It is better to have nothing than to possess much and impart it to no one.
Sextus the Pythagorean

INTRODUCTION
Last issue's focus on the Golden Section has stayed with me, and thinking back to all my work with this ratio, I recall doing a construction regarding a particular "action taken" on the ratio, 1.618… : 1. Let me explain what I mean by, "action taken".[1] As an example, an action can be taken on the golden section rectangle by rotating its long side to its opposite long side to cut off the Öf on that opposite side. This action generates both the Triangle of Price[2] and the Öf rectangle, which is much different than the original golden section rectangle that yielded it. There are only a couple of elements brought over to the new grid. One primary similarity is the making of the golden section ratio (the "golden cut") on both rectangles as a result of a key element in the grid. In the golden section rectangle, the side of the square cuts the long side of the rectangle at the golden section. In the square root of the golden section rectangle, the occult centers are at the golden sections of all four sides. Most obvious is the fact that both rectangles have the same short side length of 1.

Another action that can be taken with the rectangle is to use its diameter length (here, in the golden section rectangle, the length is 1.902…) to become the new length of the long side of the new rectangle; it's ratio then being, 1.902… to 1 (1.902…:1 is the ratio for the American flag!). The new system generated in the "1.902…" rectangle is the "Triple Square", a system that "grew" from the rotational action taken on the diagonal. (A similar rotational action taken on diagonals of rectangles is the Square Root Rectangle system.) In essence, the creative action of the geometer's procedures transforms (we say morphs now) one geometric system into a totally new one. It is totally unexpected because there is no obvious external "formula" that would appear to link the two. It was through the process of drawing and construction that the metamorphosis took place.
Before we begin the drawing, we must first develop various portions of the rectangle's anatomy:

• We will generate the triple square from the golden section rectangle and its diagonal through the rectangle's transformation into the "1. 902… : 1" ratio;
• We will be separating the rectangle into its two component parts: the square and the (reciprocal) golden section rectangle.
• We will also do the diagonals and the reciprocal systems. These become key elements in the transformation.

Here then, we proceed to the steps for our construction:

NOTES
[1] There can be infinite actions taken on any and all geometric constructions, ratios, and shapes that can and will transfer them from one geometric system to another; this is to be discussed in a future column. return to text

[2] A name given to the only right triangle whose three sides are in a geometric progression: 1, Öf , Öf. See also my article "A Comparative Geometric Analysis of the Heights and Bases of the Great Pyramid of Khufu and the Pyramid of the Sun at Teotihuacan" in the NNJ vol. 1, no. 4 (October 1999). return to text

Copyright ©2005 Kim Williams Books