As 1.618 is the magic number points out, 1.618 is the magic number — or at least the start of the magic number:
Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887…What’s the significance of this number? It’s the “golden ratio” and, arguably, it crops up in more places in art, music and so on than any number except pi. Claude Debussy used it explicitly in his music and Le Corbusier in his architecture. There are claims the number was used by Leonardo da Vinci in the painting of the Mona Lisa, by the Greeks in building the Parthenon and by ancient Egyptians in the construction of the Great Pyramid of Khufu.
What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can be produced simply by adding 1 and whose reciprocal by subtracting 1. If you take a golden rectangle — one whose length-to-breadth is in the golden ratio — and snip out a square, what remains is another, smaller golden rectangle. The golden ratio is also difficult to pin down: it’s the most difficult to express as any kind of fraction and its digits — 10 million of which were computed in 1996 — never repeat.