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Some joint work with students (Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Isoperimetric pentagonal tilings<\/a>, Notices AMS, 2012) has found least-area tilings by convex pentagons<\/em>:<\/p>\n The Cairo tiles<\/a> are popular in Cairo, as documented by David Bailey and Helen Donnelly<\/a>, also here<\/a>\u00a0and references here<\/a>. My colleague Steve Zottoli<\/a> has observed that they appear in American concrete tiling by Daniel Ogassian<\/a><\/p>\n and that the second, Prismatic tile<\/a>, often called a house, specifically looks like a corn crib:<\/p>\n (\u201cField guide to New England barns and farm buildings\u201d by Thomas Durant Visser). Zottoli also found an article on an Iranian \u201c800-year-old pentagonal tiling…\u201d by Emil Makovicky in Fivefold Symmetry by Istvan Hargittai:<\/p>\n this dargah (shrine) of the mystic Hazrat Iniyat Khan via Roger von Oech<\/a>:<\/p>\n examples from nature:<\/p>\n and from the heavens (via StarDate<\/a>), Auriga the Charioteer:<\/p>\n Another colleague showed me pentagonal faces on pyrite (“Fools’ Gold”) and a model of the dodecahedral pyritohedron<\/a>, which figured in the 1994 Weaire-Phelan counterexample<\/a> to Kelvin’s Conjecture:<\/p>\n Pentagons are everywhere.<\/p>\n Note added 4 June 2012.\u00a0<\/strong>There are 14 known types<\/a> of convex pentagons that tile the plane. Three of the latter ones were discovered in the late 70s by amateur Marjorie Rice (homemaker, another noble profession), who had read about tilings in Martin Gardner\u2019s column in her son\u2019s issue of Scientific American<\/em>. In 1985 Rolf Stein (U Dortmund) found the 14th. Bagina [B2, 2011] proves 8 edge-to-edge convex types (also Sugimoto 2012). Only in 1985 [HH] were the five types of tilings by a single equilateral convex pentagon proved complete (also [B1, 2004]); Hirschorn says they have unpublished classification for nonconvex equilateral.<\/p>\n [B1] \u00a0 O. Bagina, Tiling the Plane with Congruent Equilateral Convex Pentagons, J. Combin. Theory. Ser. A , 2(105):221\u2013232, 2004.<\/p>\n [B2] \u00a0 O. Bagina, Tilings of the plane with convex pentagons (in Russian), Vestnik KemGU 4(48): 63-73, 2011.<\/p>\n [HH] \u00a0M. D. Hirschhorn and D. C. Hunt, Equilateral convex pentagons which tile the plane, J. Comb. Theory 39 (1985), 1-18.<\/p>\n Note added 30 October 2017.\u00a0<\/strong>New type of pentagonal tile discovered<\/a> July 2015 by Casey Mann, Jennifer McLoud-Mann, and\u00a0David Von Derauc<\/a>. And that’s it, as proved\u00a0July 2017 by\u00a0Micha\u00ebl Rao<\/a>, arX,<\/a>\u00a0In a rest area bathroom I found this pentagonal tiling:<\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n