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Book: The Golden Ratio

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I'm not going to give this book a good review, so if you're looking forward to reading it, turn away now. I won't give a spoiler warning, as the book is non-fiction. How could I possibly spoil that?

Somewhere in the 253 pages of this book is a good book. In fact, I marked thirty pages or so with bookmarks, most of which I will transcribe notes on in the extended entry. However, the rest of it is mostly crap. It's the type of crap that if it were to appear in another book, it might be quite good, but condensed and contrived into the pages of this book, it's crap. It's as if the author went to Google, typed in "Golden Ratio" and all it's possible synonyms, and then arranged the search results chronologically and categorically. Then, deciding that this wasn't enough material, he went through the material and did separate searches on all the nouns so that he could use this extra material as background and segues.

I'll take a step back from this analogy to say that Livio is a smart guy that certainly didn't write this using the technique just described, but he is in dire need of good editing to point out what is a useful tangent and what is irrelevant filler. It's really cool to hear about facts like Benford's Law, but, as written, it belongs in a different book I've accused other books that similarly try to connect all of philosophy, mathematics, art, and history of being too ambitious. I'm not sure that ambition was the flaw here, as much as it was the Livio's desire to share all the information he has gathered. His background in astrophysics is also a conduit for more unnecessary material.

Here are some examples of what I deem irrelevant:
- Faust somehow gets a full page in the book (p. 196). If we follow the tenous chain, Faust draws a pentagram to trap the devil, which demonstrates the belief that magical powers are associated with pentagrams, which dates back to the Pythagoreans, who may or may not have known about the Golden Ratio, but the pentagram probably helped with the discovery of the Golden Ratio.

- The entire final chapter. It's as if the author had written an essay about mathematics and the Universe awhile back, and while writing the book, remembered this essay and decided that he could squeeze it in. To be clear, the essay is not about the Golden Ratio. It is mostly an essay on whether the universe is based on mathematics or is mathematics a human invention. On its own its an interesting read, but I'm puzzled to see it as the conclusion of this book

- p. 188-192. Livio debunks claims that the Golden Ratio appeared in Bartok's and Debussy's music. The main reason for him choosing this two composers, in my opinion, is so that he can conclude with a completely unrelated anecdote that mentions both of their names.

- Ten whole pages dedicated to Kepler, mostly so that Livio could also discuss Kepler's contributions to astronomy

There are four themes to this book, as I see it:
1. The emergence of the Golden Ratio
2. The interesting mathematical properties of the Golden Ratio
3. Debunking the appearance of the Golden Ratio in art
4. Extra stuff related to mathematics

With regards to (1), it belongs in the book and is reasonably interesting, even if I did feel that the organization of it could have been better. Livio goes a little too far back in history, back to the invention of counting. I found this part fascinating, but, as before, it belongs in a different book.

(2) was the meat and potatoes of the book. This part of the text moves really well and is very captivating. Of course I'm biased, but, then again, this is the heart of the book.

(3) overtakes most of the book. This, plus the previously discussed tendency to include too much tangential material, really hinders the book. While I appreciate Livio's debunking of the "Golden Ratio is everywhere" meme, rarely does he actually do a decent analysis. Instead, he waves his hand and looks for the next target. It's hard to get drawn in because the subject matter disappears just as soon as its introduced. The section where he discusses the Golden Ratio and paintings is thirty pages of, "Some have argued that this painting uses the Golden Ratio. Re-examining this painting this conclusion seems arbitrary. Next painting." The section on music is not much better. The only debunking section that seems to hold its own is the pyramids of Egypt.

(4) I've beat this horse enough.

So, in conclusion, between pages 78 and 120 you will find some really interesting stuff. In the other pages, you will occassionally find some interesting material, but you may wonder what it has to do with the Golden Ratio.

PS - despite the quote on the front cover, this book will not teach you anything about Picasso. I'm not sure Brown read this book based on his quote.



