How Does One Cut a 30-60-90 Triangle?

Abstract

We enumerate dissections of a 30-60-90 triangle into smaller triangles similar to itself, counting unlabeled arrangements up to similarity and reflection. There are 18 four-piece, 121 five-piece, and 844 six-piece arrangements, corresponding to 11, 47, and 235 distinct multisets of piece areas. We present illustrated catalogs organized by these area multisets and distinguish layouts obtained by subdivision and local rotations or reflections from independent layouts. A separately implemented computational search reproduces every arrangement up to nine pieces, finding 460404 arrangements. Counts through nine pieces for five other classes of triangles are included for comparison.