck on a boring phone call with a pen and notepad handy.

It’s a little trickier to envision tiling with hexagons, but imagine a honeycomb, snowflakes or (if you’re a certain nerdy sort of person) a battle map from a role-playing game.

But that’s it for the regular polygons. Now it gets complicated.

Even with irregular shapes included, it can be mathematically proven, for example, that there are exactly three kinds of convex hexagons that can tile a plane, Casey Mann said. One is regular, two irregular, meaning their angles and sides are of different sizes.

“It can also be proven that if you have seven sides or more you can’t tile a plane with them. They can’t fit around corners,” he said.

It’s possible to tile an irregular pentagon, however. It’s just not easy to figure out the correct pattern.

It also cannot be proven how many different tilings of pentagons exist, Mann said. There might be just 15. There might be an infinite number.

“The truth is we just don’t know. Pentagons really are the odd one,” he said.

The first five pentagonal tilings were discovered in 1918 by a German mathematician, Karl Reinhardt. Then Richard Kershner, of Johns Hopkins University, published a paper in 1968 that identified three more and said that was all of them.

After an article appeared in Scientific American magazine in 1975, several people took it as a challenge. One reader named Richard E. James III found a ninth pattern, and Marjorie Rice, an amateur with only a high school diploma, found four more a couple of years later.

The 14th pattern was found in 1982 by Rolf Stein of the University of Dortmund, Germany, and that was it for the next 33 years.

The latest discovery came about with advances in software modeling.

The Manns proved a mathematical theorem that showed there were a finite number of symmetrical forms in a tiling pattern.

That emphasis on symmetry was the key, Mann said. If a pattern composed of multiple irregular pentagons could be shown to be symmetrical, then it could tile the plane.

“That told us we could write a computer program to search for them,” he said.

Von Derau, who now works as a programmer and researcher for Viavi Solutions in Bothell, said he was finishing his degree in math and needed an elective credit.

His choices were geometry or independent research supervised by Mann.

“Because I didn’t want to take geometry, I asked him if he had a research project he needed help with,” Von Derau said. “It’s kind of ironic.”

The program, running on a University of Washington computer cluster, found the tiling pattern in a few hours’ time, Mann said. The symmetrical pattern is composed of 12 identical irregular pentagons.

Finding more patterns — if they exist at all — will require searching for much more complex symmetries, and that’s going to take much more computing power and time.

“We’re basically reducing the problem to the only way you’re going to find a new one is if it’s really exotic,” Mann said.

“There might end up being an infinite number of types, there might be 22 types,” he said.

The Manns soon hope to submit a paper on their discovery to ArXiv, an online scientific database.

Chris Winters: 425-374-4165; cwinters@heraldnet.com. Twitter: @Chris_At_Herald.