Initial counting was "one, two, many." (p. 15)

Some societies used base 20 (hands + feet). Sumerians used base 60 (p. 20)

When Alice says, "four times five is twelve," Carroll may just be using a different base (p. 21)

numerology + Gematria (p. 22)

- Pythagoras responsible for the terms "philosophy" ("love of wisdom") and "mathematics" ("that which is learned")
- geometric proof of Pythagorean theorem (p. 27)
- discovered harmonic progressions of musical notes
- odd numbers were masculine and considered good (light/good), even numbers were feminine and considered bad (dark/evil)
- 1 was considered the generator of all numbers and not a number itself. It also represented the point, and thus was the generator of all dimensions.
- 2 (woman) was the number of opinion and division
- 3 (man) was harmony, as it is the sum of unity + division.
- 4 was justice and order, space and matter
- 5 was love and marriage (2 + 3)
- 6 was the first perfect number (sum of all factors). 28 and 496 are also perfect numbers
- 10 represents everything: unity + division + harmony + space/matter
- pentagram was symbol of brotherhood/health (p. 34)
- possibly discovered incommensurability, which would have been devastating to them, as it means that everything cannot be reduced to whole number ratios

Great Pyramid (p. 51)

Rosicrucians (gah!) (p. 53)

- 5 Platonic solids (tetrahedron (fire), cube (earth), octahedron (air), icosahedron (water), dodecahedron (quintessence) )
- cube and octahedron are reciprocal shapes
- icosahedron and dodecahedron are reciprocal shapes
- tetrahedron is its own reciprocal

Math factoids

place-value system was created ~400-500s in the Hindu-Arabic world, while the Western world was entering the dark ages.

"algorithm" comes from the name of the Arab mathematician al-Khwarizmi, whose book "The science of restoration and reduction" is responsible for the term "algebra."

place-value made it to European society ~1200, though the abacus had been in use for awhile.

Golden ratio factoids

Golden ratio = 1.6180339887
Golden ratio^2 = 2.6180339887
1/Golden ratio = 0.6180339887

Golden triangles and Golden gnomons (p. 79)
- draw two diagonals in a pentagon from a single vertex to form a triangle. The center triangle is a golden triangle, the other two are golden gnomons.

sqrt(1 + sqrt(1 + sqrt(1 + ....)))) = golden ratio

1 + 1 / (1 + 1/ (1 + 1/ ....)))) = golden ratio (endlessly reciprocating fraction)

if you subtract a square from a golden rectangle you get a smaller golden rectangle. the diagonals of the larger and smaller golden rectangles intersect at the same point ("eye of god"). (p. 85)

Fibonacci sequence factoids

Fibonacci sequence was the first recursive sequence known in Europe (p. 97)

ratio of adjacent fibonacci numbers approaches the Golden Ratio. successive values of the continued fraction approximation of the Golden Ratio are equal to the ratios of the adjacent fibonacci numbers.

the sum of an odd number of products of consecutive Fibonacci numbers is equal to the square of the last Fibonacci number you used. e.g. 1x1 + 1x2 + 2x3 = 9 = 3^2

sum of any ten consecutive fibonacci numbers is divisible by 11

the unit digit in the Fibonacci sequence has a periodicity of 60. The last two: 300. The last three: 1500. The last four: 15,000. The last five: 150,000. The last six: 1,500,000.

The 11th number is 89. If you sum up the first ten Fibonacci numbers as:
= 0.01123595 = 1/89

Take four consecutive Fibonacci numbers. The product of the outer numbers, twice the product of the inner numbers, and the sum of the squares of the inner terms, results in a Pythagorean triple. e.g. 1, 2, 3, 5: 5, 12, 13

Binet formula based on Golden ratio calculates the value of any Fibonacci number. (p. 108)

The square of any term differs by the product of the adjacent terms by one. e.g. 3^2 = 9. 2*5 = 10. (p. 152)

Plants + Golden Ratio

Plant phyllotaxis: position of leaves on a stem. Leaves are rotated around stem at a fixed angular interval. Amount of rotation is always the ratio of two Fibonacci numbers.

Pineapples: each scale is part of three different spirals. The number of rows in each spiral is a Fibonacci number.

