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Mathematical Model

The plant design is composed of two geometric algorithms, the main one that covers the overall design concept and a second one that covers a design refinement at the towers. The algorithms produce mathematical formulas at every step where forms are defined. For example, defining the ring space for the towers in the concept design results in a ring whose width t is defined mathematically, folio 102:01, as:

t = r – l

The other feature of these algorithms is the sequencing; like the links of a chain, one form definition leads to the next one. This establishes a succession in how the form variables are identified, and a dependency among them. For example, the measure of the corner notch m comes after the measure of the footing width f is defined, which comes after the tower octagon side s is defined, which comes after the tower measure t is defined, and so on.

The geometric algorithm produce a parallel mathematical algorithm, sequences of simple formulas that derivatively define the variables that stand for the measures of the plant forms. The complete string of formulas starts from the base octagon and successively defines all the plant measures.

This ordered set of formulas makes up a mathematical model for the ground floor plant design. The geometric model was compared to a rubber ball, with the forms expanding and contracting in geometric proportions; the mathematical model has a similar unifying dependency. Once the dimension of the independent variable at the head of the chain is specified, all other form dimensions are also defined by the sequence of formulas.

A natural host for the collection of mathematical definitions is the spreadsheet, such as Microsoft Excel. Formulas are entered in their orderly sequence on successive rows and linked sequentially by spreadsheet formulas. The spreadsheet yields predicated dimensions in actual measurement units.

The spreadsheet makes the collection of formulas a singular and true mathematical model for the plant of Castel del Monte.

The value of the mathematical model is that it provides a means to study and explore the design by predicating dimensions and comparing them to actual measurements.

The geometric algorithm is proved when the various plant forms, such as the octagonal towers and the trapezoidal rooms, are reproduced in all details and dimensions. This is where the mathematical model becomes useful in the study.

Mathematical modeling has been extremely useful in this study; it has been instrumental in the final identification of the correct algorithms and the discovery of the unit of measurement.

Of course the credit goes primarily to the medieval architect that followed a strict geometric design process and the medieval builders that executed the construction so accurately to leave us with a blue print cast in stone.

The final mathematical model has been developed interactively with the geometric exploration in this study. Given the many study evolutions, it is likely that more refined versions may follow in the future.