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by H. Peter Aleff |

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To construct the "golden ratio", divide a unit square into two equal rectangles with sides 1 and 1/2. Add the length of the diagonal in such a rectangle to its short side. The new length is the "golden ratio"; the add-on rectangle as well as it plus the square are both "golden". When you add a square along the long side of a golden rectangle, the new rectangle is again golden, and the next one too, and so on forever. If you draw a quarter circle into each successive square so that each arc begins where the prior one ends, you obtain a "logarithmic" spiral of self- replicating growth.
The golden ratio yields also two golden triangles, with side lengths 1, phi, and phi for the slender one, and 1, 1, and phi for the obtuse one. These fit alongside each other to reproduce again their identical shapes. When you connect the pointed ends of each successive obtuse triangle with an arc around its apex, you obtain again a logarithmic spiral. Adding two obtuse golden triangles with their broad base along the equal sides of the slender one produces the pentagon, a rich source of golden ratios. Its diagonals are phi times its side length and divide each other in the same phi proportion to form a new pentagon.
Image copied from "Excavations at Brak and Chagar Bazar" by M. E. L. Mallowan, Iraq Vol. 9, Excavations at Brak and Chagar Bazar (1947), pp. 1-87+89-259+i-iv, Published by: British Institute for the Study of Iraq, Article Stable URL: http://www.jstor.org/stable/4199532. The above tray with the Pentagon is shown on Plate LXXIX. It dates from the Halaf period, around 5500 BCE. Please note that the three peripheral rings are each divided into 108 parts. Since this is the number of degrees for each inside angle in the pentagon this arrangement strongly suggests that the maker of this tray was already familiar with the division of the circle into 360 degrees which is required to arrive at 108 degrees for those inside angles. This division of the circle is formally attested only in the last few centuries BCE although the division of the yearly solar circle into the 360 days of the ritual year was much older both in ancient Egypt as well as in the Assyrian cultic year.
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