The Golden Ratio

by Gary B. Meisner, Rafael Araujo (Illustrator)

The Golden Ratio examines the presence of this divine number in art and architecture throughout history, as well as its ubiquity among plants, animals, and even the cosmos. This gorgeous book features clear, entertaining, and enlightening commentary alongside stunning full-color illustrations by Venezuelan artist and architect Rafael Araujo.

From the pyramids of Giza, to quasicrystals, to the proportions of the human face, the golden ratio has an infinite capacity to generate shapes with exquisite properties. 

With its lush format and layflat dimensions that closely approximate the golden ratio, this is the ultimate coffee table book for math enthusiasts, architects, designers, and fans of sacred geometry.

Genres: Science, Nonfiction, Art, Mathematics, Design, Philosophy, Nature

221 pages, Kindle Edition

Published October 23, 2018

About the author

Gary B. Meisner created "Phi 1.618 - The Golden Number" (GoldenNumber.net) in 2001, a leading website dedicated to the mathematics, appearances and applications of the golden ratio. In 2004, he developed PhiMatrix Golden Ratio Design and Analysis software (PhiMatrix.com), used by thousands of artists, designers and others in over seventy countries. In 2018, he authored "The Golden Ratio - The Divine Beauty of Mathematics." Gary’s work has been featured in interviews, exhibitions and an educational TV documentary. His sites provide a place to share and discuss findings on the golden ratio, to help others to appreciate the beauty and design around us, and to apply these same principles of design in their own creative works. This was inspired by his lifelong interests in mathematics and science, which translated into a career in finance and technology. After earning his CPA, and BS and MBA degrees from two top business schools, he spent most of his career in operational CFO/CIO roles. He is now an independent consultant, and continues his research, writing and software development.

Reviews

[Claudia · Rating 4 out of 5]

There are numerous books on Golden Ration topic, basically with the same information, more or less detailed. What stands out in this particular one is the presentation.

The book is divided in 6 sections: Golden Geometry, in which are outlined the basics of 1.618 number, Phi and Fibonacci, presenting the relation between the numbers in Fibonacci sequence and Φ, The Divine Proportion, on how the name was coined and its appliance in art, with examples from da Vinci, Rafael and Michelangelo, to name just a few, Golden Architecture and Design, showing various buildings and monuments using this ratio from the Great Pyramids to Nothe Dame, Golden Life, on how Phi is present from DNA to flowers and facial features and Golden Universe, in which we learn that Φ can be found from quantum level to planets’ orbits and black holes.

Each of these sections has diagrams and pictures, to deliver a better visual experience. It’s an exquisite read and even if you’re not fan of math, it’s an easy read, which will enchant both your eyes and your mind. Beside being a science book, it is also an art and photography album.

>>> ARC received thanks to Quarto Publishing Group – Race Point Publishing via NetGalley <<<

[Clare O'Beara · Rating 4 out of 5]

As opposed to pi, 1.342, we discuss phi, 1.618. This number is used to give a proportion and build rectangles, such that a painting may be shown to be two thirds one thing and one third another, with the more exact proportion of phi.

After a look at the Greeks and Arabian scholars and mathematicians, Kepler, Fibonacci and daVinci, we turn to paintings and architecture, then natural world occurrences. I enjoyed the many colourful graphs and photos. I am not at all sure that we are told anything new. Painters and architects constructed their work with care from outline up, so getting scale and perspective right and enabling them to transfer from a small work to a large surface like a wall, by gridwork. Photographers today use the principle of thirds up and across, dividing a view into nine. I think the imposition of some of the golden rectangles on portions of murals and photos seems arbitrary and forced in some cases.

This book will be enjoyed by those studying maths, and probably art and architecture as well. I downloaded an e-ARC from Net Galley. This is an unbiased review.

[Alina · Rating 3 out of 5]

***Note: I received a copy curtesy of Netgalley and Quarto Publishing Group – Race Point Publishing in exchange for an honest review.

