# Is Beauty Quantifiable ?

## 144 posts in this topic

[...] . . . . .number called the golden section, or the golden ratio . . . . .

In designing manufactures, and in architecture, the Golden Ratio is a way of measuring proportions in order that the small details of a design may have a perceptible ratio to elements of the design that are several steps larger. The small details may be proportionally coordinated with the large.

You would have to see the diagram of the "whirling squares" that has the Golden Ratio as one of the causes of its generation. The mathematics of it is the basis for the Golden Spiral. The proportions of the elements of the main fronts of the Parthenon are decided using the Golden Ratio.

The Golden Ratio is one of several ways that architects measure off elements that are slightly or greatly smaller and slightly or greatly larger than a base element. The elements are systematically and proportionally related in a way that is perceptible as a type of order. The psychologists of our era are too concerned with the acceptability of the likes of impulsives in the military or with the approvals granted by one social consensus or another, and they have no time for the causes of beauty. There is more that needs to be said, and especially regarding the proportional systems used in music.

The Golden Ratio is one of many systems of applied measurements. Others are, the Root and Square Root Number System, dividing things in accordance with the SI system of measurement, e.g., in tenths or one-thousandths, or using the English units involving 1/8ths, 1/16ths, and, 1/32nds, for example.

The architecture of the Roman Empire was Pragmatically imitative of the appearance of the works of the AGs, however, the Romans measured elements using feet and fractions, and the Greeks divided proportional areas using the Golden Ratio and other mathematical ratios or scales. The "look" of the architecture of the AGs was caused in good measure by the use of the proportional harmony system they called "Dynamic Symmetry" and an appreciation of life. The "authoritarian" "look" of the Roman architecture was determined in part by the use of unit measurement scales and ideas of political grandeur. The Romans had not used intellectual causes for architecture just as the science of geometry was lost on most of them.

Modern European International Style architecture divides areal proportions in tenths, while Meis van der Rohe used the greater and finer divisions of the Golden Ratio. The differences are marked. American designers mostly use 1/8ths, 1/12ths, 1/16ths, and, 1/32nds, to divide spaces, and the Europeans mostly use the tenths of the Metric System..

Are these differences perceptibile? Are these systems of measurement visible in determining the relative importances of every element of a design, or in making the design more dramatic and interesting? Yes.

Unless a design is rendered in terms of an imitation of nature or proportionless goo, a numerical geometric measurement system of some type must be used in every new design created. A purposeful design will express the relationships and importances of all elements in the design to all others in an ordered, dramatic, perceptible, proportional, and systematic way.

I have a course written on the Dynamic Symmetry system of proportional harmony, and its a highly sophisticated principled lecture and demonstration chalk talk.

* * * * * * * * * * *

Inventor

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[...] . . . . .number called the golden section, or the golden ratio . . . . .

In designing manufactures, and in architecture, the Golden Ratio is a way of measuring proportions in order that the small details of a design may have a perceptible ratio to elements of the design that are several steps larger. The small details may be proportionally coordinated with the large.

You would have to see the diagram of the "whirling squares" that has the Golden Ratio as one of the causes of its generation. The mathematics of it is the basis for the Golden Spiral. The proportions of the elements of the main fronts of the Parthenon are decided using the Golden Ratio.

This is the kind of 'analysis' that gives objective esthetics a bad name. It is dogmatic myth abusing mathematics to impress the gullible.

There are endless ways to mathematically generate repeating or nested symmetries, some of which describe repeating patterns that show up in nature through spirals, etc. ("Whirling squares" is jargon for a particular spiral.) The patterns may or may not be pleasing to look at, but the claims to base esthetics on them is a hoax that is no better than Pythagorean number mysticism. "Artists" and "architects" can generate proportions in many different ways, the methods of construction having no mathematically intrinsic significance.

In the Fibonacci sequence of numbers each successive number is the sum of the two previous numbers -- x_n+1 = x_n + x_n-1, so the sequence is 1,2,3,5,8,... It is only one pattern out of an infinity of possibilities. The so-called golden ratio is the limit of the ratio of successive terms in the Fibonacci sequence, Lim (x_n+1/x_n). That number is also the positive root of the simple quadratic equation x²=x+1, which can be easily derived as the equation for the limit of the ratios. The positive root is the irrational number (1+√5)/2 ≈ 1.618 to three decimal places.

The same number is also the ratio of the lengths of two line segments of length a and b satisfying the simple proportionality (a+b)/a = a/b -- because if you solve for the ratio a/b you see that it satisfies the same simple quadratic equation. It isn't magic.

