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WHICH RECTANGLE do you think is aesthetically more pleasing? Most people believe it’s the one on the left.
The rectangle on the left is an example of the “golden rectangle,” the proportions of which have fascinated people for more than 2000 years. These proportions, also known as the golden mean, ratio or section, appear in the Parthenon in Athens, the Cathedral in Chartres, Michaelangelo’s David in Florence—and the original iPod.
Several of these nuggets come from “Why We Love Beautiful Things,” by Lance Hosey, in The New York Times, February 17, 2013 (see http://goo.gl/HWnEH).
The golden ratio can be defined in several ways. Two values, A and B, are in this ratio if A is to B as their sum A+B is to A.
Algebraically, A/B =(A+B)/A, which (remember “cross-multiplying”?) implies that A2 = B (A + B) = A B + B2. Fooling around algebraically, this is equivalent to A2 – AB – B2 = 0.
Assign B the value 1, and this last equation becomes A2 – A – 1 = 0, a “quadratic equation.” Apply the quadratic formula, and it shows that with B set to 1, then the positive choice for A is (1 + √5)/2.
This number (1 + √5)/2 representing the golden ratio is given the Greek letter φ. As a decimal, it’s approximately 1.6180339887…. (An example of an irrational number, its decimal representation neither repeats nor ever ends.) The ratio 8/5 isn’t a bad approximation of the golden ratio, as 8/5 is 1.6.
The golden ratio also has a geometric construction. Start with a square of side S. From the midpoint of one side, draw a line to an opposing corner. Then strike an arc until it intersects the extension of that side. Call this prolongation P.
A rectangle with S as its short side and S + P as its long side is a golden rectangle. It’s a straightforward exercise (needing nothing more than the Pythagorean Theorem) to derive (1 + √5)/2 as the long side of the rectangle with short side S set to 1.
The golden rectangle has a fractal interpretation as well. (See www.wp.me/p2ETap-MV for background on these mathematical curiosities.)
Begin with a gold rectangle, then delete the short-side square. The resulting rectangle is again a golden one. And so on.
A Duke University professor studied the golden rectangle in 2009. His research indicated that our eyes scan the image of a golden rectangle more quickly than other shapes. It’s only natural that we admire these proportions so readily.
For those wanting to explore this golden relationship in more detail, http://goo.gl/hZzPE is an excellent starting point. In fact, it offers background to the following.
Howard Eves relates a tale concerning the golden mean in his book Mathematical Circles Adieu, Prindle, Weber & Schmidt, 1977. A fellow named F. A. Lonc of New York City measured the heights of 65 women and compared this to the heights of their navels. Mr. Lonc claimed this ratio of overall height to navel height equaled φ, the golden ratio.
What do you suppose his shtick was? ds
© Dennis Simanaitis, SimanaitisSays.com, 2013
Further reading: “It defines . . . the ratio between bones in the finger and even a series of frequencies in solid-state magnetic resonance. New research has shown that the golden ratio could point the way for a simple method of determining competition between drugs.
“… until now, no role for the golden ratio has been identified in pharmacology, or phi-macology as we termed it.”
does anybody have a chart of all the different mathematical rectangles?
Given that there are evidently infinitely many rectangles, no such chart exists. However, many books on popular math discuss the Golden Rectangle and its appearances in art and science.