Towers as Abutments
A critique can be raised on the pier role assigned to the towers. Were they really needed as abutments for the lateral thrusts of cross vaults?
The pointed arching of Gothic construction tends to produce smaller lateral thrusts than the rounded Roman arches. All vaults at Castel del Monte have a Gothic design.
Furthermore, the façade and courtyard walls are very deep and would likely provide substantial abutment by themselves.
These are reasonable arguments to question the role of abutments for the towers.
It is however an unwarranted question from our present knowledge of structural matters, or even from the point of view of builders that came after the Gothic architectural experience of the Middle Ages. The question is more properly assessed from the point of view of the medieval architect with its much more limited knowledge of the physical laws, the mathematical tools, and the science of building construction.
Even after the discovery of the gravitational forces by Kepler, Galileo and Newton, it took few more centuries before material science developed, and the modern structural engineering discipline arose.
For centuries, for example, the common perception was that stone is nearly un-crushable in building construction. Because of the limited height of building structures, the material strength of stone gave the impression that it can carry any load force.
Builders in antiquity generally did not worry much whether stone could carry the weight; there was no better construction material anyway. The technology of concrete died out with the fall of the Roman Empire, and lumber is flammable and not as permanent.
Fracturing and spalling of stone blocks and columns can be observed on occasion in some ancient buildings.
The medieval stone builders had the awareness that the building load forces required a certain masonry mass, but had no knowledge of how to make specific size determinations. The width of a marble column at a load point, for example, was sized primarily based on aesthetics and experience rather than an engineering determination that evaluates the load force density compared to the shearing strength of the stone material (keeping the load from crushing the column).
For example, we have seen in the concept design of the castle that the architect was very aware of the lateral thrust forces and we have seen how it dealt with these forces, focusing on the proper masonry strut location to avoid openings in the lateral stonework lineup. The architect knew that he needed a masonry mass to accompany the struts, but had no idea on how to make a rational determination on the size of the masonry. Therefore, he resorted to some other rationale, geometry in this case. Heuristic, religious, cultural clues, and experience most importantly served in most other cases.
The concept design is arguably the product of a cohort of top philosophers and mathematicians, including the King, that were knowledgeable in construction techniques, such as cross-vaulting and the structural issue of lateral thrusts, but lacked the hands-on knowledge of master masons that knew the stone properties, and that of Cistercians that knew how to translate ideas into real stone constructions.
The role of towers in the conceptualization of the design by the royal cohort was solely that of abutment piers, besides their architectural enhancement. The location of the towers at the nodes of an orderly web of lateral thrust and the choice of an octagonal plant for the locus of these nodes is much more than a coincidence. The tower design resulting in a tower that is 90% masonry, while the castle suffers for lack of service facilities, leaves no doubts. The heavy masonry tower walls could neither have been indented for defense, because the design leaves no space for the defenders.
The fact that the plant layout and some of the measures survived intact the modifications made later, indicates that there was an unalterable quality to the concept design that the people to whom the design was handed to for implementation would not dare to change, much like a design handed down as a royal edict.