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Concept Design
The first phase of the plant design shows how the design concept was developed in a methodic geometric process from beginning to end. The architect of Castel del Monte derived all plant forms from a square and all measurements from the size of that square.
The Concept Design Shows that the plant development is an strict geometric algorithm, an orderly sequence in the formulation of the geometric forms starting at the perimeter of the plant octagon and ending at the courtyard.
Using geometry alone, the architect defined in sequence: the tower octagons, the façade wall, the cross vault spaces, and finally the courtyard wall. The architect used the dependence among the geometric forms to define the form dimensions as well.
The geniality of this plan is that it weaves structural issues, the design of cross vaults, into the geometric design.
The plan is much more than a geometric exercise; it is a proposition for a new, elegant and exceptional solution to the quintessential structural issue posed by cross vault lateral thrusts, albeit in the special configuration of an octagonal plant.
The concept plan sets up key points in the design:

A tower of dimension t, and a footing width f.

A width for the façade and tower walls, a.

The choice of perfect square cross vaults to cover the trapezoidal ceiling.

The location of the cross vaults at the right angle intersections of the base octagon minor diagonals.

The dimension of the cross vault square, w, which controls the elevation design, such as the radius of the diagonal cross vault ribs.

The condition that the strut masonry for the cross vault lateral trusts be located inside the face line formed by the plant octagon minor diagonals.

The condition that the face line for strut masonry cross from the façade wall to the tower precisely at the 90ºcorners where the tower is attached to the façade wall.

The condition that the indoor face of the courtyard wall be located such as to make the cross vault area a perfect square.

The result that the courtyard is a leftover space with the perimeter delimited by the inside octagon formed by the minor diagonals anchored at the tower octagon centers.

The choice for the outdoor face of the courtyard wall to be the inner octagon formed by minor diagonals joining the center of the tower octagons; the result is a leftover space for the courtyard wall of thickness b.
With these milestones, the concept design is complete and tightly weaved. It is difficult to imagine that the plan needs any modification. The involvement of the king and his courtiers, who likely led the conceptualization of this structural plan, may have concluded at this stage. The design would have then moved on to subordinates, most likely as an edict of the court.
The real geometry of the towers is however slightly off the perfect geometric forms of the concept design. The towers do not fit the proportions of a perfect octagon and are slightly larger. Heinz Götze documents the asymmetry of the tower octagon sides, folio 101:01, and the existence of tiny spandrels.
Spandrels are triangular masonry forms found at the 16 corners where the towers are attached to the façade wall, folio 101:02.
Interestingly, the tower octagon distortions are repeated in the same exact geometric manner at each of the eight towers.
This new study shows that these seeming imperfections at the towers are in actuality part of a second design phase that tweaked the geometric design uniformly at each tower.
The redesign most likely followed in the creative wave that characterizes the concept design. It is a second geometric algorithm, one that affects all towers in a similar manner using the base octagon of the concept design as a reference.