The same simple equation turns up in many different applications. It turned up back in this post earlier in this thread as the ratio of the "computational efficiency" of two root-finding algorithms:

The computational efficiency index of the secant method is higher than regula falsi by (1+sqrt(5))/2 = the so-called Golden ratio, so rationalist Greek mystics can infer that it is also more beautiful. :-)

That was the case because it is the root of the 'characteristic equation' for the difference equation relating successive truncation errors in a root-finding algorithm, and the characteristic equation happened to be the same simple equation x²=x+1.

The golden ratio, 1.618..., is just a number, not magic with intrinsic esthetic properties. The ratio is no more or less intrinsically "esthetic" than 1.60, 1.55, or an infinity of other numbers. It's a good thing that is the case since as an irrational number it has a non-terminating sequence of decimal places -- it does not exist as a rational number, so it is a good thing we don't rely on it for "beauty" or all rectangles would be "approximately ugly".

The Golden Ratio is one of several ways that architects measure off elements that are slightly or greatly smaller and slightly or greatly larger than a base element. The elements are systematically and proportionally related in a way that is perceptible as a type of order. The psychologists of our era are too concerned with the acceptability of the likes of impulsives in the military or with the approvals granted by one social consensus or another, and they have no time for the causes of beauty. There is more that needs to be said, and especially regarding the proportional systems used in music.

No, it has nothing to do with psychologists obsessed with "impulsives in the military" or "social consensus". They simply do not attribute "causes of beauty" to numerology in accordance with mythology.

The Golden Ratio is one of many systems of applied measurements. Others are, the Root and Square Root Number System, dividing things in accordance with the SI system of measurement, e.g., in tenths or one-thousandths, or using the English units involving 1/8ths, 1/16ths, and, 1/32nds, for example.

The golden ratio is not a "system of applied measurement". It is a number. That's all.

The choice of different units of measurement has nothing to do with the esthetic proportions of the things they measure.

There is no such thing as a "Root and Square Root Number System".

The architecture of the Roman Empire was Pragmatically imitative of the appearance of the works of the AGs, however, the Romans measured elements using feet and fractions, and the Greeks divided proportional areas using the Golden Ratio and other mathematical ratios or scales. The "look" of the architecture of the AGs was caused in good measure by the use of the proportional harmony system they called "Dynamic Symmetry" and an appreciation of life. The "authoritarian" "look" of the Roman architecture was determined in part by the use of unit measurement scales and ideas of political grandeur. The Romans had not used intellectual causes for architecture just as the science of geometry was lost on most of them.

Modern European International Style architecture divides areal proportions in tenths, while Meis van der Rohe used the greater and finer divisions of the Golden Ratio. The differences are marked. American designers mostly use 1/8ths, 1/12ths, 1/16ths, and, 1/32nds, to divide spaces, and the Europeans mostly use the tenths of the Metric System..

Are these differences perceptibile? Are these systems of measurement visible in determining the relative importances of every element of a design, or in making the design more dramatic and interesting? Yes.

Unless a design is rendered in terms of an imitation of nature or proportionless goo, a numerical geometric measurement system of some type must be used in every new design created. A purposeful design will express the relationships and importances of all elements in the design to all others in an ordered, dramatic, perceptible, proportional, and systematic way.

Unless a design is rendered in terms of an imitation of nature or proportionless goo, a numerical geometric measurement system of some type must be used in every new design created. A purposeful design will express the relationships and importances of all elements in the design to all others in an ordered, dramatic, perceptible, proportional, and systematic way.

This run-on lecturing at us here on the Forum is becoming increasingly bizarre. The choice of system of measurement is irrelevant to esthetics. By subdividing the unit into smaller units as a fraction of the base unit any degree of precision can be attained starting with any base unit system and any system of subdividing. Units of measurement have nothing to do with the esthetics of Roman architecture or any one else's. Units of measurement are used to measure an object, not create it's geometry. It seems we have encountered a new version of primacy of consciousness: "unitism".

A "system of units" is not comparable to the number 1.618.

Greek architecture was not measured with the golden ratio and the Parthenon was not based on it. That is myth with no basis for the claim.

I have a course written on the Dynamic Symmetry system of proportional harmony, and its a highly sophisticated principled lecture and demonstration chalk talk.

Whatever you have, the use of the Fibonnaci sequence to generate a nested, repeated pattern is not a "sophisticated principled" basis of esthetics.

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[...] . . . . .number called the golden section, or the golden ratio . . . . .

In designing manufactures, and in architecture, the Golden Ratio is a way of measuring proportions in order that the small details of a design may have a perceptible ratio to elements of the design that are several steps larger. The small details may be proportionally coordinated with the large.

You would have to see the diagram of the "whirling squares" that has the Golden Ratio as one of the causes of its generation. The mathematics of it is the basis for the Golden Spiral. The proportions of the elements of the main fronts of the Parthenon are decided using the Golden Ratio.

