12/12/2023 0 Comments Drawing of impossible shapes petagon![]() 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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