Introduction

Castel del Monte stands out among medieval castles for its exquisite geometric forms and a missing finality for its construction. The fractal recursion of the octagon in the plant design is fascinating; scholars have endeavored to decipher its geometric design.

A new study unravels this design, showing that it is the realization of a unique solution to cross vault lateral thrusts, utilizing features of the octagon.

A review in the evolution of stone vault construction technology, leading up to Gothic architecture, provides the backdrop to understand and retrace the work of medieval architects at Castel del Monte.

The amazing plant design and the accurate construction have left a blueprint cast in stone. The design is a sequence of linked geometric operations, much like a paper origami. The complete sequence takes the form of an algorithm, one that translates into a mathematical model, because of its geometric foundation.

The stunning feature of this algorithm is that all plant forms are automatically generated and all dimensions spring from a single measure. This is the dimension of the plant octagon diagonal, which is the same as the diagonal of the base square inscribed inside the octagon. This leads to the discovery of the true unit of measurement used at Castel del Monte.

Most significant is the window that this opens in the medieval intellectual ways,
as we follow the architect in solving technical and geometric problems
within the mathematical capabilities in the 13^{th} century;
it is a lesson in architectural design at the height of the Middle Ages.

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