Why do these exhilarating magics with numbers exist in Nature? Is it natural or are they tricks of numbers the way they were invented or explored?
Classical Fibonacci Sequence = $0, 1, 0+1=1, 1+1=2, 2+1=3...$
Golden Ratio:
$\varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}$
$10 \times 3 = 30, 10 \times 7 = 70, 10 \times 88 = 880$ just a zero after the number.
$11 \times 7 = 77, 11 \times 9 = 99, 11 \times 11 = 100 + (10 \times (1+1)) + 1 = 121$ when multiplying 57 with 11 the leftmost and rightmost digits will be consecutively 5 and 7 and middle will be the 2 of (5+7) = 12 and 1 will be carried over to left most digit. So the result will be 627
Golden Ratio and Fibonacci Sequence Connection:
A B B / A, 2 3 1.5, 3 5 1.666666666..., 5
8 1.6, 8 13 1.625, 13 21 1.615384615..., ... ...
... 144 233 1.618055556..., 233 377 1.618025751..., ...
... ...,
Numbers (and all of mathematics) were invented to describe the world around us. No wonder they crop up all over the place.
Sure, people did follow them and their relations way, way beyond anything "natural" (Cantor's transfinite numbers have no relation to the real wold!).