The Golden Ratio is one of several ways that architects measure off elements that are slightly or greatly smaller and slightly or greatly larger than a base element. The elements are systematically and proportionally related in a way that is perceptible as a type of order. The psychologists of our era are too concerned with the acceptability of the likes of impulsives in the military or with the approvals granted by one social consensus or another, and they have no time for the causes of beauty. There is more that needs to be said, and especially regarding the proportional systems used in music.

The Golden Ratio is one of many systems of applied measurements. Others are, the Root and Square Root Number System, dividing things in accordance with the SI system of measurement, e.g., in tenths or one-thousandths, or using the English units involving 1/8ths, 1/16ths, and, 1/32nds, for example.

The architecture of the Roman Empire was Pragmatically imitative of the appearance of the works of the AGs, however, the Romans measured elements using feet and fractions, and the Greeks divided proportional areas using the Golden Ratio and other mathematical ratios or scales. The "look" of the architecture of the AGs was caused in good measure by the use of the proportional harmony system they called "Dynamic Symmetry" and an appreciation of life. The "authoritarian" "look" of the Roman architecture was determined in part by the use of unit measurement scales and ideas of political grandeur. The Romans had not used intellectual causes for architecture just as the science of geometry was lost on most of them.

Modern European International Style architecture divides areal proportions in tenths, while Meis van der Rohe used the greater and finer divisions of the Golden Ratio. The differences are marked. American designers mostly use 1/8ths, 1/12ths, 1/16ths, and, 1/32nds, to divide spaces, and the Europeans mostly use the tenths of the Metric System..

Are these differences perceptibile? Are these systems of measurement visible in determining the relative importances of every element of a design, or in making the design more dramatic and interesting? Yes.

Unless a design is rendered in terms of an imitation of nature or proportionless goo, a numerical geometric measurement system of some type must be used in every new design created. A purposeful design will express the relationships and importances of all elements in the design to all others in an ordered, dramatic, perceptible, proportional, and systematic way.

I have a course written on the Dynamic Symmetry system of proportional harmony, and its a highly sophisticated principled lecture and demonstration chalk talk.

* * * * * * * * * * *

Inventor

Inotherwords,

The Golden Ratio

Unites my mind

With glowing thoughts

I cannot find;

Esthetically

They're well-designed.

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[...] . . . . .number called the golden section, or the golden ratio . . . . .

The correct spelling should be, Ludwig Mies van der Rohe.

During the late 1950s I attended a lecture by Mies at IIT, Chicago, in the Crown Hall architecture and design building that he designed, on the topic of establishing the proportions of a beautiful building design. He especially discussed the use of the Golden Mean Ratio, and for example, he showed how that was used to decide the proportions of design elements for the Seagram Building on Park Avenue in New York City. It was great to see how everything in the design made sense.

I bring that up should anyone doubt that the discoveries of the Ancient Greek geometers and their predecessors regarding the Golden Mean would have any causal relationship or relevance to the beauty of the designs of buildings that exist today.

Inventor

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[...] . . . . .number called the golden section, or the golden ratio . . . . .

The correct spelling should be, Ludwig Mies van der Rohe.

Alex didn't say anything about that. You seem to be responding to your own post, not Alex's, changing a spelling while ignoring all the serious errors subsequently pointed out, which you haven't addressed. This included a strange inversion of the concept of unit of measurement into a role of creating shapes depending on the unit of measurement chosen. There is a lot more at stake here than spelling.

During the late 1950s I attended a lecture by Mies at IIT, Chicago, in the Crown Hall architecture and design building that he designed, on the topic of establishing the proportions of a beautiful building design. He especially discussed the use of the Golden Mean Ratio, and for example, he showed how that was used to decide the proportions of design elements for the Seagram Building on Park Avenue in New York City. It was great to see how everything in the design made sense.

I bring that up should anyone doubt that the discoveries of the Ancient Greek geometers and their predecessors regarding the Golden Mean would have any causal relationship or relevance to the beauty of the designs of buildings that exist today.

Anyone who objectively looks into this subject "doubts" such an alleged "causal relationship or relevance". It is number mysticism no better than the primitive mysticism practiced by ancient Pythagoreans. Following the myth, some architects in modern times have arbitrarily used, or claimed to have used, (1+√5)/2 = 1.618... as a ratio in their proportions. That does does not demonstrate any inherent esthetic superiority for it, let alone establish the claims that ancient Egyptians and Greeks used it or demonstrated through mathematics or any other means any objective quality of it in esthetics.

The lecture you heard 50 years ago may have contained a variety of interesting and valuable elements that impressed you, but whatever was said there, it is not a reason to embrace this "golden ratio" number mysticsm. Jumping into a fad and repeating a myth does not make it true.

There is nothing wrong with your interest in geometry or its role in the visual arts, but latching onto a mythology of number mysticism in the name of objectivity in art is a big fallacy.

