2012/06/05 by Michael Beeson, Beeson, Michael
Engineering · Materials Science · Mathematics · #51M20 #FOS: Mathematics #Finite Group Theory Research #Mathematical Dynamics and Fractals #Mathematics and Applications #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1206.2229
openalex publication_date 2012/06/05 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
An N-tiling of triangle ABC by triangle T (the `tile') is a way of writing ABC as a union of N copies of T overlapping only at their boundaries. Let the tile T have angles (α,β,γ), and sides (a,b,c). This paper takes up the case when 3α+ 2β= π. Then there are (as was already known) exactly five possible shapes of ABC: either ABC is isosceles with base angles α, β, or α+β, or the angles of ABC are (2α,β,α+β), or the angles of ABC are (2α, α, 2β). In each of these cases, we have discovered, and here exhibit, a family of previously unknown tilings. These are tilings that, as far as we know, have never been seen before. We also discovered, in each of the cases, a Diophantine equation involving N and the (necessarily rational) number s = a/c that has solutions if there is a tiling using tile T of some ABC not similar to T. By means of these Diophantine equations, some conclusions about the possible values of N are drawn; in particular there are no tilings possible for values of N of certain forms. We prove, for example, that there is no N-tiling with N prime when 3α+ 2β= π. These equations also imply that for each N, there is a finite set of possibilities for the tile (a,b,c) and the triangle ABC. (Usually, but not always, there is just one possible tile.) These equations provide necessary, and in three of the five cases sufficient, conditions for the existence of N-tilings.