# The Golden Ratio  ## Metadata - Author: [[Mario Livio ]] - Full Title: The Golden Ratio - Category: #mathematics #problem-solving ## Highlights - “When you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.” ([Location 70](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=70)) - A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. ([Location 113](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=113)) - “The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, no longer feel amazement, is as good as dead, a snuffed-out candle.” ([Location 125](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=125)) - One story has it that when the Greek mathematician Hippasus of Metapontum discovered, in the fifth century B.C., that the Golden Ratio is a number that is neither a whole number (like the familiar 1, 2, 3,…) nor even a ratio of two whole numbers (like the fractions ½, ⅔, ¾,…; known collectively as rational numbers), this absolutely shocked the other followers of the famous mathematician Pythagoras (the Pythagoreans). The Pythagorean worldview (which will be described in detail in Chapter 2) was based on an extreme admiration for the arithmos—the intrinsic properties of whole numbers or their ratios—and their presumed role in the cosmos. ([Location 129](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=129)) - The realization that there exist numbers, like the Golden Ratio, that go on forever without displaying any repetition or pattern caused a true philosophical crisis. ([Location 135](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=135)) - The Golden Ratio’s attractiveness stems first and foremost from the fact that it has an almost uncanny way of popping up where it is least expected. ([Location 198](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=198)) - “When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong.” ([Location 253](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=253)) - Clearly, an even bigger mental leap was required to move from the simple counting of objects to an actual understanding of numbers as abstract quantities. Thus, while the first notions of numbers might have been related primarily to contrasts, associated perhaps with survival—Is it one wolf or a pack of wolves?—the actual understanding that two hands and two nights are both manifestations of the number 2 probably took centuries to grasp. ([Location 285](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=285)) - Many studies show that the largest number we are able to capture at a glance, without counting, is about four or five. ([Location 313](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=313)) - This possibility was already raised by the Greek philosopher Aristotle (384–322 B.C.) when he wondered (in Problemata): “Why do all men, barbarians and Greek alike, count up to ten and not up to any other number?” Base 10 really offers no other superiority over, say, base 13. We could even argue theoretically that the fact that 13 is a prime number, divisible only by 1 and itself, gives it an advantage over 10, because most fractions would be irreducible in such a system. ([Location 407](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=407)) - Probably the most perplexing base found in antiquity, or at any other time for that matter, is base 60—the sexagesimal system. This was the system used by the Sumerians in Mesopotamia, and even though its origins date back to the fourth millennium B.C., this division survived to the present day in the way we represent time in hours, minutes, and seconds as well as in the degrees of the circle (and the subdivision of degrees into minutes and seconds). ([Location 427](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=427)) - In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Con-fucius. ([Location 546](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=546)) - “most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge.” ([Location 550](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=550)) - The Pythagoreans were probably the first to recognize the abstract concept that the basic forces in the universe may be expressed through the language of mathematics. ([Location 645](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=645)) - Five represented the union of the first female number, 2, with the first male number, 3, and as such it was the number of love and marriage. ([Location 702](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=702)) - The striking property of all of these figures is that if you look at line segments in order of decreasing lengths (the ones marked a, b, c, d, e, f in the figure), you can easily prove using elementary geometry that every segment is smaller than its predecessor by a factor that is precisely equal to the Golden Ratio, φ. That is, the ratio of the lengths of a to b is phi; the ratio of b to c is phi; and so on. ([Location 722](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=722)) - The idea behind the ingenious method of reductio ad absurdum is that you prove a proposition simply by proving the falsity of its contradictory. ([Location 765](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=765)) - In mathematics, reductio ad absurdum is used as follows. You start by assuming that the theorem you seek to prove true is in fact false. From that, by a series of logical steps you derive something that represents a clear logical contradiction, such as 1 = 0. You thus conclude that the original theorem could not have been false; therefore, it must be true. ([Location 770](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=770)) - According to legend, in a fight between the god Horus, the son of Osiris and Isis, and the killer of his father, Horus’ eye got torn away and broke into pieces. The god of writing and of calculations, Thoth, later found the pieces and wanted to restore the eye. However, he found only pieces that corresponded to the fractions ½, ¼, ⅛, 1/16, 1/32, and 1/64. Realizing that these fractions only add up to 63/64, Thoth produced the missing fraction of 1/64 by magic, which allowed him to complete the eye. ([Location 859](https://readwise.io/to_kindle?action=open&asin=B001L4Z6Q2&location=859))