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Mathematical Model

The plant design is composed of two geometric algorithms, a concept design and a design change at the towers. The algorithms result in mathematical formulas at every step where new forms are defined. For example, defining the ring space for the towers in the concept design results in a ring whose width t (Fig. 2) is defined mathematically:

t = r – l

Figure 2. Example of geometric construction resulting in mathematical definitions.

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 algorithms produce parallel mathematical algorithms, 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 square and successively defines all the plant measures.

This ordered set of formulas makes up a mathematical model for the ground floor design. The mathematical expressions have a linear dependency that reflects the geometry linear relationships. Like in the stretching or shortening of a rubber string, the geometric forms and their dimensions, defined in the mathematical model, increase of decrease linearly with any change to the dimension of the base square.

A natural host for the mathematical model is a spreadsheet, such as Microsoft Excel. Formulas for form dimensions are entered on successive rows of a spreadsheet in the sequence they are defined in the geometric algorithm. The dimension of the base square is the given datum at the head of this series of mathematical expressions. The spreadsheet then calculates the resulting values for each form dimension on the following rows.

The spreadsheet makes such a set of formulas a singular collection, a mathematical model for the plant design.

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

Weather on a spreadsheet nowadays or by long hand on paper 800 years ago, the geometric design model has an inextricable parallel mathematical model, because geometric forms have dimensions defined mathematically.

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 a blueprint cast in stone.

The final mathematical model has been developed interactively with the geometric exploration in this study.