On the symmetries of a Grecian urn

Abstract

While musing on the graceful sylvan sceneDepicted on a classic Grecian urn,John Keats declared that “truth” and “beauty” meanThe same, and here this statement’s worth we learn.“Thou still unravish’d bride of quietness!Thou foster-child of Silence and slow Time”,Wrote Keats [1], the um’s calm beauty to express,While he, by permutations of the rhyme,Did generate its group of symmetries,Which is dihedral and of order six.The um and its inversion therefore matchAnd all its patterns replicate in threes:So triplet dancers demonstrate their tricksWhile silent pipers play a triple catch.