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Plant Concept Design

Castel del Monte

File 140-120

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        Location of the Minor Diagonal Intersection

Having determined that the right-angle intersections of minor diagonals are the square cross vault diagonals, the next task is to measure the location of this intersection.

This is not an obvious task from the design geometry so far, folio 119:01.

The task is simplified by drawing a square inscribed inside the plant octagon with the square corners located at the mid points of the plant octagon sides, square Q, folio 119:02.

The base octagon minor diagonals trace a number of squares in the new inscribed square, folio 119:02. The larger of the inside trace squares has the same side as the base octagon, S.

The smaller trace squares in the corners have a measure that is equal to the tower octagon width u, folio 119:02.

Adding diagonals to the inscribed square highlights the similarity and equal size of the resulting right triangles A and B, folio 119:03.

These triangles have a right side that has the measure u (the width of the tower octagon) and a hypotenuse that has the measure of half of S (the side of the base octagon).

The hypotenuse of triangle B is the distance measure of the minor diagonals intersections from the side of the base octagon. This distance has the measure of half of S, folio 119:04.

The distance of the minor diagonal intersection point from the center of the plant octagon, labeled h10, is calculated from geometry, folio 119:04:

h10 = (U - S) / 2

The distance of the minor diagonal intersection point from the indoor face of the façade wall is labeled j and is calculated from geometry, folio 119:04:

j = S/2 - u - f

j = U/2 - u - f - h10 = U/2 - (U-S)/2 - u - f = U/2 - U/2 + S/2 - u - f = S/2 - u - f


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