Page 30

The Characteristic Triangle

The characteristic triangle noted in the octagon web is a right-angle triangles with the angles of 22.5º and 67.5º. It is the triangle that is obtained by splitting evenly the isoscele triangle formed in each of the eight wings of the octagon.

The footing corner triangle has the form of the characteristic triangle. The exploration of the geometry inside this triangle has revealed a number of normal triangular forms definable within the boundaries of the characteristic triangle. It is a special geometric construct.

It is theorized that this special geometric construct was known to the medieval mathematicians outside the context of the octagon. Indeed the same construct is derived by exploring the geometry inside a square.

The diagonals of a square, d, are related to the side of the square, f, by the Pythagorean relationship, folio 130:01.

d = f ∙ √2

The difference between the diagonal and the side, m, can be defined geometrically, folio 130:01.

A circle of radius m and centered on a square corner defines intersection points P and F on the sides of the square, and point H on the diagonal, folio 130:01.

A line joining a square corner and one of the intersections points just defined forms the
characteristic triangle with the square sides, triangle BEF
(T_{1}), folio 130:02.

An equilateral right-angle triangle (T_{2} )
is also definable by joining points P and
F, folio 130:03.

The horizontal line connecting point H to the side of the square defines point G, folio 130:03.

A circle centered on point G and passing through point H defines the intersection point D on the side of the square, folio 130:04.

Points D, H,
and E define a new equilateral right-angle triangle
(T_{4}),
which is essentially triangle T_{2}
rotated and laying with the hypotenuse
on the side of the square, folio 130:04.

This geometric derivation defines the geometry of triangles inside the characteristic triangle that was derived within the context of the footing corner triangle, page 129, but this time is shown as a geometric property of the square.

The geometric derivation provides a definition for the measures of the various triangle components and their dependent relationships, folio 130:05.

Knowing these geometric features and relationships provides the tools and knowledge to perform the more complex geometric manipulations in the next phase of the plant design.