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The Golden Ratio: The Divine Beauty of…
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The Golden Ratio: The Divine Beauty of Mathematics (edition 2018)

by Gary B. Meisner (Author)

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"From the pyramids of Giza to the proportion of the human face, the golden ratio has an infinite capacity to generate beautiful shapes with exquisite properties...author Gary Meisner examines the presence of this divine proportion in art and architecture, as well as its ubiquity among plants, animals, and even the cosmos." --provided by publisher.… (more)
Member:marcusstafford
Title:The Golden Ratio: The Divine Beauty of Mathematics
Authors:Gary B. Meisner (Author)
Info:Race Point Publishing (2018), 224 pages
Collections:Your library
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Tags:non-fiction, maths

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The Golden Ratio: The Divine Beauty of Mathematics by Gary B. Meisner

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I've long been a sucker for books like this that purport to demonstrate the secrets of the universe in a mathematical formula. Unfortunately, I've always been let down. Not because the golden ratio has some remarkable features and, maybe, a lot of coincidences when comparing it with measurements of the world around us .....but mainly because the authors seem very good at drawing diagrams showing golden ratios but much less able to come up with explanations about why they occur. And that's what I'm really interested in. Why do we get the two way spiral of seeds in a sunflower? Surely it must be related in some way to a packing or developmental sequence but this is not explained for even postulated.
And so it was with the current book. There is a lot of interesting stuff there like:
The golden ratio wasn’t “golden” until the 1800s. It is believed that German mathematician Martin Ohm (1792–1872) was the first person to use the term “golden” in reference to it when he published in 1835 the second edition of the book Die Reine Elementar-Mathematik (The Pure Elementary Mathematics), famed for containing the first known usage of goldener schnitt (golden section) in a footnote. The first known use of the term golden ratio in English was in an 1875 Encyclopedia Britannica article by James Sulley on aesthetics. But the term didn’t appear in a mathematical context until Scottish mathematician George Chrystal’s 1898 book Introduction to Algebra.
History records the ancient Greek mathematician Euclid as describing it first—and perhaps best—in Book VI of his mathematics treatise Elements: “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 less.” Euclid demonstrates a lot of ways of deriving this ratio. But here's an easy way of deriving the ratio:
1. With the aid of a compass, draw a circle. Then inscribe an equilateral triangle inside it.
2. Draw a single line through the midpoint of two sides of the triangle, (A and B) extending the line to the edge of the circle, (C).
3. Then the ratio of the length of that line between the sides of the triangle (AB) to the total length of the line (AC) is the golden ratio 1.618
The golden ratio is known as phi but phi has many unique mathematical properties. For example, it is the only number whose reciprocal is one less than itself, as 1 / 1.618 = 0.618. Stated more simply and elegantly: 1 / Φ = Φ–1 And, as 1.6182 squared = 2.618, phi is also the only number whose square is one more than itself; that is: Φ2 = Φ + 1.
In the Fibonacci sequence, the ratio of each successive pair of numbers converges on phi. ...With the fortieth number in the sequence—102,334,155—the resulting ratio matches phi to 15 decimal places: 1.618033988749895. What I find that I'm missing here is some explanation of why this happens.
You can also develop a spiral from a series of golden rectangles. A a similar spiral for the Fibonacci series (using squares whose side lengths are the successive numbers of the Fibonacci sequence). Technically speaking, none of these are spirals. They’re called volutes. The difference is almost imperceptible, but a true golden spiral is a unique, equiangular (that is, logarithmic) spiral that expands at a constant rate.
Meisner then moves into a long section which basically looks for golden ratios in the real world. He has software to look for this with a reasonable set of "rules" though, I confess, I'm not satisfied with his rules. I've seen this sort of thing before and seen golden ratios superimposed on (for example) the stones in one of the stone gardens in a Zen temple in Kyoto. But when I applied the technique myself, I realised tht one could juggle the measurements to make them fit. Because the stones were not single points it was easy to pick a spot, somewhere on the pile that made the golden ratio work. And, I think, given any photo with a bit of complexity in it, one could be drawing all sorts of golden ratios. However, that has not stopped Meisner. But, to give him his due he does apply a little scepticism: Viz
Da Vinci’s most famous painting is La Joconde, or the Mona Lisa. Application of the divine proportion to this painting is also the most subject to interpretation and debate. Unlike The Last Supper and Annunciation, the Mona Lisa has few straight lines, or architectural elements, to use as reference points in making this determination. Search the internet for “Mona Lisa golden ratio,” and you’ll find some very creative interpretations of golden ratios in the Mona Lisa, with golden spiral overlays of varying positions, orientations, and sizes. This can seem very arbitrary and inconsistent, and they cannot all be right. It’s unlikely that Leonardo ever used the golden spiral that is now so closely associated with the golden ratio, since such logarithmic spirals were first described more than one hundred years later by mathematician René Descartes (1596–1650). Although it may be difficult to know da Vinci’s original intent in his composition, the simplest and most objective approach is to overlay golden ratio lines based on the height and width of the canvas, and the few available reference points of her head, neckline, and hands. Here we find that her left eye is precisely centered in the painting, and her hair is roughly bounded by golden ratio lines from the painting’s center to the sides of the canvas. We also find possible golden ratio proportions between the top of her head and her arm at her chin and neckline. Actually, it's not unreasonable to look for golden ratios applied to art because if the artist was aware of the supposed "beauty" of such a ratio they would be likely to introduce it into their work. (Just as photographers use the "rule of thirds").