Even if it is written for people who don’t have much knowledge about mathematics, some effort is still required to understand the concept and the way it’s applied. Though I’m sure it can be enjoyable as a surface read also.

The book is comprised of 6 chapters, each of them offering more information on the concept of phi (Φ ≈ 1.618; aka the golden ration | golden number | golden cut | mean ratio | divine proportion), its relationship with the Fibonacci sequence, its use in art (Da Vinci, Michelangelo, Botticeli, Rafael), architecture (pyramids, temples, cathedrals), design (musical instruments, brand logos, cars, fashion), and other sciences, its presence in nature and the Universe. The text is accompanied by diagrams, photos and illustrations to explain and support the examples.

Even if I was amazed about phi's properties and mostly about its occurence in nature, and I deeply enjoyed the art and the historical voyage, I'm still not convinced about its intentional use in all the given examples, and I especially didn't like the divine note in which the book ended.

P.S. The author has developed a site/software for phi analysys: PhiMatrix

[Logan · Rating 5 out of 5]

I was introduced to this topic in geometry class with no explanation about it whatsoever, so I decided to do some research on it myself and found this book. This book is a great read, by teaching you about Phi and the golden ratio throughout history like in renaissance paintings, DNA, the parthenon, and more. Overall this book has introduced me to this topic and has given me many examples about how this ratio is used, but I would still like more answers on how to apply it.

[Rahul · Rating 4 out of 5]

4 Stars

Can't say much about phi and its various interpretation and application by humans. I just skimmed through this book. But anyways, this book contains world famous paintings, drawings, pictures and illustrations, which are extremely beautiful.

[Elentarri · Rating 5 out of 5]

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”.

________________________________________
This book was received from NetGalley in exchange for a review. This is my honest opinion of the book.
________________________________________

[Jacob · Rating 2 out of 5]

The author divides the world into phi skeptics and phi believers. He believes he can make you a believer by overlaying rectangles on images and saying "see, here it is again".

As nature randomly advances and increases in complexity, various optima will be stumbled upon. Some problems will have a solution that involves the golden ratio. I wish the book had been an exploration of why that may be the case for various examples. Instead, the book is simply a list of examples.

[وسام عبده · Rating 5 out of 5]

Fascinating voyage in art, mystic and history in quest of finding the mean and influence of divine number. Its look like a very interested novel for Dan Brown, and a series history for Hobsbawm.
No less than five stars.

[Annie · Rating 4 out of 5]

Originally published on my blog: Nonstop Reader.

The Golden Ratio is a book on mathematics, written by Gary B. Meisner, an applied math guy who kept finding instances of phi (φ) in everything from sunflowers to renaissance paintings.

From his website:

The inspiration for the site was a deepening awareness and appreciation of the beauty and design in life. The more I explored, the more I discovered that the number called Phi, or 1.6180339…, appears as a pervasive constant of design in many aspects of our existence.

This book is ostensibly aimed at the layman. That being said, it's not going to be a free ride. Much of the content covers concepts which require a modicum of effort and thought. It is enjoyable on a surface read. The illustrations and accompanying notes are appealing and historically interesting, however, the meat of the book requires some effort and probably some fiddling with pen and paper, following the proofs in the book. The description of Pythagoras and the Kepler triangle made me smile. (It's both gorgeous and elegant).

The book is arranged in chapters which introduce and develop the idea of phi, explain the golden ratio and progress to a discussion of occurrences of the golden ratio in art, mathematics, biology, engineering, and astronomy amongst others. The text is accompanied by illustrations and stock photography which support and illustrate the examples. The book ends with appendices, a bibliography and further reading list along with notes from the author, image credits and acknowledgements and finishes with a cross referenced index.

This edition was released 18th Sept 2018 by Quarto's Race Point imprint, it's 224 pages and available in hardcover (and possibly ebook) formats.

