It's a famous claim that the golden ratio was both known to and deliberately used by artists and/or architects, for example in the Parthenon. I doubt this, and suspect that any observations about the golden ratio existing in art or architecture were retrofitted later. What contemporaneous, direct evidence is there for the deliberate use of the golden ratio in art or architecture before the 19th century?

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    An interesting question. I guess one should find pretty precise construction lines in plans and studies or below the surface of paintings, if they where made intentionally. I didn't knew it was disputed to be an antique construction method. Mar 5, 2011 at 5:02
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    see also cogsci.stackexchange.com/q/1627/52 Sep 10, 2012 at 12:37

4 Answers 4


There is no strong evidence for this, and plenty of evidence that ancient and Renaissance artists used other systems — for example, ratios of integers. That is, it is not as if we have lost all writing from classical times on ideal numerical systems for art and architecture, but this idea does not appear. Therefore, it is likely that the systems which were documented were the ones used by artists and architects — not the golden ratio, which was not written about in this way until its modern popularization.

Knowledge of the ratio dates back to at least Euclid (300 BC), but he just noted it as interesting in mathematics, not aesthetics. Then, it doesn't seem to really turn up in writing until about 1500, when Italian mathematician Luca Pacioli wrote about it as "the divine proportion" — and while his mystical writing touches on aesthetics, there's no evidence that anyone (even da Vinci!) took it to heart.

The idea that the number was important in aesthetics comes from a psychologist named Adolph Zeising in 19th century Germany. There is credible evidence that this is the start of the modern idea. Zeising's theory — that there was a number behind beauty — matched the times, as scientists worked to turn psychology into a quantifiable empirical science (see more in the article Architecture and the Golden Section), and perhaps this helped the idea spread.

(More on Zeising: here, here, here, and here.)

Just because Zeising popularized the idea doesn't mean he's wrong. However, I'm coming up empty coming up with anything credible before the 19th century — which suggests that even if he's right about any special aesthetic qualities (a separate question, really), he was wrong about its actual use by prior artists (just as he was wrong about its frequency in nature).

The Zeising approach of finding some natural or ancient thing which appears to fit the theorized measurements is not satisfactory, for two reasons:

  • First, this is subject to selection bias — it stands to reason that if we measure all of the things, some will appear to fit by coincidence.

  • Second, it is usually the case that such measurements are done very shoddily in order to make them match.

    The images Sklivvz has provided in another answer are perfect examples of this: sure, there are convincing-looking lines and measurements at first glance, but if you look more closely, it's unclear what actual building features they correspond to. The rectangle around the Parthenon doesn't even line up with the borders of building in the diagram — and in fact only comes close to the golden ratio if you start at the second of four steps. In short, the idea that the Parthenon corresponds in this way has been well debunked.

Most ancient writing on proportions in design seems to follow a system of ratios of integers — for example Vitruvius, on who's work da Vinci's famous illustration is based. This is the case with the Parthenon:

The Parthenon, like a statue, exemplifies a certain symmetria. Its symmetria largely depends upon the 9:4 ratio, which is present in various dimensions of the building—the length of the stylobate [the platform that forms the base of the building] to the width of the stylobate, the width of the stylobate to the height of the column and entablature [the top section between the columns and roof] together.

Nova, the Glorious Parthenon

In all, this 4 – 6 – 9 theme pervades the entire Parthenon: in the symmetry of the architectural elements it leads to the geometric proportion 4:6 = 6:9. In the sculptural program of the temple's east front it constitutes the core of the 4 – (6 - 7) – 9 symmetric number symbolism.

Anne Bulckens, Parthenon's main design proportion and its meaning

Bulkens connects these proportions to Vitruvius' writing on the proportions of Doric temples, This aligns with Vitruvius's writing on architecture, which is extensive — Vitruvius's Ten Books on Architecture. A particular section is worth quoting a large chunk from:

  1. The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in Greek [Greek: analogia]. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man.
  2. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.
  3. Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.
  4. Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever.

(Emphasis added to highlight the numbers.)

This text — not the golden ratio! — is the inspiration for da Vinci's Vitruvian man. Now, the actual influence of Vitruvius on architecture before the Italian Renaissance is itself in doubt, but an example of an intended-to-be-comprehensive ancient survey of practices in architecture, and it's all about the integer proportions. The golden ratio isn't mentioned.

If we move forward to the Renaissance, the most common example cited is Luca Pacioli's Divine Proportion. But, only the first part of this book is concerned with the golden ratio, and is primarily about the relationship between religion and math — the word "divine" here is intended to be literal. Pacioli's writing is deeply mystical, and connected with ideas about an indivisible God, the trinity, and so on (see this for more). He does discuss Platonic solids and more complex derivatives and their relation to architecture, but the connection from that to direct use of the golden ratio as a design principle just isn't there.

