February 16, 1923
St. Petersburg, Florida
|Died||July 2, 2017 (aged 94)|
San Diego, California
|Known for||Discovery of new tessellating pentagons|
Marjorie Rice (née Jeuck) was an American amateur mathematician most famous for her discoveries in geometry. Rice was born February 16, 1923 in St. Petersburg, Florida, and died July 2, 2017 in San Diego, California, where she was living with her son and his wife.
Marjorie Rice was a mother of five, who had become an ardent follower of Martin Gardner's long-running column, "Mathematical Games", which appeared monthly, 1957–1986, in the pages of Scientific American magazine. By the 1970s, Gardner was a popular science writer and amateur mathematician. Rice said later that she would rush to grab each issue from the mail before anyone else could get it, especially her son who subscribed to the magazine.
In 1975, Rice read Gardner's column that discussed what kinds of convex polygons can fit together perfectly without any overlaps or gaps to fill the plane. In his column, Gardner indicated that "the task of finding all convex polygons that tile the plane …. was not completed until 1967 when Richard Brandon Kershner … found three pentagonal tilers that had been missed by all predecessors who had worked on the problem". Gardner was repeating Kershner's claim that the list of convex pentagon tilers was complete. But within a month, Gardner received an example of a new convex pentagon tiler, and published this news in his December 1975 column.
Inspired by this new discovery, Rice decided to try to find other new pentagon tilers. Despite having only a high-school education, she began devoting her free time to discovering new pentagonal tilings, ways to tile a plane using pentagons. She worked on the problem in her free time and through the 1975 holiday season "by drawing diagrams on the kitchen table when no one was around and hiding them when her husband and children came home, or when friends stopped by". She even developed her own system of notation to represent the constraints on and relationships between the sides and angles of the pentagons.
By February 1976, she had discovered a new pentagon type and its variations in shape and drew up several tessellations by these pentagon tiles. She mailed her discoveries to Gardner using her own home-made notation. He, in turn, sent Rice's work to Doris Schattschneider, an expert in tiling patterns, who was skeptical at first, saying that Rice's peculiar notation system seemed odd, like "hieroglyphics". But with careful examination, she was able to validate Rice's results.
By October 1976, Rice had discovered 58 pentagon tilings that needed two pentagons stuck together in order to tile "transitively" (most of them previously unknown), which she arranged into 12 classes. By December 1976, she had discovered two additional new types of tessellating pentagons and over 75 distinct tessellations by pentagons that were in blocks that could be seen as "double hexagons". In December 1977, she made her fourth discovery of a new type of pentagon tiler and by then had enumerated 103 "2-block transitive" pentagon tilings. In the following decade, she discovered several more tiling patterns by pentagons and explored aperiodic tilings.
Rice had a keen interest in art, and had completed half of a correspondence course in commercial art before she married. Throughout her investigations, she explored how to use pentagonal tilings as grids on which to overlay beautiful tessellations of flowers, shells, butterflies and bees.
Rice's discoveries were never published in Gardner's Scientific American columns, but were revealed in an addendum to his original column that was included in his 1988 collection of columns, where he declared her discoveries "fantastic achievements".
Representative tilings by four tiling pentagons discovered by Marjorie Rice
|Type 9||Type 11||Type 12||Type 13|
b = c = d = e
2A + C = D + 2E = 360°
2a + c = d = e
A = 90°, 2B + C = 360°
C + E = 180°
2a = d = c + e
A = 90°, 2B + C = 360°
C + E = 180°
d = 2a = 2e
B = E = 90°, 2A + D = 360°
In 1995, at a regional meeting of the Mathematical Association of America held in Los Angeles, Schattschneider convinced Rice and her husband to attend her lecture on Rice's work. Before concluding her talk, Schattschneider introduced the amateur mathematician who had advanced the study of tessellation. "And everybody in the room . . . gave her a standing ovation."
- In 1999, one of the tilings discovered by Rice became a reality in glazed ceramic tiles, and was installed in the foyer of the headquarters of the Mathematical Association of America in Washington, D.C.
- Rice's papers and materials in support of her mathematical discoveries are preserved at the Eugène Strens Recreational Mathematics Collection at the University of Calgary Library, Alberta, Canada.
- Wolchover, Natalie (July 11, 2017). "Marjorie Rice's Secret Pentagons". Quanta Magazine.
- Schattschneider, Doris (Spring 1996). "Perplexing Pentagons". Discovering Geometry Newsletter. 7 (1). OCLC 1001465604. Archived from the original on August 13, 2016.
- Cole, K. C. (March 11, 1998). "Beating the Pros to the Punch". Los Angeles Times. p. 1. ProQuest 421258615.
- Gardner, Martin, "On tessellating the plane with convex polygon tiles", Scientific American (July 1975), pp. 112–117
- Gardner, Martin, "A random assortment of puzzles, together with reader responses to earlier problems", Scientific American (December 1975), pp. 116–119
- Schattschneider, Doris (1981). "In Praise of Amateurs". In Klarner, David A. (ed.). The Mathematical Gardner (PDF). Boston: Prindle, Weber & Schmidt. pp. 140–166. doi:10.1007/978-1-4684-6686-7_16. ISBN 978-1-4684-6688-1. Reprinted as Mathematical Recreations: A Collection in Honor of Martin Gardner, Mineloa, NY: Dover, 1998
- Schattschneider, Doris (2018). "Marjorie Rice and the MAA tiling". Journal of Mathematics and the Arts. 12 (2–3): 114–127. doi:10.1080/17513472.2018.1453740. S2CID 125461075.
- Schattschneider, Doris (2017). "Marjorie Rice (16 February 1923–2 July 2017)". Journal of Mathematics and the Arts. 12 (1): 51–54. doi:10.1080/17513472.2017.1399680.
- Gardner, Martin, "Tiling with Convex polygons", Chapter 13 in Time Travel and Other Mathematical Bewilderments, W.H. Freeman, New York, 1988, pp. 163–176
- Rice, Marjorie (2003). "Escher-Like Patterns from Pentagonal Tilings". In Schattschneider, Doris; Emmer, Michele (eds.). M.C. Escher's Legacy: A Centennial Celebration. Berlin: Springer. pp. 244–251. doi:10.1007/3-540-28849-X_24. ISBN 978-3-540-20100-7.
- Schattschneider, Doris (1978). "Tiling the Plane with Congruent Pentagons" (PDF). Mathematics Magazine. 51 (1): 29–44. doi:10.1080/0025570X.1978.11976672. JSTOR 2689644. Reprinted with Afterword in The Harmony of the World: 75 Years of Mathematics Magazine, eds. Gerald Alexanderson and Peter Ross, Mathematical Association of America, 2007, pp. 175–190
- Schattschneider, Doris (1978). "Errata". News & Letters. Mathematics Magazine. 51 (4): 256. doi:10.1080/0025570X.1978.11976726. JSTOR 2689477.
- Schattschneider, Doris (2018). "Marjorie Rice and Her Pentagonal Tilings" (PDF). Bridges Stockholm 2018. Phoenix: Tessellations Publishing. pp. 1–2. ISBN 978-1-938664-27-4.