Meisner also looks at that old favourite....the Egyptian pyramids ...and sure enough finds some golden ratios there. Maybe!
Then he moves onto the biological world: In 1979, German mathematician Helmut Vogel devised an equation to represent the Fibonacci spiral pattern with florets, where θ is the polar angle and n is the index number of the floret in question: Θ = n × 137.5º In this model, 137.5º is the angle of rotation, also known as the golden angle. Why 137.5? As it turns out, when you divide the degrees of a circle (360) by the golden ratio (1.618), the value you obtain for this arc is 222.5º. That makes the smaller segment of the circle 137.5º. What seems to be missing here is some explanation of why a plant does this. What's constraining it ?.......presumably something to do with packaging geometry and the timing of emergence of florets.
Meisner points out that the beauty and common appearance of logarithmic spirals is, unfortunately, a source of much confusion. Many people incorrectly assume that all logarithmic spirals are golden spirals expanding continuously by a factor of 1.618. In fact, the golden spiral is an unusual example of a logarithmic spiral—much like an apple being a special member of the fruit family, or a pentagon being a special member of the polygon family. All true golden spirals are logarithmic spirals, but not all logarithmic spirals are golden spirals, just as all apples are fruits, but not all fruits are apples. The same applies to the nautilus shell (beloved of mathematical-nature books). But Meisner points out that in the classic golden spiral, the width of each section expands by 1.618 with every quarter (90-degree) turn, and its proportions bear little resemblance to those of the nautilus spiral. However, another spiral exists that is just as golden. This spiral expands by a factor of 1.618 with every 180-degree rotation. Note how it expands much more gradually. Clearly, a golden spiral based on a 180-degree rotation is much more similar to the nautilus spiral than a golden spiral based on a 90-degree rotation. (Though, I guess you could keep doing this sort of thing with a spiral expansion of 1.618 every 80 degrees or every 96 degrees etc. until it fitted what you wanted).
He goes on to explore the golden ratio with insect morphology and with human faces. I remain to be convinced.
To be fair to the author he ends up saying: If you explore this topic in more depth, you’ll find some people who will tell you that the golden ratio is a universal constant that defines everything. You’ll find others saying the even the evidence that I’ve presented in this book does not exist at all. This is your golden opportunity to carefully consider what you’ve seen and learned, come to your own thoughtful conclusions, and then ponder the implications. In its own unique way, phi touches upon some of the most fundamental questions of philosophy and the meaning of life. When we discover common threads in the mathematical design of things in our world, especially where it seems unexpected or unexplained, it can beg the question of whether there could be something more than chance at work—a grander plan of design with some guiding purpose, or even a designer. Others may seek to explain these same observations as coincidences arising from natural processes in adaptions and optimizations.
Though this bit of speculation really seems to me to undermine his whole project. I'd like to see the biological-chemical- mechanical linkages which actually underpin the growth of biological phenomena and this should be possible. I'd also like to see the faces techniques applied to a greater range of faces (does it apply to children for example) or to Australian Aborigines as well as to Tibetans? I remember sculpting human portraits with clay and we were given an actual skull to work from as a foundation but I pretty quickly realised that not all skulls were the same..and hence doubt that his face measuring software is really going to work ...but if it does...what is the mechanism driving this. That's what's lacking in the book. Three stars from me. ( )
  booktsunami | Dec 21, 2023 |
Sumptuous large format book with lovely illustrations. Text is kind of goofy, though.
  themulhern | Mar 19, 2021 |
The Golden Ration by Gary Meisner is an exquisitely illustration, beautifully and clearly written introductory book about the Golden Ratio and related subjects. There are lovely full-colour illustrations and photographs on nearly every page. The book begins with the unique properties of the golden ratio and then continues on to its appearance in art and design, architecture (pyramids, cathedrals, musical instruments), nature (leaf and petal arrangements, fractals, spirals, facial proportions, buckyballs, quantum physics, golden DNA, the nautilus controversy), and many other interesting mathematical goodies such as tessellations, platonic solids, the Fibonacci sequence, Pascal’s Triangles etc. The book also includes appendices that deal with critical thinking, notes and further reading, and “Golden Constructions”. There are a number of equations and geometrical illustrations, but nothing particularly complicated. In the author’s own words: “not everything is based on the golden ratio, but the number of places in which it seems to appear is truly amazing and we are sure to uncover it more and more as technology advances and out knowledge of the physical universe expands”.

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This book was received from NetGalley in exchange for a review. This is my honest opinion of the book.
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  ElentarriLT | Mar 24, 2020 |
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"From the pyramids of Giza to the proportion of the human face, the golden ratio has an infinite capacity to generate beautiful shapes with exquisite properties...author Gary Meisner examines the presence of this divine proportion in art and architecture, as well as its ubiquity among plants, animals, and even the cosmos." --provided by publisher.

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