The second part of Pacioli's book covers architecture, and discusses the integer-based Vitruvian system as applied there — not the golden ratio! Furthermore, despite da Vinci's illustrations, Pacioli was not an artist, and I see no contemporary evidence that his ideas caught on (in art or otherwise) until Zeising. Da Vinci made extensive notes and sketches for his own work, and none of these make reference to the golden ratio — but there are examples of other proportions. The idea that he believed in this theory but left it out is an extraordinary claim with no support.

There is, however, significant counter-evidence in a 1876 study by Gustav Fechner — a strong believer in Zeising's ideas. He surveyed 10,558 paintings from from 22 European art galleries and analyzed their aspect ratios. Expecting to find the golden ratio, he instead found a general disregard for it:

Page 291 of volume II of Vorschule der Aesthetik

Of the three types of paintings analyzed, those in "portrait" orientation have an average aspect ratio of about 1:1.25 (that is, 4:5) and those in "landscape" orientation average about 1:1.37 — which is close to 3:4 but nowhere near phi. Of course, in such a large sample, one can surely find some examples which approach the golden ratio, but were it in any way special to the artists of the time, one would expect at least a pattern of use — and there isn't one.

For more analysis comprehensive paper on some "conventional wisdom" examples which do not hold up, see George Markowski's paper Misconceptions about the Golden Ratio, which covers the Great Pyramid, the Parthenon, da Vinci, This paper itself contains many further references if you're unconvinced.

Ancient (Pythagoras, Vitruvius) and Renaissance (Pacioli) authors wrote about the beauty, importance, and mysticism of small integer ratios, and this is matches various very precise measurements of surviving structures. That is, the writings match the apparent result. To the contrary, there seem to be no ancient writings espousing the use of the golden ratio in art or architecture, and few if any examples of precise measurement of artifacts.

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    Saying "there's no strong evidence" is a dubious answer on this site: given that answers need references/evidence, what's your evidence for saying "there's no strong evidence"? See e.g. this opinion about "negative searches". Instead it might be on-topic to say that integer rations were known and used (do you have evidence/reference for that claim?) and that 5:3 is indistinguishable from (i.e. within 1% of) phi.
    – ChrisW
    Jul 3, 2015 at 17:38
  • For every/any reference in your answer, I'd prefer to see at least one sentence (if not a paragraph or two) from that reference block-quoted into your answer.
    – ChrisW
    Jul 3, 2015 at 17:43
  • Wherever you have a hyperlink to an answer I'd also prefer to see the title of what you're referencing included in the hyperlink: see How should we deal with broken links on posts? for detail.
    – ChrisW
    Jul 3, 2015 at 17:45
  • @ChrisW: sure, I'll update with hyperlinks and quotes when I get a chance. I can soften the negative claim, although based on what I've seen, there's a strong body of expert opinion in support, so I'll at least reference that.
    – mattdm
    Jul 3, 2015 at 22:46
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    @LennartRegebro A case where the entire design is clearly based on 5:8 might be tantalizing even without written supporting evidence. But a case where various small-integer ratios are used and 5:8 just happens to appear is clearly not meaningful.
    – mattdm
    Jul 6, 2015 at 12:53

I think that a lot of the claims are quite believable.

Let's start with this tidbit I found (source below)

In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias.

The measurements of the Parthenon are quite objectively due to phi. As you can notice in the image below, all major measurements are strictly related to the golden ratio.

There are many university, or otherwise reputable sites corroborating this:

For example.

It has always been a problem to undertand how the Greek architect and his consruction workers managed to incorporate into the design of large-scale temples like the Parthenon the "irrational" measurements which the Golden Mean requires. The Greeks had no system for handling irrational numbers in a theoretical manner, let alone applying irrational measurements to an actual conctruction project. Extending the numbers of the GM proportion from one place to another on a building in the process of construction would seem to have been impossible.

But the proportions are clearly there in fact.


Throughout history, the ratio for length to width of rectangles of 1.61803 39887 49894 84820 has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the collumns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece.

He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. You can take a slide show visit to the Parthenon which is pictured above. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

Parthenon spiral

Phi temple

I find way more believable that the Greek would have used some symmetrically pleasing geometric construction, such as phi, to design the Parthenon, than assuming it's a mere coincidence. The figures match to within 5%.

Regarding the Renaissance, the artists that extensively used the golden section during the Reinassance are:

  • Leon Battista Alberti (1404-1472)

    L.B. Alberti

    See this design and the following quote:

    Di questi numeri si servono gli architetti non confusamente e alla mescolata, ma in modo che corrispondano e consentano da ogni banda all’armonia


    The architects use these numbers without confusion or messily, but so that they correspond, and allow from every band, to harmony.

    What he means is that architects should use integers to scale the various parts. To be honest he never explained how he chose the small integers used in his designs so that the composition was harmonious, but subsequent studies have shown he must have used the golden ratio extensively.

