(Continuation of part 1)
How precise can N/L be measured? We need to know this before we can distinguish the irrational number φ=0.61803... from simple fractions like 3/5=0.6 or 5/8=0.625.
Both for L and N, we turn to medical experts. Osteoporosis specialists need precise height measurements in order to assess height loss in their patients. In 2005, Osteoporosis Update (here, pp.4-5) described tools and techniques to be used for accurate height measurement. We learn from it (p.5) that the precision error in height measurement is surprisingly large (up to 3.0 cm), that the clinically relevant threshold is a precision error less than 2 cm, and that the average of three separate measurements, recorded to the closed mm, is to be registered. Surprising indeed! There has been some internet debate about Albert Einstein's height, which has been given by Swiss authorities as 171.5 cm on one occasion and 175 cm on another. Both may have been equally right and perfectly in line with Swiss precision. Osteoporosis Update even suggests to measure height at the same time of day, to reduce diurnal effects!
For navels we have to consult plastic surgeons. In this paper, dealing with young adults, we learn that there are several forms of navels, and that they extend vertically over several centimeters. The average round female navel in this sample has a diameter of 3 cm, but they may extend vertically up to an unexpected 5 cm. (Note that we deal with facts, not esthetics.)
It follows that it would be unreasonable to claim L or N with more precision than ±1 cm. These bounds are definitely underestimated. Measuring a navel's height, for instance, is technically more involved and indirect than measuring total height, and sucking in the belly or breathing heavily has a noticeable effect.
For a person of "175 cm", with a navel at "106 cm", let's agree that we know that N lies between 105 and 107, and L between 174 and 176. This implies that N/L lies between 104/176 (which is 0.59090...) and 106/174 (which is 0.60919...). These bounds differ by 0.01828..., and this exceeds the difference φ-3/5=0.018.... Therefore, such a measurement would not distinguish one from the other.
In the sequel, whenever N/L is presented as a number with three decimals, one should take care to add '±0.009'.
Modern statistics (1)
We could only find three experiments worth reporting. First we consider
T. Antony Davis and Rudolf Altevogt, Golden mean of the human body, The Fibonacci Quarterly 17, No 4, 1979 (here).The paper is very flawed, as we intend to show. Two entirely different samples are investigated: German school children of both sexes, and Indian young men (Calcutta).
As for precision, we are told (p. 341) that the measurements were done with a set-square and a vertical pole marked in centimeters. Inspecting the results recorded with three decimals, there is room for some suspicion.
The authors record A,B,C,D,E hoping to recover the golden mean—announced in the title—from A/B and/or B/C and/or C/D and/or D/E. With some luck, C/D and D/E would turn out to be identical, because that's exactly what the golden ratio means. It's not clear what mystic properties were expected from ratios involving B (distance from nipples to top of head) because this physical measurement is highly dependent on gravity, and—as everybody knows—evolves with age. Anyway, these are their findings, with some miscalculations corrected in red.
GT = German bisex sample, tallest quarter
GS = German bisex sample, shortest quarter
G = German bisex sample
GG = German girls
BG = German boys
IT = Indian young men, tallest quarter
IS = Indian young men, shortest quarter
I = Indian young men.
Strange! The entry D/E for German boys (for example) is not the average of the individual values of D/E, but the average of D divided by the average of E. For comparison: instead of the average of 1/2 and 1/3, which is 5/12=0.416..., we would be given (1+1)/(2+3), which is 2/5=0.4. Not a single N/L is provided, only these (strangely defined) averages, which range from 0.611 to 0.636. We feel highly uncomfortable about this procedure.
Anyhow, if the intrinsic imprecision of ±0.009 is taken into account, nothing relevant is seen to emerge, but you can judge for yourself.
It's 'approximately φ' all right, but everything is 'approximately φ' within a certain precision. My own guess for would be:
N/L = 0.62 ± 0.01
The justification is that 31/50 (0.62) is both simple and precise enough, and 0.01 is about the right precision. The guess is empirical, and intended for Homo Sapiens (all sexes, races and ages) as is. After all, Man was not designed by God according to some intricate plan, but he's an evolved quadruped. It's not clear why the navel would miraculously divide the length head + body + one hind leg, strictly aligned according to some remarkable, let alone divine, proportion.
The navel is hardly ever on the midline, but predominantly to the left of it (here and here p.9 and p.10), females have lower situated navels than men (here, p.10) and it sinks to a lower position as age advances (here, p.1).
The best we can hope for is some empirical relation holding statistically for the right gender, race and age.
Modern statistics (2)
Next, this contribution to Journal of Recreational Mathematics, vol. 24 (1992), pp.26-29. Being "recreational", the paper is not to be taken too seriously, and the secondary information surrounding the experiment is the usual crap quoted from unreliable sources. Concerning precision, we are only told that measurement error was a ticklish problem. The golden truth to be uncovered was 1/φ, being 1.61803... The average for the 161 male students was 1.654 and for the 158 females 1.646. Hence, the navels measured (admittedly with low precision) are statistically situated below the golden one, and closer to da Vinci's navel height (which would give 5/3=1.666... here) than to the golden expectation.
Modern statistics (3)
Finally, this worksheet of the University of Colorado, Department of Mathematics, Fall 2010, Math 1310, CSM. Data are restricted to only 30 students, and nothing is told about sex or race. Again, it is 1/φ, being 1.61803..., that should turn up. This being a classroom exercise in statistics, it would be unreasonable to expect anything precise, and L/N (the reverse of N/L) is given with only two decimals. The values range from 1.50 to 2.01, with
mean value = 1.629 ; standard deviation = 0.095.
In part 6, the students had to statistically test the hypothesis that the population mean equals 1/φ. The answer was YES. Wow, scientific confirmation at last! Wait, wait. (We hurriedly switch to LaTeX because html-math is too clumsy.)
Did you ever notice that arm span over body height (investigated here) is approximately π/3 (which is 1.04....)? Try it, π/3 may well pass the statistical test, and reveal yet another unexpected mystery of the human body! Want to switch from the φ-mystery to the π-mystery? just use the approximate equality (up to 0.001)
√φ = π/4