Four stars for the engaging text and enthusiastic prose along with the lovely illustrations. The author's passion for the subject shines throughout the book.

Disclosure: I received an ARC at no cost from the author/publisher for review purposes

[Sally F · Rating 5 out of 5]

This is a lovely book which explains and examines the Golden Ratio/Divine Proportion using clear and engaging text accompanied by a kaleidoscopic array of beautiful imagery from the worlds of art, architecture, and nature. This book is sure to be appreciated by anyone who loves math, design, or classical art, as well as by those who are merely curious about the title topic.

I am one who was merely curious. I haven't studied any math since high school (eons past), but I was able to grasp (with a bit of concentration) the very basic of the mathematical concepts presented in this book. I was actually quite pleased with myself when I encountered and even remembered the Pythagorean Theorem on page 27! While most of the mathematics in the book was over my head, I nevertheless found much to like in its pages.

For example, nearly 100 pages are devoted to a presentation of the appearance of the Divine Proportion in art and architecture. I thoroughly enjoyed looking at the masterworks of DaVinci, Michelango, Botticelli, and more, and seeing evidence of the Golden Ratio in these famous art pieces. I was especially drawn to a section of the book which focuses on the design and construction of the great cathedrals of Europe. Photographs (with PhiMatrix overlays) of the rose windows at Notre Dame and Chartes are impressive.

The last part of the book takes a look at the appearance of the Divine Proportion in the natural world. Again, I am not a mathematician, but I had fun counting the plant spirals on photographs of pine cone bases, and looking for the "beauty of fives" in pictures of flowers and fruits.

Overall, I enjoyed spending time in the pages of this book. It has something for everyone. And, it looks pretty impressive on my coffee table; if I were hunting a mate, I might impress a super successful STEM dude—or perhaps an artsy guy--with this book!

[Leslie · Rating 5 out of 5]

Hermosamente ilustrado

[Yannick · Rating 3 out of 5]

2,5/5

Ik ben altijd al gefascineerd geweest door de Gulden Snede en Fibonacci. Dit boek kreeg ik daarom cadeau. Helaas ben ik na het lezen van dit boek iets minder stoked geworden over de Gulden Snede..

Het boek is mooi vormgegeven en heeft mooie tekeningen en veel voorbeelden, toch had het veel beter kunnen zijn. Fibonacci wordt inconsistent weergegeven: de ene keer met één 1 in de reeks, de andere keer met twee 1-en. De enige juiste is uiteraard die met twee 1-en. Sommige andere (kleine) berekingen zijn ook fout. Daarnaast is de PhiMatrix die de auteur zelf over verscheidene foto's en afbeeldingen heeft gelegd niet consistent toegepast. De auteur vraagt de lezer zelf een opinie te vormen over de Gulden Snede en het geloven van de waarde ervan. Het is lastig positief te zijn als de gegeven informatie zo slordig wordt gegeven.

Ik heb kort contact gehad met de auteur (Gary Meisner) die toegeeft dat het wat duidelijker had kunnen zijn (geeft geen fout toe), en gaf als excuus voor de slordigheid dat hij weinig tijd had om er aan te werken.. I rest my case.

[Rachel Noel · Rating 3 out of 5]

*Book provided via NetGalley for an honest review.

A lot of this book had me feeling like I really didn't understand math or art. Meisner does a wonderful job at the beginning of explaining what the golden ratio is, what makes it so special, and what its history is. But the application of it to the artwork was less than clear to me. A lot of times it really felt like the software he designed to find golden ratios in artwork was just finding coincidence rather than purposeful use. Some of it, like the use of a canvas sized to match the golden ratio, does apply. Others it felt pretty forced. That being said, I do give Meisner full credit for including a section on the controversy of the golden ratio and how some people argue that it doesn't exist or doesn't have as much influence as others claim, etc. Overall, it was a good beginner's book for those interested in the golden ratio.

