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Modified Design Epilogue
The plant design modification is a geometric algorithm that starts with a quantum enlargement of the tower and then follows trough to ensure that:

Key forms retain the measurements determined in the concept design

The plant minor diagonals that mark the path for cross vault lateral thrusts meet the dual condition of:

Marking the cross vault location with their intersections

Passing through the critical corners M, the 90ºintersection points between the tower and the façade wall

The design changes are imperceptible overall; scholars that take notice explain them as deviations for an otherwise perfect geometry.
There are indeed some minor and localized deviations in the construction. H. Götze, for example, highlights the imperfection in the octagonal form of the courtyard perimeter.
The tower design modification was an intended planning change that was implemented with extreme precision. It is a testimonial to the high level of mathematical precision and the high mastery of stone construction at the height of the Middle Ages.
The geometric manipulations in the modified design show the same masterly hand at work that was seen in the concept design, but with two distinctions. The first is a design work highly focused on geometry that requires a sophisticated knowledge of geometry even for medieval intellectuals. Historians do mention that the court of Frederick II attracted the best minds from Europe and the Middle East, and Arabic science was very advanced in geometry and mathematics at that time.
One example is the feature triangle and the collection of smaller isosceles and rightangle triangles that populate its area. It is a subject of almost no interests nowadays, but obviously an area of great intellectual concentration to medieval mathematicians. It is an axiomatic geometric knowledge, like the Pythagorean relationship, but much less important. It is the geometric tool that the medieval architect rediscovered in the web of geometric forms inside the octagon as used to develop the geometric design algorithms for Castel del Monte.
The second distinguishing element is that the modified design worked with preset conditions and limitations set by the concept design. The modified design consists of routine analyses and calculations; it does not include any grand ideas as it was the case in the concept design.
On the contrary, the modified design transudes a consciousness of obedience and respect for the concept design; this is the response expectable by royal subjects not too high in the court hierarchy. These circumstances indicate that this work was done by lower level courtiers that had the sophisticated knowledge of mathematics and the detailed facts on the concept design, but were close enough to the king for the final plan to receive royal approval.
This modification may have come some time later after the concept design was finalized and possibly some strategic planning actions had been taken. One such action is the ordering of longlead items, such as the prefabrication of the groin stone arches at the quarries. This design modification was not, with most certainty, a modification made after the erection of the structures was started, because it affects the foundations.
With regard to the motivation for this change, the enlargement of the open space inside the tower is the only net change that came out of this elaborate modification. The process, which is based on quantized geometric movements much like the jumping through the lines and nodes of a web, is a fascinating revelation itself. Because of the limited enlargement achieved in the end, the whole development may have been a circumstantial consideration where the opportunity for some space enlargement may have come from a parallel and independent study of the geometric properties in the final concept design.
The other amazing fact is the role that the footing dimension plays in this complex geometric algorithm. While it is the feature triangle that drives the whole geometric algorithm, it is the footing dimension, the side f of this triangle that ends up affecting the measure of all other forms. This impact is much more determinative in the modified design phase than it was in the concept design phase. The footing width f branches out into subelements, such as the measures m and n of the feature triangle, and become key operating factors in the modified design.
The footing dimension was chosen as oneeight the measure of the tower octagon side in the concept design. It is an odd measure both in modern times (0.38 m) and in medieval times; yet it was a key reference measure, most likely determined with extreme precision.
The footing feature is present only at the tower; there is no footing for the façade wall. Other castles built by Frederick II have similar towerfooting features. It is interesting to probe the geometric design role of the tower footing in these other cases, because this may provide clues to the medieval thinking and perception of structural issues, such as the role of footings in the dispersion of lateral thrusts to ground. But Castel del Monte seems to be a unique structure in the castlebuilding of Frederick II; so too may have been the geometric design of this structure and the unique role that was given to the footing dimension in this exceptional design.
The courtyard wall thickness has increased by the same amount in the outward direction (away from the plant octagon center).
The plant octagon radius has increased by the measure k+i (70.2 cm).
The plant octagon side, S", has increased by the measure 2m+2n (53.7 cm).
The tower octagon side, s", has increased by the measure m (15.7 cm).
The tower octagon has an apparent nonregular octagonal form because the two visible sides abutting the façade wall are longer that the other visible sides by the measure m (15.7 cm), which is H. Götze’s spandrel.
The plant minor diagonals that define the cross vault location and pass through the 90º critical intersection corners M terminate inside the tower (at the concept design footing corners EE), not at the tower outer corner E", which was the case in the concept design.
These are all slight spatial changes that do not modify substantively the overall dimension and shape of the plant design.
The modified design is a second geometric algorithm, a peculiar and unimaginable procedure. Its inspiration may have been the peculiar geometric circumstances presented by the results of the concept design.
A story is weaved in this presentation to explain and rationalize the various steps in this procedure. The actual steps in the algorithm may have been somewhat different and the actual creative story may have been something else. We have only the castle as the historical documentation, but the results and the details are cast in stone and are certain.
It is the best theory at this point of the research. As pointed out in the concept design presentation, the theory of the castle towers intended as abutment piers is groundbreaking, ignored and shunned by scholars.
Castel del Monte is an enigma that can be explained by its construction. These two geometric algorithms, the concept design and the modified tower design, explain all of the features of the plant design in all details.
The final proof is given in the dimension and mathematical model analysis, which provides calculated results based on these algoorithms that match the actual measurements with an amazing variance of less than 0.5% overall. Given the complexity of these geometric algorithms, it is inconceivable that such results are just a coincidence. These stories are the most rationale and plausible theories for the plant design of Castel del Monte.