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Greek architecture was not measured with the golden ratio and the Parthenon was not based on it. That is myth with no basis for the claim.

http://milan.milanovic.org/math/english/golden/golden4.html

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Greek architecture was not measured with the golden ratio and the Parthenon was not based on it. That is myth with no basis for the claim.

http://milan.milanovic.org/math/english/golden/golden4.html

You can find any number of web pages repeating this myth. Anyone can say anything on the internet. And they do. Especially when they are naively repeating myth following a fad.

Jumping into a fad and repeating a myth does not make it true.

Furthermore, again -- contrary to your previous post -- the 'golden ratio' is a number, not a system of measurement, and choices of units of measurement do not create the shapes they measure.

The rectangle overlay in your link does not even have the right proportions to be a 'golden ratio'. Keep juggling different rectangles in different places until you find one with the proportion you want. It still won't distinguish the 'golden ratio' = 1.61803398875... from other numbers such as the simple ratio 8/5 = 1.6 and doesn't prove anything about what the designers of the Parthenon actually used, intended or thought they were doing, of which you have no evidence at all.

Even if someone had once used a certain proportion -- for which you have no evidence -- simply because it was something they knew how to construct, it would say nothing about esthetic criterion, which requires conscious application for that purpose, and it says nothing about some alleged unique and inherent esthetic superiority for the 'golden ratio', which is number mysticism and nothing more.

You continue to ignore all the rebuttals to your claims as if they had never been written, while continuing to insist on unsubstantiated claims. That is a sign of true religious faith over reason, belief in the absence of or contrary to reason. You know better than that.

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It is clear enough that Greek geometry does not establish any inherent esthetic superiority to the so-called "golden ratio" - (1+√5)/2 = 1.618... as a required ratio of proportions, but the mythology surrounding this number mysticism extends beyond the misuse of geometry into widespread bogus history as well. There is no evidence that ancient Greeks or Egyptians themselves used the "golden ratio" as an esthetic design principle, even with the influence of Pythagorean number mysticism. Historic claims to the contrary turn out to be a much more modern mythology, a movement begun by 19th century irrationalist German intellectuals, not the ancient Greeks.

The Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, 1994, contains an interesting and revealing 9 page entry in which the common claims of "golden numberism" are distinguished from legitimate mathematics, debunked, and its source and cause tracked down. The article distinguishes between the mathematical history of the ratio (beginning with its introduction in Euclid, the relationship to the Fibonacci sequence, etc.) versus other claims made by "golden numberism".

The section on Golden Numberism (p. 1580, vol 2) begins with the account of its founding in mid 1850s Germany:

The expression 'golden numberism' refers to the ensemble of doctrines and claims relating GN ['golden number'] to natural phenomena or the use of GN in human constructs. Examples of these claims can be found in Zeising 1854 and Röber 1855, the cofounders of golden numberism. Zeising saw in the GN a primary morphological quantity to which various dimensions (e.g. the relative position of the navel) of the human body were related. Röber claimed that the GN was used to design almost all the pyramids of Egypt (but not the Great Pyramid)...

At present, golden numberism is flourishing, and even in some otherwise austere and serious mathematics books we find, often in connection with the Fibonacci numbers or Fibonocci search, the completely false assertion that the ancient Greeks used the GN in designing the Parthenon and other temples... or the statement that the GN has special aesthetic properties.

Here is more on the history of golden numberism and how the first hints of it did not start with the Greeks:

The earliest examples that can be said to be a form of golden numberism are found in the works of Kepler. [Kepler was a raving mystic aside from his contribution to astronomy.] These are his use (1596) of the icosahedron and the dodecahedron, and thus the GN, in his polyhedral model of the Solar System, and perhaps (interpretation is difficult because of other remarks) his statement (1608) that the ratio of the orbital periods for Earth and Venus 'comes very close to the divine proportion', and some remarks (1608, 1619) relating the 'laws of generation' to the GN.

After Kepler is the claim made in the second edition (1799) of the Montucla-Lalande Histoire des mathématiques that Pascioli had advocated the use of the GN in determining the proportions of works of art and architecture. Not only is this oft-repeated claim false, but, to the contrary, Pascioli explicitly states that simple ratios are best.

Thus, historically, the situation is this: there does not seem to be the slightest piece of documentary evidence that anyone before Zeising and Röber, apart from Kepler and Montucla-Lalande, wrote or did anything which could be interpreted as golden numberism. As far as the name 'golden number' and related forms ('section', 'cut', and so on) are concerned, the earliest attested use - abeit in a strictly mathematical context -- is by Martin Ohm in 1835.

