Golden Maths — Elementary arithmetic

The Golden Number hype is mainly humbug, but its core does contain some mathematical facts. It is a long standing tradition to blow these up to pompous mysteries. The founding father of this dubious practice is Luca Pacioli, who in his Divine [sic] Proportion (1509) gives all kinds of overblown, even theological, epithets to mathematical properties known since Euclid. Today is hardly different, and one may find simple mathematical facts described at length and analysed in depth as if the golden number were unique in having those properties. Most people seem unaware of the fact that the golden number is but one of a kind, and that there is nothing very special about preserving its decimal part under inversion (wow, 1/0.61803...=1.61803...), or being the limit of a simple sequence of fractions (wow, approximately a quotient of Fibonacci numbers).

This episode of Golden Maths is devoted to Elementary arithmetic. Actually, by 'arithmetic' we mean 'non-geometry'; limits, for instance, are included. And 'elementary' implies there is less elementary to follow.  For now, we consider the golden number as senior member of the family of metal numbers. In recent times there is some tendency to extend the golden epithet (of 19th century German origin, as we know) to silver, bronze and generally metal(lic). It's funny, and therefore recommendable in view of the deadly (I would say leaden) seriousness of the golden prophets.

So here is the victory stand with the gold, silver and bronze finalists in the grand metal race:

There are infinitely many more competitors (all staying in Hilbert's hotel, BTW 😄), and they look very much alike. All you will ever need to know about them is on the blackboard below. For gold, silver, bronze plug in T=1,2,3.

The (elementary) proofs are in this single sheet:

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Golden Maths — Geometry

In the Golden Maths posts I'll gather mathematical facts about the golden ratio/section/number. The series will not be very extensive, because the golden number only rarely pops up in mathematics. People are led to believe otherwise, but that is one of the many Golden Myths. The latter series will be bulky indeed.

We start with geometry, which is where the golden saga began. There are four items: the decagon, the pentagon, the icosahedron and the dodecahedron. Only the decagon leads unavoidably to the golden number. The proofs are short and elementary, and how could it be otherwise? The golden ratio is just an elementary application of the Pythagorean theorem!

In fact, all you'll ever need of golden geometry is in this illustration:

or (equivalently, and more useful) in these formulas:

Read all about it here.

 

Sunflower heads and other parabolic spirals

My interest in sunflower heads arose as part of my crusade against golden ratio misconceptions. Most of the claims involving the golden ratio are easily dismissed, but sunflowers are in a different league. There is no denying that they prefer Fibonacci numbers (see here), and this calls for an explanation. Botanists like Ridley (reference below) assure us that a fixed divergence angle, let alone a 'golden' one, is not the answer (see also here), but what is the answer then? In our paper (here) we contribute something to this fascinating question. Our text is self-contained, but more context is found in

1. Helmut Vogel, A Better Way to Construct the Sunflower Head, Mathematical Biosciences 44 (1979), 179-189.
2. J.N. Ridley, Packing Efficiency in Sunflower Heads, Mathematical Biosciences 58 (1982), 129-139.
3. Ivan Niven, Herbert S. Zuckerman  and Hugh L. Montgomery, An introduction to the theory of numbers, Fifth Edition, 1991, Chapter 7.
4. Oskar Perron, Die Lehre von den Kettenbrüchen, Band I. Dritte Auflage, 1954. 

Ridley was the main source of inspiration. 

It will turn out that the key mathematical notion is that of continued fraction, that Fibonacci and Lucas numbers impose themselves for botanical reasons, and that the golden ratio is not mentioned. As a matter of fact, there is no need to mention any irrational number. A FIDIPAS (finite discrete parabolic spiral) is completely characterised by three positive integers: besides the number N of points, the numerator p and denominator q of the fraction 𝛿 which determines how much of a turn is needed to get from one point to the next.

From the paper we only display the very last illustration, where a real sunflower of some 920 seeds (#56 of this collection) is compared to its mathematical counterpart of 920 points. Everything about the spirals is perfectly and quantitatively predictable.