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January 1967 Radio-Electronics
 [Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from Radio-Electronics,
published 1930-1988. All copyrights hereby acknowledged.
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Mr. Milton Badt submitted this "Ladder Lingo"
response to the "What's
Your EQ?" feature in the June 1966 issue of Radio-Electronics magazine.
Interestingly, while he pointed out the significance to the relation to phi (φ),
defined as (1+√5)/2, he did not also note that the fraction is commonly referred
to as the Golden
Ratio, and its result, 1.618034... is called the Golden Number. A rectangle
with side lengths who's proportions are according to a/b = φ is called a Golden
Rectangle.Correspondence: Ladder Lingo
 Dear Editor:
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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