The measure j is the distance of the cross vault from the façade wall.
The measure j seems an incidental measure, but was a special reference to the medieval architect. Together with the measure f, the footing dimension, it serves to determine the room width, but there is more information in the definition of j.
The importance of the measure j comes from the singular geometric form of an equilateral right-angle triangle that ties two adjacent towers and the cross vault center, folio 127:01.
The examination focuses on the base octagon minor diagonals, because they are the critical markers for the lateral thrusts masonry. They mark the limiting position beyond which the lateral thrust masonry has no more use. For a stone mason arranging the stonework related to lateral thrusts, this is the outside face of any lateral thrust stonework.
The measure j is the distance of the minor diagonal right-angle intersection from the indoor face of the façade wall, Folio 127:02.
An equilateral right-angle triangle is formed by joining this intersection with the outside corners of two adjacent towers, a triangle labeled P1, folio 127:02.
Triangle P1 has an hypotenuse that is the side of the base octagon, S, and a height that is half the side of the base octagon, S/2, folio 127:02.
This triangle is divisible into two equal and similar equilateral right-angle triangles, labeled P2 and P3, folio 127:03.
Triangle P3 in turn can be divided into a square and two more equilateral right-angle triangles, P4 and P5, folio 127:04.
It is much like the fractal geometry of nested and ever smaller, repeating and similar right-angle triangles.
All geometric measures for this singular web of triangles can be determined from the established geometrical measures, s, f, and S, folio 127:05.
An inspection of the geometric forms in folio 128:05 leads to the determination that j is the measure of the tower octagon side, s, reduced by the size of the footing width, f:
j = s - f
It is also noted that;
y = s - 2 f
This implies that:
y = j - f
j and y are both measures of open room spaces inside the castle; they are both dependent variables to the measure of the tower octagon side, s, adjusted by the measure of the footing width, f.
j is smaller than s by the measure f, and y is smaller than j by another measure f.
The measure f appears in many of the geometrical definitions much like an “atomic” component.
The interconnections of f, j, and y may very well have had some physical rationalization in the mind of the medieval architect of Castel del Monte, with some mystical connection.
Mysticism in the Middle Ages was more than a spiritual connection; it was the intellectual component that together with philosophy and the limited state of science and mathematics formed the knowledge basis that the medieval architect had at its disposal to resolve real life problems.
Another observation that will become relevant in the second phase of the design is the variability of j as a function of the tower size and location.
In the concept design, all plant forms are geometrically interconnected and all variables are mathematically interconnected . The interdependency is like the situation of the features of a figure on a rubber balloon that expand or shrink in proportional manner as the size of the balloon changes.
In this model, if the size of the tower changes, everything else changes in proportional manner. The location and size of the tower is constrained by the geometric tower ring in this model.
But what happens if all the towers are arbitrarily moved or resized in a uniform procedure, after the concept plan is finalized?
Increasing arbitrarily the size of the tower octagon causes the measure j to decrease with the minor diagonals affixed and moving with the tower outer corners. Moving the towers along the major diagonals and away from the center of the plant octagon causes j to increase.
This is easily ascertainable from the geometry, folio 128:05. It is likely that the medieval architect may have used physical models to study the variability of the geometric forms within an octagonal frame.