A benefit of geometric forms is that they translate into mathematical relationships. It is this mathematical attribute of geometric models that makes possible to study and investigate the plant design.
Naming the plant forms as mathematical variables and organizing the mathematical relationships for these variables into a sequential collection makes up a mathematical algorithm. This mathematical model matches the steps in the geometric model with sequential mathematical definitions.
The mathematical formulation of the geometric algorithms makes possible to compare predicated dimensions based on a theorized model to actual measurements, drawing conclusions, refining models, and evaluating different theories.
The mathematical model that equates to and is developed as a pair to the geometric modeling has made possible to successfully decode the plant design of Castel del Monte and reveal the complex design.
Going beyond decoding and verification, the mathematical model allows to study and identify with great certainty the unit of measurement used at Castel del Monte.
Besides being an essential tool to study the plant design, the mathematical model reveals a high level of mathematical sophistication at the height of the Middle Ages. It shows a rigorous and scientific approach in the structural design of an edifice, albeit within the framework of the limited scientific knowledge of the times.
The complex design algorithms and the peculiar geometric manipulations are too sophisticated to have happened in one genial inspirational moment or as the fruit of masonry construction tryouts. They were the result of iterative design processes, working with geometric models, supported by mathematical formulations, and possibly including some physical models as well, folio 101:01.
Scholars that have instinctively described Castel del Monte as a model of mathematical precision do not know how true that characterization is, because there is a mathematical model at the basis of the plant design.