  • Piero della Francesca (1416-1492)

    A very notable painter, he was also a famous geometer at the time and wrote a book on perspective and proportions which was heavily referenced by Pacioli in his magnum opus. I cannot find an online version but it's cited on Wikipedia (FWIW).

  • Luca Pacioli (1445-1517) He literally wrote a book on the golden ratio in architecture.

    De divina proportione (On the Divine Proportion) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, composed around 1498 in Milan and first printed in 1509. Its subject was mathematical proportions (the title refers to the golden ratio) and their applications to geometry, visual art and architecture.

There are skeptics -- like George Markowski -- which seem to dismiss the claim based on believability. I personally don't find his argument compelling: it's based on the fact that phi is close to 5/8ths, and the measurements are not accurate enough. On the other hand, since phi is irrational, one can always claim that any work is not close enough, because the precision "required" to sustain or dismiss this claim seems to be set arbitrarily by the author.

In any case, I've only scratched the surface of examples: there's a Wikipedia page dedicated to listing all the works of art which use the Golden Ratio which gives around 12 examples, and many more can be found in art history books.

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    @Sklivvz, thanks for your work here, but unfortunately, none of these solidly meet the criteria of what I'm looking for. I'll update the question shortly to elaborate and clarify.
    – mattdm
    Mar 6, 2011 at 21:49
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    ofcourse it is possible to find this relation in a lot of measurements. the problem is that we activly seek this data points.you can find more relations between various measurements of buildings. are the relations that fall in the range of phi more present in estaticly pleasing buildings?
    – oɔɯǝɹ
    Mar 19, 2011 at 22:57
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    Regarding the Parthenon example, you have to be careful - if you pay attention to the bottom examples of phi, they seem to be pretty arbitrary (from the edges to the outer side of the centre-most columns) and if the standard is "within 5%", then with 8 columns evenly spaced with roughly half-the-spacing for width, you'd find an approximate golden ratio lining up with the edge simply by the property of 8 (namely, that it would line up with either an edge or a centre of a column or space with accuracy 1/32). Indeed, 5/3 is a fibonacci fraction, which makes it a natural approximation to phi.
    – Glen O
    Apr 8, 2015 at 15:55
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    @GlenO Right! The Parthenon has many examples of integer ratios throughout, so it's not surprising at all when 5:3 appears.
    – mattdm
    Jul 3, 2015 at 13:29
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    Deeply unconvincing: anything which explicitly involves integer ratios is not the golden ratio, which has been described mathematically as the most irrational number. Nor is the Parthenon convincing when the height/width ratio depends on drawing a line down an arbitrary number of steps. Although Euclid and other geometers know explicitly about the ratio, I am not aware of any explicit reference to it in art before Pacioli in 1496.
    – Henry
    Jul 6, 2015 at 20:06

"Misconceptions about the Golden Ratio"

George Markowsky

College Mathematics Journal: Volume 23, Number 1, Pages: 2-19 1992

first paragraph...

The golden ratio, also called by different authors the golden section [Cox], golden number [Fi4], golden mean [Lin], divine proportion [Hun], and division in extreme and mean ratios [Smi], has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks such as [Ser] make many incorrect statements about the golden ratio.

So I guess we have to look at Markowsky's references.


The golden ratio or golden section was known to Euclid and clearly mentioned in his treaties.

Of course the ancients used fractions to approximate irrational numbers as the decimal system had not been invented. In fact, irrational numbers were frowned upon by the Greeks; but not the Babylonians who had expressed the square root of two over a thousand years prior to its purported introduction by Pythagoras.

There are no drawings extant that show any schematic design diagrams for any Ancient Greek or Egyptian architecture or painting. There are from the middle ages on and in particular the Italian Renaissance due to the celebration of Plato's Timaeus.

There is now and has always been a split between the proportional "theory" of a building and its actual construction. There is a reason all materials have tolerances as does all construction activity.

The only way to demonstrate any form of adherence to a design principle is to physically measure the actual thing as built. This has been done for the Parthenon and Pyramids to a great extent.

We have no actual written parchment that says: "I did it this way, because...." until we get to the 15th century. Viollet le Duc drew and pronounced but did not explicate.

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    Welcome to Skeptics! Please provide some references to support your claims. How do we know you are right about Euclid, Plato, le Duc, the Ancient Greeks, etc?
    – Oddthinking
    May 12, 2023 at 13:25
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    It seem like "materials and construction activity have tolerances, so theory and construction won't match" and "physical measurement can show the design principle" are at odds.
    – mattdm
    May 18, 2023 at 14:46
  • "the ancients used fractions to approximate irrational numbers as the decimal system had not been invented": fractions and decimals can both approximate irrational numbers with arbitrary precision. However close any fraction is to a given irrational number, there's always another one that's closer. And finite decimals decimals are all expressible as fractions (as are repeating decimals).
    – phoog
    May 25, 2023 at 0:09

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