[Jeffrey Romine · Rating 4 out of 5]

Remarkable. Amazing. Astounding. Incredible. Perhaps not even these words are sufficient to describe the feeling that comes over the person who sees it, gets it, appreciates it, and marvels at it. Euclid saw it. Kepler saw it. We need to see it too. There are a number of YouTube videos that discuss the Golden ratio along with the Fibonacci sequence. Seeing it on paper and plugging numbers into a calculator for oneself is the most convincing way to study it. Meisner does a nice job. I found myself going back to review the relationships a number of times.

[Steve · Rating 3 out of 5]

Coffee-table book for mathematics

This is a beautiful book, at least as I can tell in a PDF. I liked the art re-creations, photos and the examples of the golden ratio in biology, although I found that tying these into the golden ratio at times a stretch. While the author was clear that some of these connections were tenuous, I felt that the author was trying too hard to make these connections.

[Lloyd Downey · Rating 3 out of 5]

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.

[James Harbaugh · Rating 5 out of 5]

The book is a visual masterpiece of masterpieces containing the ratio itself and it's inverse in the Fibonacci Sequence, platonic forms, all kinds of applications, and the a correction one example that is best known but is incorrect in the form of snail shells (though he does have some close fits with more complex algorithms including the ratio). It's wonderful to look since so much of math is left brained with algebraic expressions that are precise but are best coupled with the more right brained geometries of old Greek proofs (which he includes in a few places with Euclid). That said, I'd like to tag on a long tangent with some neuroscience gibberish to why this and other forms are "beautiful"


Theory of mathematical beauty from neuroscience from nature from mathematics-

1) Pleasure & Pain -The first point relies on limbic system of the brain and it's function as a prediction network with or Timothy Leary's Oral Circuit as outlined in RAW's "Prometheus Rising." The basis is pleasure (and accurate or better than predicted) or pain (the an error in a prediction) for the basic Prediction/Expectation-Observation-Error Model. A large part would be in the Anterior Cingular Cortex that deals with general pain in this form with other network nodes and is integrated into tactile pain, social pain, but more general forms of "rightness" or "wrongness." Observe the cringe you get when you're listen to a song you know well and a note is off or when your's watching a movie and the voices are way out of sync with the moving mouths. In Seth Horowitz's "The Universal Sense" he describes how nails on a chalk board and a metal rack on pavement are so unpleasant because that are so close to being a predictable note but are scholastically always off of a prediction and thus irritating and an extreme case of the unpleasant or ugly in a more solidified form (since habituation is not an option unlike something like heavy metal).

2) Symmetry - Using this model, an easy example for a predictions would be chirality or how well one side mirrors the other as one side would provide a prediction for the other. In Julia Smith's "Evil" more symetrial faces are a general heuristic for more beautify faces in this sense though they are not the most trusted which tends to go to the most average of all the faces in the group (nor does this include social beauty with things like eyebrows, beauty marks, etc which would fall under more complex conditioned standards of beauty). In Dan Ariely's "Predictable Irrationality" he takes to different faces that were originally hard to compare in terms of attractiveness and he later added a Photoshoped version of a less symmetrical version of one of the faces. After wards the original was seen as better than asymmetrical recreation and with the irrational transitive property that the tie was broken with the manufactured ordinal pair also giving weight above the once hard to judge apples-to-oranges competitor (the dirty tactic of the wing woman or man to to make a person look good... a roll I played all too often in my late teens and early twenties).

