Geometric Construction of the Plant Octagon
The theorized geometric construction of the octagon is based on bisecting circles, starting from the base square.
- The corners of the base square define four of the eight corners of the octagon, corners A in folio 105:01
- The diagonals inside the base square define the square center at point C, folio 105:01
- Point C is also the center of a circle inscribed inside the base square, folio 105:01
- The four intersections of the inscribed circle with the square diagonals define the centers for the new circles, points D in folio 105:01
- The radius of the inscribed circle is half the side of the base square
- Points D are the centers of four circles of the same radius as the inscribed circle, folio 105:01
- The new circles define four new intersections, points B in folio 105:01
- The intersection points B are the other four missing corners of the octagon, folio 105:02
- The base octagon is formed by drawing line segments joining adjacent points A and B, folio 105:03
Points B define another square of equal size but rotated 45° compared to the base square, see folio 105:02.
Scholars such as Heinz Götze report that an ancient method to construct an octagon was to start by first drawing a square, rotating a copy of this square 45°, and then joining the corners of the two squares.
It is difficult to imagine that the physical rotation of squares whether on paper or in the field, using for example a square wooden frame, would be easier and more precise than the procedure outlined above. The geometric construction described above is simpler, easier and more precise to perform on paper and duplicate in the field just using ropes.
Aside the question of how to obtain a rotated square, the fact is that geometrically the octagon is formed by the corners of two squares, one rotated 45°compared to the other, folio 105.03.
More importantly, the size of the base octagon is defined by the size of the base square, and the diagonals of the base square are also diagonals of the base octagon, folio 105:04. These are important features whose significance becomes more evident later on.
One measure controls the size of the base square and base octagon; it is the measure of the side of the square, L, folio 105.04. The diameter T of the base square and base octagon are tied geometrically to the measure L by the Pythagorean relationship.
Besides the diameter T, other measures of the plant octagon include the octagon side S, the octagon width U (minor diagonal), and the derivative measures r and l, folio 105:05.
The same reference letters are used to refer to the same feature dimensions of the octagon, whether the object is the tower or base octagon. Capital letter are used for the base octagon while lower letters are used for the tower octagon.