Archive for January 2015

New Optimal Pentagonal Tilings

Updated with new discoveries 31 January —11 February 2015 and 3 April 2019; first published 27 May 2014. (Incidentally, new type of pentagonal tile discovered July 2015 by Casey Mann, Jennifer McLoud-Mann, and David Von Derauc. And that’s it, as proved July 2017 by Michaël Rao, arX). For these examples and a proof that symmetry groups with order three rotations cannot occur, see John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng, Symmetries of Cairo-Prismatic tilings, Rose-Hulman Und. Math. J. 17 (2016),

A joint paper [C1] with my SMALL undergraduate research Geometry Group found least-perimeter pentagonal unit-area tiles, Cairo and Prismatic:

They proved that mixtures of unit-area convex pentagonal tiles can do no better, but found many examples of Cairo-Prismatic tilings that do equally well [C1, C2], one appearing in the new Math Library at Williams:

Since their work nine more have been discovered.  Continue reading ‘New Optimal Pentagonal Tilings’ »