3) Glamour - To barrow whole sale from Lieberman and Long's "The Molecule of More" (The definition of glamour is a beautiful illusion with a quote from Virginia Postel on pages 9-10) “The word itself originally meant literal magic spell and promises to transcend ordinary life and make the ideal real. It depends on a special combination of mystery and grace. Too much information breaks the spell." There are levels of Woo and I like to go for the simplest first while acknowledging that there are layers and contexts. So first up is an old summery from Scientific America's Book of the Brain dealing with Homosexuality. The optic nerves carrying raw data from the eyes to the processing levels in the visual cortex (V1-V5) make a crisscross at a sight known as the optic chasm which is close to the hypothalamus which is a master node in the endocrine system (hormones). In autopsy of gay man and area in this region (fill in later) tended to be smaller and comparable to heterosexual women as opposed to the greatly enlarged bump of heterosexual men (remembering that straight men tend to be largely visual in lust). As additional evidence that this is subconscious pattern recognition first is Leonard Mlodinow's Subliminal where the occiptial lobe was damaged (from poorly fitting helmets in WW1 allowing bullets to get this area in British Soldiers) but the eyes and optic nerves were fine resulting in a blind sight that allowed for subliminal recognition of rudimentary facial expressions and objects but no more as an indication that some of the basic patterns are before the visual cortex. Next is the opposite with Daniel Dennet's "Consciousness Explained" where blind patients with damaged eyes but functioning occipital lobes were fitted with computerized tactile grids so they could see with their skin (guessing merkle cells). As explained in Lisa Feldman Barrett's "7 1/2 Lessons about the Brain" blind brail uses or artificially blind brial users integrate the sensations from there skin into the tightly knitted mapping structure of the occipital lobe for sensory imaging (which it think goes through the parietal lobe and probably somatosensory cortex) but doesn't pass the optic chasm area mentioned earlier. So main thing returning to Dennet's findings what that the blind patents with the tactical imaging had pornographic images press do there skin (with a system of dull pins in a grid) but were disappointed that they didn't get aroused (they were also straight men). So that said, there is something called the waist to hip ratio outlined in David McRaney's "You're Not So Smart" which is around 7 to 10 as a min with more exaggerated hips for a given waist eliciting a stronger subliminal response (the male jewel beetle example in the same chapter is a great analogy for how such heuristics can greatly reduce a man's intelligence in the form of glamour even if that man is an insect which feels fitting in the moment... like a moth to flame). Just a guess but they's probably a nerual structure tying the hypothalmus to the optic chasm that is excited with raw data fed to the V1-3 with that is looking for this ratio in pixils width to then trigger the hormones of rut in heterosexual men. So that would be a different layer of "beauty" for mating purposes though it's led to terrible outcomes with manipulations in corsets, plastic surgeries, etc (McRaney mentions how Marilyn Monroe would be plus sized in the modeling world now despite the ratio heuristic accommodating much thicker women). Although women tend to rely on other factor such as the complex sense of smell with the sweat of a potential partner and the difference of the Major Histocompatibility Complex (also a natural safe guard against potential recessive mutations so long as the system is not tricked by birth control at mate selection) as outlined Rachel Herz's "The Scent of Desire" there is also a lesser role for the visual and other lust factors. In Kathleen McAuliffe "This is your Brain on Parasites" she introduces a straightforward link on how deformities (asymmetries) could be a subliminal sign of pathogens or bad genes for a harsh environment and thus a subliminally repelling nature along with matted or unkempt hair and a general lack of hygiene or other proxies for physical or mental aliments which would the opposite of fitness in such times (the latter more of a conditioning complex rather than in the mathematical or geometric theme though fitting into cultural exceptions of a general model and diverse conditioned subcultures).

