Ed Pegg Jr | Searcharxiv

arXiv2026

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

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.

math.GM↗

arXiv2019

There are no Cubic Graphs on 26 Vertices with Crossing Number 10 or 11

We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find good embeddings of each remaining graph. In the cases where the embedding found has 10 or more crossings, the heuristic is re-run with a different settings of parameters until an embedding with fewer than 10 crossings is found. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal.

math.CO↗