Distinct equilateral triangle dissections of convex regions. (English) Zbl 1265.05091

Donovan, Diane M.; Lefevre, James G.; McCourt, Thomas A.; Cavenagh, Nicholas J.

Commentat. Math. Univ. Carol. 53, No. 2, 189-210 (2012).

The dissection (tiling) of an integer sided equilateral triangle into smaller, integer sided equilateral triangles is a classic problem considered by W. T. Tutte in [Proc. Lond. Math. Soc., II. Ser. 50, 137–149 (1948; Zbl 0030.08102)]. In the paper under review an extra restriction is imposed on the dissection. If such a dissection satisfies the condition that no point is a vertex of more than three of the smaller triangles, it is called a proper triangulation.
In the paper necessary and sufficient conditions for the existence of a proper triangulation of a convex region are given. Moreover, it is established precisely when at least two such equilateral triangle dissections exist. Also the authors provide necessary and sufficient conditions for some convex regions with up to four sides to have either one, or at least two, proper triangulations when an internal triangle is specified.
The results can be applied to latin trades.

Reviewer: Elizaveta Zamorzaeva (Chisinau)

MSC:

05B45	Combinatorial aspects of tessellation and tiling problems
05B15	Orthogonal arrays, Latin squares, Room squares

Keywords:

equilateral triangle dissection; convex region; latin trade

Citations:

Zbl 0030.08102