Corbelled Domes in Two and Three Dimensions: The Treasury
Nicole A. Lazar, Joseph B. Kadane, Fang Chen, William
G. Cavanagh and Clifford D. Litton
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.