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Correspondence: Ladder Lingo |
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Correspondence: Ladder Lingo
About the problem on the resistance of an infinite ladder network, which appeared in "What's Your EQ?" in the June issue. This has an interesting sidelight, especially for electronics men interested in mathematics. The answer, (-1 + √5)/2, is equal to the reciprocal of "tau" (also called "phi"), which appears many times over in geometric design, architecture, and art, as well as in mathematics. Evidently we may now add electronics to the list. Briefly stated, tau is the ratio of length to width (aspect ratio) of a rectangle, the semiperimeter of which bears the same ratio to its length, as its length bears to its width. Tau is approximately equal to 1.618034. A curious property of tau is that it is the only positive number which becomes its own reciprocal by subtracting 1 from it. In view of the ladder network problem this means that if you add a single 1-ohm resistor in series with the input, the new network has a resistance equal to tau, or (1 + √5)/2. A little manipulation will verify that this is the reciprocal of the original expression above. Another interesting aspect of tau, which for the astute observer will tie it in directly with the ladder network problem, is the following alternate way of expressing it: IMAGE HERE [Tau or phi - it's Greek to me. - Editor] Milton Badt, Jr. APO, N.Y.
Posted April 4, 2023 |
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