This reciprocity is also illustrated by one of the classical constructions of the golden section, with the help of a square inscribed into a semicircle", (next figure),

"A circle drawn from the center of the square's base touching the opposite corners of the square (radius r) will produce the golden section's proportions along both sides of the extended baseline.

If the square's sides are 1 unit long, then each of the extensions will be 0.618 unit long and the 1 x 0.618 rectangles on either side of the square will be golden rectangles. Each of these combined with the square for a larger golden rectangle, 1 x 1.618. These larger rectangles and the smaller ones are each other's reciprocals, in the sense that the larger side of the small ones and the smaller side of the large ones are the same. The total length of these reciprocal golden rectangles is 2.236 units, this number being identical with sq.rt5".

The Power of Limits
By Doczi
BH 301. C84 D62

© Copyright. Robert Grace. 1999