I don’t know that anyone apart from me calls these Rice pentagonal tilings, but I think everyone ought to.

What is a Rice pentagonal tiling?

In 1975, Martin Gardner wrote a Scientific American column about the kinds of polygons that can tile the plane. He reported an assertion from a 1967 paper that all of the classes of convex pentagons that tile the plane perfectly had been discovered; within a month, a reader (Richard James III) had written to say “hang on, they missed this one.”

Fellow reader Marjorie Rice read this and thought “if they’ve missed one, maybe they missed others” and went to work looking for others. Within a couple of months, she’d developed her own notation for the problem and told Gardner about her first discovery; she would eventually come up with four distinct classes of pentagons that could be arranged to cover the plane. (Before Gardner’s column, only eight were known. The fourteenth – and I believe final ((My attention has been drawn to the fifteenth and final class, discovered by Mann, McLoud and Von Derau in 2015; Michael Rao proved that these were the only possibilities in 2017. Thanks to Adam for letting me know.)) – class was discovered by Stein in 1985).

Cat voice So What is a Rice pentagonal tiling?

Rice’s four tilings have something in common: they cleverly combine a pair of convex pentagons into a concave hexagon, which then tile the plane. I think this is they key insight: all of the tilings can be considered as sets of more or less complex larger shapes that fit together neatly. The patterns are neat, but I think the real beauty is in the maths.

Who was Marjorie Rice?

You’d be justified in asking “so what?”.” People discover things all the time. Rice tilings are famous largely because of her mathematical background, which is to say “none to speak of”. Her post-high-school education had consisted of half of a correspondence art course before she settled down to raise a family; she famously worked on the problem in her free time, hiding her notes when her family showed up.

She was born in St Petersburg, Florida on February 16th, 1923 and died in San Diego, California on July 2nd, 2017.