3) Back to the Book - So with a smattering of factoids from researcher as a base for why any ratio would seem good in a shallow sense and not a deeper Socratic sense, I hope the rest will be clearer. From the bottom up (a zero point to a gauge or a gauge to a standard plus infinity) the Fibonacci sequence is a growth function not only for the original rabbits but also a cellular level (A growth pattern as there are others but it shows up a lot given this). So in terms of food or mates, using this heuristic ratio could approximate health as well as something organic since it would naturally have grown to this ratio in it's full size (the gauge if the of 1.618 if the Fibonacci sequence started as a zero point or a standard infinity shift the infinite string from the decimal point if starting at the gauge though cells and molecules would be the point or gauge in this case). So the tighter the ratio (indicating that was the intended growth pattern) the healthier the growth without possibility harmful interruptions with unpleasant surprises lurking in taste, food poisoning, or the higher child mortality as examples. So with that in most likely ingrained the pattern has some other interesting properties. Symmetry and repeatability can also be boring after a while due to habituation (listening to a new song over and over, then getting tired of it, and then liking it occasional when it comes on but also with conditioned memories of it playing... oh Mickey you're so fi... you know). However, Phi doesn't repeat in decrease terms, has really odd power rules, and allows for novelty with which is also dopamine dependent when something is new and sticks out which would play into deeper levels of glamour/beauty like an asymmetric beauty mark... once again it's shallow and more primitive in the brain all kinds of other variables, parameters, contexts, conditions, etc but more complex models need basic building blocks. Tool has an interesting song titled Lateralus that uses the Fibonacci sequence as music is another area with mathematics very close to it's application though Pythagoras pined down much of the basics long ago in that field.

[Maggie Macnab · Rating 5 out of 5]

The engaging golden proportion/sequence is found in spirals, stars, triangles, fractals and natural forms all around us. This mathematical proportion supports all kinds of 2 and 3D design in the beauty expressed by its universal aesthetic appeal, and in function by being a prevalent pattern of nature. Non-mathematicians might find themselves a bit challenged (even after years of incorporating it into design teaching and writing about it myself I still need to think about its presentation), but the language is beautifully balanced with illustrations from Rafael Araujo and the many images of art, architecture and biological/natural systems that incorporate the phi proportion. The visuals engage your natural aesthetics with analytical skills to help you to understand and absorb the principles described.

Some of the proportional overlays might be difficult to discern from the background because of contrast issues, and some of the written explanations might take more than one read because while math is elegant it isn't necessarily simple: translating math into words is an interpretation. Like anything worthwhile it takes some reciprocal investment and patience. As I would tell my students, get out a piece of paper, pencil and compass, and create the ratio yourself by hand (there are many how-to videos on the internet). Experience is the best teacher. Learning about it will only elevate your appreciation of beauty and its intentional structure, but like all expansive concepts you need to be both receptive and active to fully appreciate and incorporate it. The Golden Ratio is a wonderful place to start your investigation. A worthwhile investment!

Disclosure: I received a complimentary copy of this book via Race Point Publishing for review purposes.

[Debbie · Rating 3 out of 5]

"The Golden Ratio" is a look at phi and if it really is found in art, architecture, and nature as much as is claimed. The author started with a history of the development of phi and how it relates to pi. I enjoy math, but the author didn't spend much time explaining phi at a common person's level, so his points and formulas were sometimes lost on me. I assume the target audience is mathematicians and such who already understand how phi is used.

The author then talked about how the Golden Ratio has been used and can be found in things like art, architecture, nature, logos and even at a molecular level. He talked about his Golden Ratio finding software and how this has been used to examine art, architecture, etc., to see if the Golden Ratio really is found. He would show a picture of the object with these ratios overlaid as outline-only boxes of different colors.

Unfortunately, the boxes often weren't clear either due to overlaps (several boxes starting along the same line but only one color being shown) or the box's line color blending in with the background color. He strongly made the point that the Golden Ratio was found in these things. But unless he explained the starting and ratio points in the text (and sometimes he did), I often couldn't clearly see what the boxes were showing and so couldn't appreciate the full impact. Perhaps these boxes will be easier to see in the printed book. While a lovely book, I felt like I wasn't understanding enough of what was explained or shown in the book.

I received an ebook review copy of this book from the publisher through NetGalley.