Golden numberism first appeared in the US in the early 20th century, along with other German intellectual influences in this country beginning in the late 1800s:

Golden numberism sees to have been limited to Germany until the early part of this century, but after about 1910 it spread rapidly to other countries as 'a sudden and devastating disease which has shown no signs of stopping' (Cook, 1922). An impetus was given in the USA by J. Hambridge ('dynamic symmetry', circa 1920) and in France by M. Ghyka (from 1927). Under the latter's influence the architect Le Corbusier, previously anti-GN, published drawings indicating that an already constructed building -- for which he had used another system -- had been designed using the GN! His book Le Modulor (1948), which described an architectural design system based on the GN, was in turn influential in the spread of golden numberism...

The "Dynamic Symmety" and "whirling-square rectangle" of Hambridge, operating under German intellectual influence in the US around 1920 was previously referred to in this thread by Inventor, who claimed to "have a course written on the Dynamic Symmetry system of proportional harmony, and its a highly sophisticated principled lecture and demonstration chalk talk." Inventor also mis-attributed "Dynamic Symmety" to the Greeks, claiming that "The 'look' of the architecture of the AGs was caused in good measure by the use of the proportional harmony system they called 'Dynamic Symmetry'".

But it did not come from the Greeks and there is nothing "sophisticated" about it. It's no accident that American intellectuals in the early twentieth century had been influenced by German philosophy and its Hegelian mystical metaphysics and statism: American scholars in the late 1800s had been travelling to Germany and returning to this country to dominate and develop the American university system under that influence -- See Leonard Peikoff's The Ominous Parallels and Arthur Ekirch's The Decline of American Liberalism. German irrationalism imported to America is hardly something to embrace in the name of "sophistication" -- in art, education, progressive politics or anything else.

Inventor, you have been had. This fad has nothing to do with mathematical or philosophical objectivity in art and was not created by Greek geometry.

The cause of the spreading of the myth despite its actual origins and lack of foundation are not hard to understand:

The 'success' of golden numberism seems to be due to several factors. One is perhaps the connotations of the adjective 'golden', which is also used in many other contexts. The geometrical construction, which is simple yet not mundane, and the defining quadratic equation, which its irrational and closed-form solution, were accessible to the general public. The infinite-reproduction property [in Euclid]... and the related connection with the Fibonacci numbers also increase its mathematical and exotic appeal. A scientific basis for golden numberism has been claimed by references to aesthetic experiments, usually misinterpreted in the GN literature, performed in the 1870s by the psycho-physicist Gustav Fechner. Research has indicated that there is simply no GN basis for the very complicated processes involved in aesthetic preference (Zusne 1970: 399).

And more from the article showing how the fallacy has been rationalized:

Röber did not claim that the GN had been used to design the Great Pyramid, having, in this case, accepted a competing theory. Indeed, there are at least nine theories concerning the shape of the Great Pyramid, four of which agree with the observed value to the first decimal place. Two of the nine theories involve the GN, one based on a non-existent passage from Herodotus that gives rise to the GN in a most surprising way...
The GN literature abounds with claims wherein the author does not realize that approximate measurements or drawings do not suffice to prove the claim. A proof requires some sort of documentary evidence that the designer of the object in question had the GN in mind as a theoretical basis. Sometimes, as with the painting Le Cirque by Georges Seurat (1890-91), it is known that the simple ration 8/5 was used, yet a claim is made that the GN is involved. Similarly, the cubist painter Juan Gris (circa 1920) explicitly stated that he had not used the GN; yet claims to the contrary, based on measurements, have been made... Incorrect treatment of data has also sometimes led to an apparent clustering of observed values around the GN.

The author of this encyclopedia article, Roger Herz-Fischler, has also published two books related to this subject in far more detail:

The Shape of the Great Pyramid

A Mathematical History of the Golden Number

You can look at the tables of contents of the books at those links to get an idea of how much there is on the history of both the legitimate mathematics of the "golden ratio" and the phenomenon of "golden numberism".

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Here are excerpts from another professional article debunking the myth of the "golden ratio", this time from the Mathematical Association of America. There are many interesting genuine mathematical facts related to the 'golden ratio', but the topic emphasized here is the bogus number mysticism of 'golden numberism' and its pseudo-role in esthetics, in accordance with the topic of this thread.

The Myth That Will Not Go Away

Devlin's Angle, May 2007

Keith Devlin, Executive Director of the Center for the Study of Language and Information, Stanford University

Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like "which the ancient Greeks and others believed to have divine and mystical properties." Almost as compulsive is the urge to add a second factoid along the lines of "Leonardo Da Vinci believed that the human form displays the golden ratio."

There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on...

My first suspicions that all was not right with some of the claims made about the aesthetic appeal of the golden ratio were aroused when I admitted to myself that I personally did not find the golden rectangle the most pleasing among all rectangles. My doubts grew when tests I performed on several classes of students revealed that few people, when presented with a page of rectangles of various aspect ratios, picked out as the one they found most pleasing the golden rectangle. (Actually, given that the aspect ratio of any actual rectangle you draw can be only an approximation to a theoretical ideal, a more accurate description of my experiment would be that few people picked the rectangle that most closely approximated the theoretical ideal of a golden rectangle.)

