Page 29

Footing Corner Triangle

The extensions of the tower octagon minor diagonals intersect the footing octagon and with the major diagonals define tiny triangles in the footing space; these are the footing corner triangles, folio 129:01.

Enlarging the footing relative to the tower provides a clear view of a footing corner
triangle, which is labeled T_{1}, folio 129:02.

The footing corner triangle has the same form of the octagon characteristic triangle: a right angle triangle with angles of 22.5º and 67.5º.

One of the right angle sides is the footing width, f. The diagonal side, k, is the measure of the footing width along the major diagonal, the corner direction in the octagon. The remaining side is labeled m. These other sides, k and m, are dependent functions of f:

k = f / cos (22.5º)

m = f ∙ tan (22.5º)

A footing octagon minor diagonal cuts across triangle T_{1}
dividing it into two more triangles, T_{2}
and T_{3}, folio 129:03.

Triangle T_{2} is a right-angle triangle sharing
the side m with triangle T_{1}, folio 129:04.

A duplication of triangle of T_{2} can be fitted
geometrically in the footing corner triangle
with its hypotenuse laying along the footing width f, folio 129:05.

This leads to the definition of the other two triangles T_{4}
and T_{5} inside the footing
corner triangle T_{1}, folio 129:05.

It is a menagerie of triangles inside the footing corner triangle, geometrically interrelated, folio 129:06.

Triangle T_{2} can be divided into two more right-angle
triangles (T_{6}) with sides
n, n and m, folio 129:06.

n = m / √2

As a result, the footing width f can be defined from the measures of its triangle subcomponents, folio 129:06.

f = m + 2 n

A further right-angle triangle, T_{7},
can be formed inside the footing corner triangle by the
hypotenuse of triangle T_{5}, folio 129:07.

The footing corner triangle has the sides f, k and m. The additional geometric measures n, e and i are defined inside the footing corner triangle, folio 129:07.

These geometric measures and the geometric construct in which they are framed form the design tools and building blocks used for the modification of the plant concept design in the second phase of the design process.

The detailed examination inside the footing corner triangle reveals geometric details that are irrelevant in the concept design, but are critical to understand the next design phase.

The footing corner triangle and the menagerie of triangle sub-elements are a geometric features of what has been referred to as the octagon characteristic triangle. While it seems to us a geometric curiosity, it was most likely a key geometric reference concept in the medieval knowledge.

Geometry was an evolving science in the Middle Ages, and mathematicians were exploring all kind of features and properties. This one is a handy one, because of its practical usability, but it is also a universal geometric construct that transcends the octagonal geometry within which we have noted its emergence.

Further exploration reveals that the same geometric construct of the characteristic triangle is derived geometrically from a square. The following page shows the geometric derivation of the characteristic triangle and the set of triangle sub-components from a square.

It is a worthwhile exploration to understand better the knowledge and mindset of the medieval architect.