MLD to the more popular chair tiling, this version allows a simple translation into a coloured lattice: Replace each square of type i (1,2,3, or 4) with its midpoint, and assign to it colour i. Then each set of all points of colour i is a model set with internal p-adic space with p=2. This was first shown in [BMS98], a general framework is given in [LMS03].
Lee, J E S and Moody, R V and Solomyak, B
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
Discrete and Computational Geometry 2003, 29, pp. 525-560, MR1702375