Sunflowers also exhibit spirals

Successive leaves separated by the 137.5 Golden Angle (p. 112). This angle is an energy minimizing state (the Golden Ratio is the "most irrational" of irrational numbers -- the continued fraction converges more slowly than other continued fractions)

Golden Ratio is well suited to properties of homogeneity and self-similarity (p. 114)

Dishful of magentic fluid over a magnetic field disturbed by silicone oil converges on a Golden Angle arrangement.

Logarithmic spiral

Self-similar: same at all scales

Nautilus, sunflowers, seashells, whirlpools, hurricanes, and galaxies exhibit logarithmic spiral

Archimedean spiral: constant spacing in spirals

Logarithmic spiral can be inscribed in embedded golden rectangles and golden triangles.

Logarithmic spiral is an equiangular spiral. If you draw a line from the center to any point on the spiral, it meets the curve at the same angle.

Falcons use logarithmic spiral when approaching prey. Allows them to keep head at same angle to prey, which is important b/c eyes are on the sides of their heads. (p. 120)

Painters + Mathematicians

Piero della Francesca

used Golden Ratio in mathematical problems, and used vanishing point, but Livio does not show any examples of Golden Ratio in paintings.

Luca Pacioli

published Summa, considered to be the first book on accounting
five reasons why Golden Ratio = Divine Ratio

1) "That it is one only and not more."
2) Golden Ratio definition uses three lengths <=> Holy Trinity
3) incommensurable ratio <=> ineffable God
4) self-similarity <=> omniprescence and invariability of God
5) even more Platonic model of universe than Plato described. Golden Ratio + dodecahedron <=> God + quintessence

Vitruvian man (p. 134)
- man is defined by whole number ratios.

- some of the earliest examples of polyhedra nets (sheets that can be assembled into 3-D polyehdra)
- use of rhombohedron in painting

- tried to create model of planetary orbits based on five platonic solids. Model was way off, but he put forth the notion of trying to come up with mathematical model of universe, helping advancing the introduction of the scientific method (p. 148)
- Kepler's Golden Ratio triangle (p. 149)
- Kepler's First Law: orbits are ellipses, with the Sun at one focus
- Kepler's Second Law: oribital speed is proportional to distance to sun. Derived this before gravity was known.
- no supernovas have been witnessed in Milky Way since Kepler and Brahe witnessed one each
- discovered that the ratio of consequtive Fibonacci numbers converges to the Golden Ratio

Ruth's missing square puzzle (p. 152)


Appearance of Golden Ratios in paintings is complicated by the its closeness in value to 8/5, a nice whole number ratio

Madonnas at the Uffizi
- some claim there is the Golden Ratio
- Livio argues this is arbitrary. Also, no evidence painters planned paintings using Golden Ratio

Leonardo da Vinci
- certainly was familiar with Golden Ratio from his work with Pacioli
- no conclusive evidence that he used Golden Ratio to design his paintings

- probably did not use Golden Ratio in paintings, despite claims to the contrary

Serusier (p. 168)
- post-Impressionist
- perhaps first painter to conclusively use Golden Ratio. Used it to "verify, and occassionally to check, his invention of shapes and his composition."

Russian cubist Marevna used Golden Ratio to layout paintings

Le Corbusier
- proponent of standardized proportions based on Golden Ratio. Modulor Man (p. 173)

Mondrian, despite assertions, most likely did not use Golden Ratio. However, there are so many rectangles in his paintings that it's easy to find a Golden Rectangle


No study has demonstrated a psychological basis for asserting that the Golden Ratio is more pleasing to the human eye


- ratio of lengths of rhythm compositions may have used Golden Ratio. Not conclusive.

