That child died but five other children survived.įor Rice, math was an indulgence. Marjorie Rice worked for a time as a commercial artist, until the couple moved to San Diego with their infant son. In 1945, she married Gilbert Rice, a deeply Christian conscientious objector, and they moved to Washington, D.C., where Gilbert was to work in a military hospital. Though she loved learning and particularly her brief exposure to math, poverty and cultural norms prevented her family from even considering that she might attend college. As I report in Quanta today, a new computer-assisted proof by the French mathematician Michaël Rao establishes that there are precisely 15 families of convex pentagons that tile the plane - including the four that Rice discovered.īorn Marjorie Jeuck in Florida, Rice went to a one-room country school where she skipped two grades and studied with the older kids. Dementia prevented her from learning that the pentagon tiling story has finally come to a close, decades after Gardner first called it. As we continue our computerized enumerations, we also hope to gather enough data to start making specific predictions that can be tested.Rice died on July 2 at the age of 94. “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. “Many structures that we see in nature, from crystals to viruses, are comprised of building blocks that are forced by geometry and other dynamics to fit together to form the larger scale structure,” he added. Study of pentagonal tilings is interesting also because of its potential applications. “The problem also has a rich history, connecting back to the 18th of David Hilbert’s famous 23 problems.” “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,” said Casey. But full classification of the pentagons is still an open area of research. And no convex heptagon, octagon, or anything else-gon tiles the plane. It was proved in 1963 that there are exactly three types of convex hexagon that tile the plane. Pentagons remain the area of most mathematical interest when it comes to tilings since it is the only of the ‘-gons’ that is not yet totally understood.Īs mentioned above, all triangles and quadrilaterals tile the plane. “We were of course very excited and a bit surprised to find the new type of pentagon. “We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey. Until last month, when Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell announced last week that they had discovered this little beauty: An amateur mathematician, Rice developed her own notation and method and over the next few years discovered another four types of pentagon that tile the plane. That same year an unlikely mathematical pioneer entered the fray: Marjorie Rice, a San Diego housewife in her 50s, who had read about James’ discovery in Scientific American. Richard James brought the number of types of pentagonal tile up to nine in 1975. Most people assumed Reinhardt had the complete list until half a century later in 1968 when R. And for further clarification, we are talking about convex pentagons, which are most people’s understanding of a pentagon in that every corner sticks out.) He discovered five classes of pentagon that can each be described by an equation. (To clarify, he did not find five single pentagons. The hunt to find and classify the pentagons that can tile the plane has been a century-long mathematical quest, begun by the German mathematician Karl Reinhardt, who in 1918 discovered five types of pentagon that do tile the plane. (A regular pentagon has equal side lengths and equal angles between sides, like, say, a cross section of okra, or, erm, the Pentagon). The regular pentagon cannot tile the plane. Every four-sided shape can also tile the plane. If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to tile the plane.Įvery triangle can tile the plane.
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