He sees his pentagon proof as an early milestone on this much larger quest. Its name has nothing to do with the famous physicist but rather is German for “one stone.” “For everybody who works on tiling, this is a kind of holy grail,” Rao said. Rao, like many tiling specialists, seeks the elusive “einstein,” a hypothetical shape that can only tile the plane nonperiodically, in a pattern of tile orientations that never repeats. Rao still has to submit the proof for peer-reviewed publication, but Hales already feels confident that it holds up.Īs one journey - the classification of all convex polygon tessellations - ends, another is just beginning. Thomas Hales, a professor of mathematics at the University of Pittsburgh and a leader in using computer programming to solve problems in geometry, has independently reproduced the most important half of Rao’s proof, indicating that there isn’t a bug. “Rao beat us to the punch,” he said, adding wryly, “Which is great, because it saves us a lot of work.” Mann said he and his collaborators had been working on taking a partial step toward an exhaustive proof when they heard the news from France. (No tilings by convex polygons with seven or more sides exist.) His proof closes the field of convex polygons that tile the plane at 15 pentagons, three types of hexagons - all identified by Reinhardt in his 1918 thesis - and all quadrilaterals and triangles. In the end, his algorithm determined that only the 15 known pentagon families can do it. In his new computer-assisted proof, Rao identified 371 possible scenarios for how corners of pentagons might come together in a tiling, and then he checked them all. When Rao heard about Mann and his team’s discovery, he set out to do an exhaustive search that would complete the classification of tessellating convex pentagons once and for all. Then, in 2015, Casey Mann, an associate professor of mathematics at the University of Washington, Bothell, and collaborators used a computer search to discover a 15th type of tessellating convex pentagon.
(Rice found four and a computer programmer named Richard James found one.) The list of families grew to 13 and, in 1985, to 14.
But soon after, lay readers like Marjorie Rice, a San Diego housewife with a high school math education, discovered new tessellating pentagon families beyond those known to Kershner. News of Kershner’s pentagon claim spread to the masses in 1975 when it appeared in Martin Gardner’s popular math column in Scientific American. But Kershner’s paper left out the proof that his list was exhaustive “for the excellent reason,” reads an introductory note, “that a complete proof would require a rather large book.” Then, in 1968, Richard Kershner of Johns Hopkins University discovered three more types of tessellating convex pentagons and claimed to have proved that no others existed. Reinhardt didn’t know whether his five families completed the list, and progress stalled for 50 years. In his 1918 doctoral thesis, the German mathematician Karl Reinhardt identified five types of irregular convex pentagons that tile the plane: They were families defined by common rules, such as “side a equals side b,” “ c equals d,” and “angles A and C both equal 90 degrees.” But squash and stretch a pentagon into an irregular shape and tilings become possible.
The ancient Greeks proved that the only regular polygons that tile are triangles, quadrilaterals and hexagons (as now seen on many a bathroom floor). Try placing regular pentagons - those with equal angles and sides - edge to edge and gaps soon form they do not tile.
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