- may have used Golden Ratio to construct compositions

- used Fibonacci intervals for successive notes


Fibonacci sequence appeared in poetry before Fibonacci
- Sanskrit and Prakit poetry known as matra-vrttas
- each meter is the sum of the two earlier meters


Penrose tilings (p. 204)

quasi-crystals of al-cu-iron and al-palladium-mn alloys demonstrated Golden Ratio in the heights of their layers


Golden Sequence
- demonstrates many levels of self-similarity
- start with 1, replace each 1 by 10, each 0 by 1:
- Ratio of number of 1s to 0s approaches Golden Ratio
- append the preceding line to the current line to get the next line
- Similarity at different scales: (1) whenever you encounter a 1, mark three symbols, for zeros mark two. From every group of three retain the first two, from groups of two retain the first. This is equal to the Golden Sequence

Koch snowflake (p. 217)

Fractal dimension
- no necessary to have whole number dimensions
- introduced by Mandelbrot
- 1-D: divide in half produces two objects
- 2-D: divide in half produces four objects
- 3-D: divide in half produces 8 objects
- Koch snowflake has fractal dimension of 1.2619

Golden Tree fractal
- reduction factor of 1/phi produces closest packing without overlap

Pocket universes (p. 222)

Stock market (p. 224)

Viswanath's number: 1.13198824 (p. 227)
- based on random coin tosses combined with Fibonacci sequence rules.

Benford's Law
- if you select random distributions of data, and a random selection of data from them, the first digit will be 1 30% of the time, followed by the digit 2 18% of the time. 9 is the first digit less than 5% of the time.

Prime number factoids
- 1,234,567,891 is a prime number
- the 230th largest prime is composed of 6,399 9s and one 8
- with the exception of 3, every Fibonacci number that is a prime has a prime subscript

Einstein: "How is it possible that mathematics, a product of huan thought that is independent of experience, fits so excellently the objects of physical reality?"

Comments (6)


Very helpful for my homework assinment. Thanks!!


After reading "The Golden Ratio" I must agree with the above negative comments: although containing lots of interesting mathematical factoids in different areas, the part of this book that is really relevant to the golden section is very small.
Below I listed some additional remarks:
- In spite of Brown's quotation on the cover that "math-buffs" will very much enjoy reading this book, all of the Fibonacci number factoids (listed above) remain without proof, although they can be quite easily proven (I did so myself). Why did Livio omit this in appendix? Also a good intuitive explanation for Benford's law exists ( ), but is not included.
- Livio gives an intuitive proof for the property of phi to be the "most irrational number": because its continued fraction expansion contains only ones, successive convergents converge more slowly to phi than for any other irrational number. But the denominators of these convergents also increase the slowest, and this could cancel the property that phi is the worst approximated by rational numbers. Although Livio's explanation is widespread, it is wrong.
- The only connection between the golden ratio and logarithmic spirals is that the golden ratio can be used to define specific examples of logarithmic spirals. But none of the many examples of logarithmic spirals given in nature (except perhaps the spirals in the sunflower) are related to the golden section. See also .


Thanks for the additional commentary and links!


Thank you for all the information. It was very useful!

I really enjoyed Livio's book, for what it was.

He does repeat the error of seeing the spiral parastechies of phyllotaxis as logarithmic, when they're closer to Fermat spirals. Dixon makes this point and ironically, is usually there in every bibliography..

Also, shame he missed the more contemporary artists who have used the ratio in consistent and contrasting ways, eg Don Judd or Mario Merz (both celebrated figures).

I think you were too hard on Livio!


In general, I agree with the analysis of Livio's GR book as being a survey of the subject. He has no real understanding of how this proportion functions in nature, so his assessment is one of only bringing up a scattering of occurences and "debunking" them when the occurences are not precise.
My research into the interference pattern of the harmonic series identifies Phi as a natural damping function that suppresses the emergence of fractional waves. The temporal proportions of the Fibonacci series spirals within the harmonic series to Phi/phi proportions to create a "halo" of slack around this proportion in any standing wave system, thereby suppressing a range of fractional wave formations. As example, this is why speaker enclosures suppress standing wave formation up to 1.625 rather than exact 1.61803... and why the occurences of Phi are imprecise in nature along with historical human usage.
Because of Livio's simple lack of knowledge about the actual physical manifestation and utility of the golden ratio in nature, he misinterpretes most of its occurences.
Anyone who wants further detail on this can email me and I'll be happy to share what I know on this. There is a lot more to it.

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