Corbelled Domes in Two and Three Dimensions: The Treasury of Atreus
Before the development of the true dome, many ancient cultures used the technique of corbelling to roof spaces. Recently, a series of related statistical models have been proposed in the literature for explaining how corbelled domes might have been constructed. The most sophisticated of these models is based on a piecewise linear structure, with an unknown number of changepoints, to guide the building process. This model is analyzed by the reversible jump Markov Chain Monte Carlo (MCMC) technique. All models considered to date have been two-dimensional, that is, they have taken a single cross section through the dome; even when more extensive data, in the form of measurements on multiple slices through the dome, have been available, these have been averaged together for the purposes of analysis. In this paper, we extend the two-dimensional analysis to a three-dimensional analysis, that takes full advantage of the data collected by the archaeologists and of the rotational symmetries inherent in the structure. We also explore ways of graphically presenting the results from a complex, reversible jump MCMC implementation, in order to check convergence, good mixing, and appropriate exploration of the (high dimensional and varying dimension) parameter space. The model and the graphical techniques are demonstrated on the Treasury of Atreus in Mycenae, Greece, one of the finest extant examples of the corbelling method.