Then I read an excellent article by the University of Maine mathematician George Markowsky, titled "Misconceptions about the golden ratio", published in the College Mathematics Journal in January 1992. In his article, Markowsky subjected many of the common claims about the golden ratio to a fairly rigorous review, and found that quite a few of them come up decidedly short. Further evidence against many of the common claims you see made about the golden ratio were provided by writer Mario Livio in his 2002 book The Golden Ratio: The Story of PHI, the World's Most Astonishing Number.

In my June 2004 "Devlin's Angle" and in an article I wrote for the June 2004 issue of Discover magazine, I added my own contribution to Markowsky and Livio's valiant attempts to inject some journalistic ethics into the scene (to wit, checking facts before going into print), but by all appearances the three of us have had little success. Particularly when novelist Dan Brown repeated many of the most ridiculous golden ratio chestnuts in his huge bestseller The Da Vinci Code. (No, I'm not giving a live link to that!)

With so many wonderful things to say about the golden ratio that are true and may be substantiated, why oh why do those myths keep going the rounds? Why do we so want to believe that, say, the ancient Greeks designed the Parthenon based on the golden ratio? (For the record, they did not; which is to say, there is not a shred of evidence that they did any such thing, and good reason to believe they did not.)...

Undaunted, let me sally forth once again into this mire of misinformation and try to set the record straight....

Euclid, in his book Elements, described the above construction and showed how to calculate the ratio. But he made absolutely no claims about visual aesthetics, and in fact gave the answer the decidedly unromantic name "extreme and mean ratio". The term "Divine Proportion," which is often used to refer to GR, first appeared with the publication of the three volume work by that name by the 15th century mathematician Luca Pacioli. (He has a lot to answer for!) Calling GR "golden" is even more recent: 1835, in fact, in a book written by the mathematician Martin Ohm (whose physicist brother discovered Ohm's law).

There is no doubt that the GR has some interesting mathematical properties...

It's when you leave the mathematical world and the natural world, however, that the falsehoods start to come thick and fast.

Numerous tests have failed to show up any one rectangle that most observers prefer, and preferences are easily influenced by other factors. As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the golden ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle - usually red or yellow - is drawn and ask yourself: why put it exactly there and why make the lines so thick?)

Another spurious claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense. When you measure the human body, there is a lot of variation. True, the answers are always fairly close to 1.6. But there's nothing special about 1.6. Why not say the answer is 1.603? Besides, there's no reason to divide the human body by the navel. If you spend a half an hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6, or 1.5, or whatever you want...

Then there are the claims that the Egyptian Pyramids and some Egyptian tombs were constructed using the golden ratio. There is no evidence to support these claims. Likewise there is no evidence to support the claim that some stone tablets show the Babylonians knew about the golden ratio, and in fact there is good reason to conclude that it's false...
Just as happened last time, I anticipate receiving some truly ANGRY emails from readers incensed that I should dare question their long cherished beliefs about this particular number. (Maybe that's why we say it is irrational?) And there we scratch the surface of what I think is a fascinating aspect of human nature. People, at least many people, seem to really WANT there to be numbers with mystical properties. So much so that they are prepared to put aside their otherwise wise insistence on evidence or proof. (Many of the emails I got last time demanded that I give proof that the golden ratio is not in the design of the Parthenon, for instance. Which is of course to get the scientific method completely backwards. The hypothesis in that case is that the GR did figure in the Greeks' design, and that is what needs justification - and does not get. Ditto all the other spurious GR claims.) This almost religious attachment to a number, or to numbers in general, has a long history, going back at least as far as the Pythagoreans...

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Just as happened last time, I anticipate receiving some truly ANGRY emails from readers incensed that I should dare question their long cherished beliefs about this particular number. (Maybe that's why we say it is irrational?)

[...]

Now, there's a thought.

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One of the few people who supposedly used it in their designs were the violin makers Guarneri (Joseph "del Gesu"), Amati (the first to make the modern violin) and Stradivari.

Here are a few famous violins from them - notice how the shape varies quite significantly:

Gibson Stradivarius (played by Josh Bell):

Il Canone Guarnerius (played by Paganini):

And one by Nicolo Amati: http://orgs.usd.edu/nmm/Violins/AmatiNicol...matiViolin.html

Considering the wide variations in proportions, it becomes difficult to think of a unique ratio being used for all. All of the above violins are exceptional, generation-defining instruments.

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Considering the wide variations in proportions, it becomes difficult to think of a unique ratio being used for all.

