Separation Shortage between Cross Vault and Façade Wall
The separation distance between the cross vault and the façade wall, j, is affected by the tower enlargement.
The tower enlargement causes a displacement of the anchoring corner for the plant minor diagonals along the plant major diagonal of the measure k, from the tower corner E to the footing corner EE in the concept design, the latter relabeled corner E' in the tower enlargement, folio 115:01.
This displacement k along the plant octagon major diagonal has a corresponding displacement of the minor diagonal intersection that marks the cross vault location, reducing the separation distance j of the cross vault from the façade wall.
The displacement k is along the plant octagon corner direction while the displacement for j is along the plant octagon side (wing) direction. Therefore, the displacement k along the plant corner direction is transformed geometrically into a displacement for j along the plant wing direction. The displacement k is the slant side of the footing corner triangle.
The footing corner triangle and an equilateral right-angle triangle built on one of its sides (m) provides the solution for the displacement affecting j, folio 115:01. These two triangles are highlighted in the enlarged detail of folio 115:01.
The displacement k along the plant major diagonal is equivalent to a displacement m along a line that is at 45º from the plant side direction, triangle T1 in the detail on folio 115:01.
The displacement along the plant wing direction is therefore the diagonal of the equilateral right-angle triangle with sides m, triangle T2 in the detail of folio 115:01.
This equilateral right-angle triangle T2 is geometrically divisible in two more equilateral right-angle triangles, T6, folio 115:01.
The measures are:
m = n ∙ √2
Hypotenuse of triangle T2 = m ∙ √2 = n ∙ √2 ∙ √2 = 2 ∙ n
Therefore the minor diagonal intersection has moved closer to the façade wall by the measure 2n further adjusted by the finishing layer (m-n).
j‘ = j - 2n - (m-n) = j - (m+n)
The measure (m-n) is the variable e of triangle T7, a feature triangle inside the footing corner triangle T1, folio 113:06.