How to strike gold with a simple transformation

This post was suggested by

Roger Fischler, How to find the "golden number" without really trying (1981, here),

who himself got the idea from a remark made by Dalzell (1963, here).

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To the general audience, the "golden number" is Φ=1.618... rather than φ=0.618... This is strange, because using 1+φ instead of φ is not in line with mathematical tradition, which prefers minimal definitions. Also, the golden number originates geometrically from splitting a segment into the parts 0.618... and 0.381..., both of which would be reasonable choices for a mathematical constant. But no, it had to be 1.681...  The only reason we can think of to explain this weird preference is the geometrical configuration of the regular pentagon. If you start with sides of length 1, the diagonals are Φ and this seems more natural than diagonals 1 and sides φ.

But there is more, and less innocent. As explained here, the founding father of Golden Numberism, Adolf Zeising, also discovered gold in the remains of the Parthenon.

 His argumentation relies on two coarse measurements: m (blue, 65 ft) and n (red, 107 ft). Nothing would be simpler than to compute m/n, which is 0.607... — not satisfactory, compared to φ=0.618.... Instead, Zeising considers (m+n)*φ which is 106.30...ft,  and considers this "only a minute fraction" removed from the observed 106 ft. Strange detour! If numbers m<M are to be compared, why turn to (m+M)? Dalzell was the first to observe that golden numberists tend to prefer M/(m+M) over m/M. Both expressions are the same if m/M=φ, but this is no longer true if m/M is only approximately φ.

Set x=m/M, so that 1+x=M/(m+M).

(1) Elementary arithmetic. If  x varies from 1/2 to 3/4 (say, the aspect ratio of not too eccentric rectangles), then the arithmetical average is 0.625, and the maximum deviation 0.125. But 1/(1+x) varies from 0.571 to 0.667, with average 0.619 and maximum deviation 0.048. Now compare the poor

m/M=0.625±0.125 

with the impressive

M/(m+M)=0.619±0.048!

 (2) Quantitative estimates. Let's consider these three formulas:

The first is trivial, given that φ=1/(1+φ), and if x>0 it implies the second. We learn from it that, as long as m/M is positive, M/(m+M) will be considerably closer to φ than m/M is. In the last formula, x≈φ and thereby 1/1+x≈1/1+φ=φ. Hence the factor φ²≈0.38, pulling M/(m+M) still closer to φ.

(3) Statistical consequences for φ-spotting.

Fischler [p.408] gives the following theorem: 

if the random variable X is uniformly distributed on a small interval about φ, then about two and a half times [in fact, Φ²=2.618... times] as many values of the transformed data  1/(1+X) as of the untransformed values will lie in that interval

and concludes [p.409]

the various seemingly impressive results in the literature are really due to an invalid transformation of data from a more or less uniform distribution.

Our example under (1), choosing any number between 1/2 and 3/4 without any preference, yet finding 0.619±0.048, may serve as a convincing prototype.

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