... or which part of the complex geometry is to be singled out as somehow the essential, to be based on a magic number. The design and successful functioning of these instruments, like all others, depends on the physics of acoustics, including the role of all the geometric shapes and dimensions of the different parts, the different materials used, and the psycho-acoustics of the perception of the sounds. The different components of this science and its history are described in the fascinating book by Fletcher and Rossing, The Physics of Musical Instruments, Chapter 10, "Bowed String Instruments" in Part III "String Instruments", Springer-Verlag, 1991. No mention of the "golden ratio" or number mysticism there.

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Considering the wide variations in proportions, it becomes difficult to think of a unique ratio being used for all. All of the above violins are exceptional, generation-defining instruments.

rtg24:

You gave a delightful counterpoint to the heavy math.

Artistic designers of products and buildings have free will. Their artistic feelings exist in a finite world of measurement, and the proportions and size of the intellectual / directly perceptible entities they create is of the utmost importance to them.

The designer will select the epistemological, say geometric, principles that will govern the design according to the structure of his own self-developed sensibilities and constructions. Product design is a branch of architecture, and all the principles of design are available, and they are applied where appropriate for a certain design intention. The designer selects the measurement system that best expresses, measures, and makes possible, the design idea, and the specific measurements for the design are applied to the design.

No one measurement system is mandated, except where the laws of physics, properties of material, or characteristics of methods of making and tools are involved. The measurement scheme selected will be appropriate to the task.

Mathematical number systems are numerous, and they are most useful for deciding the proportions of the model of the item. Concepts of number are used, and these may be based upon the simple counting of whole numbers. Geometric divisions and mathematical principles may be used to generate a number series or spirals, for subdivisions, that are best for a design.

For example, some concepts and principles of number that are useful tools for designing may be,

Phi sprial - Whirling Phi Rectangles ....

Root rectangle series: 1^2, 2^2, 3^2, 4^2, ....

Triangular / square number series, ....

Additive numbers: 1, 2, 3, 4, 5, 6,

Multiples and submultiples: 10000, 1000, 100, 10, 1, .01, .001, ....

fx

Conics

B-spline curves

and many more.

Not to forget the architectural and musical,

1/2, 1/2, 1/3, 1/4, 1/5, ....

Was there something Pythagorean in the ratios? Maybe not.

Inventor

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Not to forget the architectural and musical,

1/2, 1/2, 1/3, 1/4, 1/5, ....

"Not to forget the architectural divisions and musical ratios,

1/1, 1/2, 1/3, 1/4, 1/5, .... "

The single unit of measurement, 1/1, is important.

Inventor

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"Not to forget the architectural divisions and musical ratios,

1/1, 1/2, 1/3, 1/4, 1/5, .... "

The single unit of measurement, 1/1, is important.

Measurement is a concept that identifies a property of particulars, and the particulars may be metaphysical concepts, say of physical things, or they may be epistemological concepts, say of ideas.

Measurement science, a sub-classification of physics, or of mathematics depending upon the context, is important to the acoustics of music. In music measurements are made of physical, that is, material, functionings.

Measurement science cannot be applied to the universe. Conventional conservative [religious] science says that there are dimensional and temporal limits to the universe. That presumes that the universe is a single entity that can be measured. That is not so. Nor can the universe be identified, except in terms of the three axioms of, Existence, Identity, and Consciousness, to be anything other than substance or functioning (Aris.), e.g., that the stuff of the universe exists and continues to function. The universe is a plurality of particular things, and only those particular things may be measured.

The universe is not made of numbers (which the anti-Pythagoreans claim Pythagoras purported).

Rather the universe, that is everything there is, is identified in terms of scientific principles. The implication here is that Pythagoras held that principles identify the facts and principles of particular existents, and relationships of same, in the universe. And, not particular numbers.

This path of thought is based upon previous conclusions that the "Universe is made of numbers" concept that is attributed to Pythagoras is due to a wrong interpretation and translation of the AG term for "scientific concept", which was mis-translated to be, "magnitude", by translators of philosophy and geometry somewhere in Classical AG, Roman, anti-rational period, or in modern history. The concept, "magnitude", meant to the AG, nearly the same as the concept, "measurement" [regarding properties and amounts of things] to Ayn Rand [in ITOE]. Her understanding was Aristotelean in terms of principles, and did not deal with the concept, measurement, in terms of particulars only, e.g., in terms of number.

The universe is a continuing plurality of physical existents, and each discrete existent has specific properties. Those properties may be identified in terms of universal principles [Pythagoras], and where appropriate, measured and given number.

Particular existents may have their properties identified, and if amounts are required, the particular existents may be measured.

In the universe, only particular existents exist.

Only particular existents may be measured, their properties identified and given number.

The universe is a plurality of existents, and it is not one thing.

Therefore, the universe, in particular, may not be identified or measured [Aris.].

