A Magic Number Worth Its Weight In Gold
Introduction
I have been thinking about writing and posting an article on the Golden Ratio for quite some time and had the beginnings of a draft going. But just recently, I purchased a beautiful piece of wall art for a couple of friends of mine:
It was marketed as “Nautilus and The Golden Ratio Fibonacci Spiral Metal Wall Art”, and as soon as I saw that I knew I would have to get to work and finish and post this article. I hope my reasons will become clear in what follows.
Before posting, I searched Medium for previous articles on the Golden Ratio. Thinking I would not find that many, I supposed that this contribution could be a useful addition. Instead, I found dozens of them before abandoning the search, so perhaps in such a vast sea this teardrop of an article will do only small harm. It may even provide some insight in other areas, especially the connection with construction of geometric figures using only the three permitted tools of Euclidian geometry, more on this later.
The first known mention of the Golden Ratio ϕ is from around 300 BCE in Euclid’s Elements, the Classical Greek work on mathematics and geometry, where it is referred to as the “extreme and mean ratio”. It is also known as the Golden Section and the Golden Mean.
Derivation of the Golden Ratio
Take the line segment in Fig. 1 and divide it into two unequal pieces a and b. The Golden Ratio is defined by a proportion, where the ratio of the length of the whole line L = a + b to its long segment a equals the ratio of the long segment a to the short segment b.
We lose no generality if we let the whole line length L = 1. Let us take a = x, then b = 1 – x, then a + b = x + 1 — x = 1, as assumed.
Substituting into the definition, we get
At right, we cross multiply to get 1 – x = x² or x² + x – 1 = 0. This is a quadratic equation in the standard form with coefficients a = 1, b = 1, and c = –1. Please note that here the coefficients a and b are arbitrary and are not the same as the line segment lengths above. The solution is given by the quadratic formula,
where we have chosen the positive root since x is a length, hence positive. Then by definition,
The Golden Ratio is an irrational number which neither terminates nor repeats, but it is not a transcendental one (like π), since it is the solution to a polynomial equation.
The Golden Rectangle
Euclid’s Elements is arguably one of the most influential works in mathematics in all of human history. Its logical development of geometry and other branches of mathematics is the cornerstone of the entire corpus of modern mathematics and science. Euclidean geometric constructions rely on just three tools, a straightedge, a compass, and a pencil.
We begin by showing a diagram of the construction of a Golden Rectangle using only the three permitted Euclidean tools, followed by a derivation of the Golden Ratio from it, independent of the Euclidean prescription.
Then by the Pythagorean Theorem the diagonal DE, the hypotenuse of the right triangle EBD, is given by
since these two segments are symmetric diagonals. Then
Therefore, segments AB and BH divide the line in extreme and mean ratio equal to ϕ. The construction of the upper segment is the same by symmetry, and with the rectangle height GH = 1, the rectangle is a Golden Rectangle with the ratio of length to height equal to ϕ. Golden rectangles can be divided into a square and a smaller rectangle which by definition is itself golden. Successively dividing each smaller Golden Rectangle produces a series of smaller and smaller Golden Rectangles which can be used to define a Golden logarithmic spiral.
I have prepared a quite lengthy video (about 40 minutes) to show how the two amazing geometric figures presented above can be constructed using only the tools permitted by Euclid in the Elements. Click here to view it if you have any interest at all, as well as the stomach for it. The video could have been significantly shorter — I should have edited out some missteps and consequent revisions I had to make and may do so in the future. But even the things that went sideways may prove instructive as process. On second thought, the video size is about 75 Megabytes, so it probably will be best to download it to your hard drive and view it from there.
The Fibonacci Sequence
The Golden Ratio is intimately connected with a series of numbers called the Fibonacci sequence, where the first two members of the sequence are given (1, 1) and each succeeding number is generated by summing the previous two numbers. They seem to have first been discussed by Indian mathematicians as early as 200 BC, but they were introduced to the Western world by the Italian mathematician Leonardo Pisano Bigollo (1180–1250), known as Leonardo of Pisa or more succinctly as Fibonacci (meaning “son of Bonacci”, where “Bonacci” means “good man”).
