Page 115

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 T_{1} 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 T_{2}
in the detail of folio 115:01.

This equilateral right-angle triangle T_{2} is geometrically divisible in two more equilateral
right-angle triangles, T_{6}, folio 115:01.

The measures are:

m = n ∙ √2

Hypotenuse of triangle T_{2} = 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 T_{7}, a feature triangle inside the footing
corner triangle T_{1}, folio 113:06.