A violin string is particular, material, and finite, and its properties of mass and length permit specific functionings of sonorous vibrations. The so called "Music of the spheres" attributed to Pythagoras refers to the actions of selected entities that function according to certain principles, or "magnitudes". The "spheres" to him meant the "functioning magnitudes" of things in the universe, and Aristotle also relays that concept to us. To the Pythagorean AG the stars, sun, and moon are identified to exist functioning in their "spheres" of principles of operation. The entities were not known in terms of numerous physical properties and characteristics, however, their principles of functioning were known to be separate. Hence the idea that the universe is made of spheres.

That means that "spheres" to the AG are what we know to be specific laws of nature. The moon functions according to different principles than the stars, for example.

A violin string functions in accordance with the principles that are appropriate to its nature, and the results may be given number in addition to being sensed and known.

To Pythagoras the musical string may be known in particular, and, the universe may be known only in terms of particulars.

Inventor

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

[text omitted]

c^2 = (a+b )^2 - 4 x 0.5 x ab = (a^2 + 2ab + b^2) - 2ab = a^2 + b^2.

In my high school geometry textbook (published in the 1950s), this is presented as an algebraic proof of the Pythagorean Theorem.

I'm not strong in math, however, it seems to me that there are so many errors in the equations that science stopped.

1. Formal error - proof.

a. No proper format for a proof has been followed, e.g., equations are proved by reasoning from universals to particulars, and no particulars have been found.

b. No proper first and middle premises and conclusion have been placed.

2. Formal error - ambiguous structure.

a. Its a run-on sentence that has three [=] verbs.

3. Formal error:

a. The terms are not equal as required by the [=] sign.

b. If it were broken down into three premises required for a proof it might be:

c^2 = (a+b )^2 - 4 x 0.5 x ab

= (a^2 + 2ab + b^2) - 2ab

= a^2 + b^2

b. The first premise is not true. [Make a=b, a=1, solve.]

c. The second premise is not true. [The term, "-2ab" is arbitrary.]

d. The conclusion is wrong. [it lacks the term, "+2ab".]

e. Substitutions of numbers in the premises do not yield consistent results when, e.g., a=b, a=2 [or, e.g., a=b, a=1]

2. Logical errors.

a. The first premise is so not true that it cannot equal the conclusion. That leads to the fallacy of Non Sequitur.

b. The first premise with its numbers is more particular than the conclusion; that's a fallacy of distribution.

c. Both the second premise and the conclusion should equal "c^2", and they do not.

d. Nor does the second premise equal the conclusion, as specified by the "=" sign.

If I were more skilled in math, I would have seen the faults and I wouldn't have bothered. I may have been time-trolled.

Inventor

The only thing I can gather from this is that Inventor doesn't actually understand algebra, and it's bad enough that he was impolite to System Builder who politely pointed out things that didn't make sense.

Can this thread be killed, or ask Inventor to stop posting in it? ewv and Bob have demolished the validity of his points, but he continues to post and ramble on ignoring them. At this point, soon their logical statements will be submerged beneath a sea of Inentor's confusing posts, who apparently refuses to recognize that nothing he is saying makes any sense at all.

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Can this thread be killed, or ask Inventor to stop posting in it? ewv and Bob have demolished the validity of his points, but he continues to post and ramble on ignoring them. At this point, soon their logical statements will be submerged beneath a sea of Inventor's confusing posts, who apparently refuses to recognize that nothing he is saying makes any sense at all.

As long as Inventor is speaking only for himself, not misrepresenting Objectivism or treating other FORUM members disrespectfully, I'll let him post. Other FORUM members are free to ignore his posts or disagree with him as they see fit.

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...

If I were more skilled in math, I would have seen the faults and I wouldn't have bothered. I may have been time-trolled.

The only thing I can gather from this is that Inventor doesn't actually understand algebra, and it's bad enough that he was impolite to System Builder who politely pointed out things that didn't make sense.

Can this thread be killed, or ask Inventor to stop posting in it? ewv and Bob have demolished the validity of his points, but he continues to post and ramble on ignoring them. At this point, soon their logical statements will be submerged beneath a sea of Inentor's confusing posts, who apparently refuses to recognize that nothing he is saying makes any sense at all.

I don't think he means to be impolite. Despite his enthusiasm for the subject he clearly does not understand it and does not realize that. He had an early background in art, with some success, and doesn't realize he has picked up a lot of nonsense from the art world and other sources of subjectivism.

Despite the frequent ramblilng and nonsensical statements found throughout the thread, the topic itself is important because of how widespread the myths about Pythagoras and about the golden ratio are and because of the widespread confusion over the distinction between objectivity in esthetic criteria versus the use of mathematics for optional methods (including the role of numerical measurement and construction) and misuse of mathematics for baseless, arbitrary criteria. The latter includes, but is not restricted to, number mysticism.

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I suspect that there is modern math nihilism at work, and confusion.