The question that has bothered me for some time was: Why? Why the Golden Ratio? Why it is everywhere? In other words, why do the dimension ratios we find all around us conform to the ratios in the Fibonacci sequence? For what it's worth, here is what I think: Imagine that the huge, solid rock mentioned by me at the beginning of this article is the subject of some single process that breaks it first in two pieces, and then breaks each resulting piece into smaller two pieces and so on, smaller and smaller pieces until all that is left is a pile of sand grains. So what should the ratio be of the two pieces resulting from each step of being broken by the process? Remember that we are talking about one specific and well-defined process, so each act of breaking should result in two pieces with the same ratio, a ratio which is characteristic of that process. Well, it seems that only the Golden Ratio will do the trick for us. So if each piece C breaks into pieces A (the smaller) and B, then B is 61.8% of the C and A is 38.2%. If B breaks again in the same ratio, then the resulting pieces B1 (smaller) and B2 (larger) will be respectively 38.2% and 61.8% of B. Please observe that B2 is 61.8% of B, which itself is 61.8% of C. Therefore B2 will be 61.8% * 61.8% = 38.2% of C. The B2 chip of B is the same size as A. Both are 38.2% of the initial element C. These pieces are identical in size despite being produced by two different branches of the process. One was the result of chipping 38.2% from C (this became A), and the other piece was the result of chipping 61.8% of B (which became B2). Different breaking paths resulted in chips A and B2 ending up with exactly the same size. In diagrammatic form, this is what happens:

Why an obscure mathematical formula from Middle Ages is critical to the world around us