Beyond favorite numbers exists the exquisite world of beautiful relationships between mathematical quantities.
Most folks with some education are aware of the so-called golden ratio, which is reputed to provide the most pleasing size of rectangles. It's value can be calculated by solving the equation...
1/x = x-1
to obtain x = [1 + sqrt(5)] / 2 = 1.61803...
Also well known is the Fibonacci series. Starting with 0 , 1 each succeeding term is obtained as the sum of the previous two terms, e.g.
The series occurs in nature, manifesting itself in the layout of branches on plants and petals on flowers among other forms.
Now, what is truly amazing is the fact that, if you take the ratio of adjacent terms in the Fibonnaci series, they will approach ever closer to the golden ratio...
3/2 = 1.5
5/3 = 1.67
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538
34/21 = 1.61905
55/34 = 1.61765
89/55 = 1.61818
144/89 = 1.61798
233/144 = 1.61806
377/233 = 1.61803...
If you've ever been puzzled hearing a mathematician describe something mathematical as "beautiful", this is the sort of thing they're talking about.
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DIYSwede (Jan 7, 2021)
I so admire Mr. K's infinite ability of textually diagramming intricate subjects. Even more pleased his many decades dedicated to 'our side'. Talk about directing interest in serious topics, laid down comprehensibly......Evidence? Look at his contributions linked in his forum signature.
Yes, admittedly I'm one of those with some [barely] education. The Fibonacci as a term was new to me, around the eighties or so, coming across 'their' divider/ mechanism/ tool. That an identifiable person realized it, then sat down to extract it; epic. We can visualize how either calculating OR device came first, to prove the other. Found myself wanting to illustrate that as "helping" to prove the other, but help is a weak term. It proves itself, in various ways.
But visual factor was something I seemed to use regularly in photography, labeled "Rule of Thirds". Now while the ratios aren't exactly alike, nor any photo bugs running calculations, it makes itself known. Peer through the viewfinder, it tells you the ideal frame. The Golden Ratio, very prevalent with architecture, in use thousands of years. For some perhaps it's subliminal, but it's definitely not universal. Airy and Bessel come to mind, lol, different solutions to outwardly same physics.
So yes, the math IS beautiful, unalterable, predictable, not unlike a mechanism; but the visual mystery more interesting.
Same for music; one jumble of notes gives us chills, the same in different order, abominable. To another, the combination might reverse!
Yet musical scales are a quite simple mathematical progression. And there are studies tying intelligence quotients to levels of intricacy preferred in music.
All in all, good thing we are individuals, sharing that what we have in common.
...we'll learn more by wandering than searching...
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