[Vijayender Karnaty · Rating 4 out of 5]

It's a good read. The chapters on the mathematical properties and construction of golden ratio were quite a delightful read - by throwing light on the numerous way we keep coming across phi, by the numerous properties coming out of phi and its close cousin fibonacci sequence and painting the historical perspective introducing the legends of mathematics and scientific exploration in general.

I think school students with an interest in math will find this content very fascinating and enjoyable. I wish I had come across it at a time in school when number theory was quite a fascinating topic. And this would have helped get a historical perspective and opened more avenues for exploration.

Chapter on modern design and fashion was a little boring and felt a tad bit preachy. But the following ones on life and universe has offset the damage.

Overall it does a fair job in instilling a mystery that kept explorers busy through the millenia!

[Jason Baldauf · Rating 4 out of 5]

Meisner gives a nice introduction into the discovery of the golden ratio and the Fibonacci sequence. At times it delves into mathematical concepts that are a bit intense, but given the subject, it's expected. It then moves to how it can be found in nature, architecture, art and space. A short read with good pictures that gives an overview of the subject if you're starting to dive into sacred geometry. It doesn't take a mathematical inclination to appreciate the elegant beauty of it's intriguing role in the universe.

[Naomi · Rating 2 out of 5]

While it has some good background information on the mathematics of the Golden Ratio and good visuals, I found this book to be highly biased. It searches for and finds the Golden Ratio (often using convoluted methods and arguments like stacking the Moon in top of the Earth) and argues intent to all of the instances found, which isn’t backed by reliable sources. Clearly a sales pitch to use the author’s website.

[J Booth · Rating 5 out of 5]

Simply a marvel of mathematics

l have read much on the topic of Golden Ratio. I am in agreement with the belief it deserves credit as one of the greatest elements of mathematics and natural science. The logial proof is convincing on its own. The multitude of evidence across so many disciplines is only ignored at the peril of opening future doors of yet to be found truths. What an adventure this study opens! KUDOS!

[Stan · Rating 3 out of 5]

Fun book. Not too long. Discusses how Phi (the Golden Ratio) came to prominence. Then in the concluding chapters he shows examples of Phi in nature, art, architecture, etc. It was really enlightening. I thought the book was just the perfect length. I'm sure he could have rattled on for many pages, listing phi examples, but he keeps it to a moderate length. Just enough to keep you interested, without boring you.

If you like math, and you like art and nature, this is worth the read.

[Roshell Bissett · Rating 3 out of 5]

I liked this book. I wanted to love it, but I couldn't. The images are beautiful and I'm sure I will look back at them, but somehow I found the content lacking. At times I found it difficult to follow and I wanted more explanation as to the history and "meaning" of the golden ratio. I would still recommend it though and am glad I read it.

[Autumn Kearney · Rating 5 out of 5]

This is a beautiful book. Who would have thought that I would enjoy a math book? I struggle with math on a daily basis. However, I am a visual learner. Therefore, I am drawn to books containing illustrations and color photographs. They help me to understand abstract concepts. I highly recommend this book to anyone else in a similar situation.

[Jeff · Rating 4 out of 5]

This book is an interesting look at math history with intersections in art, architecture, engineering, natural science, biology, chemistry, astrophysics, etc. Meisner looks at the natural occurrence of phi as well as the human use of the irrational number in history.

[John · Rating 3 out of 5]

A good overview

A very good overview of the golden ratio. However, the presentation of the evidence by a series of examples became monotonous. I found myself skipping ahead when I felt the point had been made.

[Elizabeth · Rating 5 out of 5]

Art and geometry, the patterns of nature. How we see and what we miss that create beauty of what we see. I am not a math person but I still loved this book. So much visual information is helpful and illuminating.

[Viking One · Rating 3 out of 5]

From the visual point of view, this book is spot on. Excellent graphs, printing, drawings, explanations. Sometimes it takes the presence of phi in nature a little bit too far, as when discussing some astronomic coincidences. Overall a good starting point in the subject.