Footing Corner Triangle
The tower enlargement is by the measure of the footing width f in the side direction, which corresponds to the measure k in the corner direction. Both f and k are the sides of the footing corner triangle.
The footing width is a key measure in this geometric process and is a constant; significant is the role it plays in the geometric design.
The footing corner triangle is in the form of the feature triangle of the octagon, a right-angle triangle with the other two angles of 22.5º and 67.5º. Its three sides are f, k and m, folio 110:01.
This triangle can be seen in all corners of the footing space, in all kind of positions framed between the tower and footing octagons and the plant minor diagonals, folio 110:01.
The footing triangle assumes the role of a quantum, the smallest discrete geometric form that is used as a building and calculating tool in the medieval design process focused on geometric forms.
Moving and changing geometric forms in terms of this quantum results in a geometric synchronization that at times seems amazing coincidence. It allows for forms to “snap” onto lines and nodes of the geometric web inside the base octagon.
One practical benefit is that changes in the forms can be measured in terms of f, k and m.
The plant octagon radius (r') is incremented by a k unit.
The tower major diagonal (t') is incremented by a k unit.
The tower increment between parallel sides (u') is incremented by an f unit.
The tower octagon side (s') is incremented by an m unit.
The footing octagon side (ss') is incremented by 3m units.
The open space inside the tower (y') is incremented by an f unit.
The other practical usefulness of this ubiquitous triangle is the ratios among these measures, which are referred to as sine, cosine and tangent in trigonometry. These ratios serve to transform a change in one direction to a change in another direction, such as from wing to corner direction.
These ratios have to be calculated once in the overall design work for this project and then they become calculation factors in the rest of the analysis. Trigonometry was in its infancy in the Middle Ages; these ratios were not yet standardized and had to be addressed laboriously when the triangle shape changed.