The terms of the sequence less than 1,000 are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
where the … means that the sequence goes on forever, never terminating. For the interested (or possibly captive) reader, what is the largest term less than 10,000? Show that it is the Fibonacci number F₂₁ and is equal to 6,765. So what, if anything, does this have to do with the Golden Ratio? If we take any Fibonacci number and divide it by the one preceding it, that ratio very quickly approaches ϕ = 1.618034… to an arbitrary number of decimal places the farther out we go in the sequence:
1 ÷ 1 = 1, 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 5 ÷ 3 = 1.667…, 13 ÷ 8 = 1.625, 21 ÷ 13 = 1.615…, etc., and for the last two of our 3–digit numbers, 987 ÷ 610 = 1.618033…, where we already have reproduced the first seven digits to better than one part in a million.
The exact convergence of the series of ratios of Fibonacci numbers to the Golden Ratio ϕ in the limit as the number of terms approaches infinity can be rigorously proved by using Binet’s formula for computing the nᵗʰ Fibonacci number. We omit the proof in the hope that at least some readers will still be awake at this point. For some reason the formula became named after French mathematician Jacques Philippe Marie Binet (1786–1856), although it had been discussed long before Binet by Abraham de Moivre (1667–1754) and Daniel Bernoulli (1700–1782).
The Golden Ratio in the Physical Universe
The Golden Ratio is manifested in many areas of mathematics, dynamical physical systems, and biological systems. Examples range from construction of regular polygons and Platonic solids such as the pentagon and dodecahedron, flower petals, seed heads, pine cones, and the branching patterns of trees and lightning bolts, to the shapes of spiral galaxies and hurricanes.
One particularly beautiful example is (or at first glance seems to be) the internal shell pattern of the chambered nautilus. This ancient animal has existed for nearly five hundred million years, having existed alongside their cousins the ammonites which went extinct about the same time as the dinosaurs.
The nautilus is a member of the Cephalopod taxonomic family of the genus Mollusca and is closely related to its fellow cephalopods the octopus, squid, and cuttlefish. However, unlike its cephalopod cousins, it is the only one to have retained an external shell. It also has many more tentacles surrounding its mouth (90 or more) compared with the octopus (8) or the squid and cuttlefish (10). The tentacles are textured rather than lined with suckers, and they are covered with a sticky secretion to help find and capture prey animals.
The shell of the nautilus comprises many individual chambers. As the animal grows, it secretes a larger outermost chamber and seals off a new, smaller chamber. These chambers are partially filled with gas and are connected to neighboring chambers by small tubes called siphuncles. By redistributing gas and water among its many chambers the nautilus is able to control its buoyancy and move up and down in the water column as it searches for food and eludes predators. A longitudinal (sagittal) section of its shell reveals the beautifully symmetric, spiral pattern of the connected bases of individual chambers.
The shape of the spiral can be shown to be a logarithmic spiral (thin blue line in the image), although it is not a Golden logarithmic spiral, as one sees so repeatedly and so very incorrectly stated in many sources, including ones that should know better. Logarithmic spirals are characterized by a constant factor that governs how fast the radius of the spiral grows for a given angle of rotation, r = ae^(bθ). The growth factor b for the Golden spiral is the Golden ratio ϕ, while the growth factor for the nautilus spiral is a related constant called the Meta–Golden Ratio χ = 1.355674…, which can be derived from ϕ and shares many of its mathematical properties.
Conclusion
My former Space Telescope Science Institute colleague, mentor, and friend Mario Livio calls ϕ “the world’s most astonishing number”, and for good reason. For anyone wanting to learn more (much more!) about its history and properties, and of the equally astonishing breadth and depth of its connections with other branches of mathematics, physics, and the natural world, I highly recommend Mario’s authoritative and mesmerizing book on the subject, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, available at Amazon and elsewhere. My own copy is resting on my bedside table, and it is one of the books I return to over and over again. Like so many fascinating and articulately written works of literature, one cannot possibly grasp its entire content on one reading, nor even after several. Be prepared to pick up pencil and paper and get to work!
All the best,
Rex the Astrodoc
From Broomall, PA
On Monday, April 1, 2024 at 9 PM
No Foolin’!
