Beauty or Geometry?

Beauty or Geometry? Leonardo da Vinci’s Vitruvian Man (1492) paid homage to the remarkable proportions of the human body: arm span equals height, and the ratio of the lengths (foot to navel) to (navel to head) was the Golden Ratio (1.618, the ratio of adjacent numbers in the Fibonacci Series).

Reconstructive surgeon, Stephen Marquardt, took this much further. After extensive analysis of faces deemed to be beautiful, he found the golden ratio everywhere: mouths were 1.618 times wider than the noses, and the widest part of the nose was 1.618 times wider than the tip. He even found the width of a supermodel’s front two teeth to be precisely 1.618 times the height of each tooth! If one must find an evolutionary significance to a preference for facial symmetry, it has been speculated to be an indicator of freedom from disease, and worthiness for mating.

Now, what’s the math behind inner beauty?

Sources: For an interactive Flash game on mathematical beauty (or not) of some famous faces, check out

Marilyn Monroe’s Golden Mask photo:

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33 Responses to Beauty or Geometry?

  1. Marian Wirth says:

    Geometry, schmeometry. What happened to the notion that beauty comes from within? 😉

  2. Mark Thomas says:

    We constantly deny the role of biology in in our own personalities and preferences. While biology doesn’t determine our path, it does give context to our choices. Our view of ourselves and others does not stem from the lofty notions we like to believe. Darwin has a lot to do with what we call beauty.

  3. So to add to what Marian Wirth brought up… does inner beauty contain the Golden Ratio? 😉

  4. Rajini Rao says:

    Most of my posts, Marian Wirth , showcase our inner beauty in a very literal sense, do they not? What shapes our view of ourselves, Mark Thomas ? There is research that shows that we often choose mates who resemble, wait for it….our parents 🙂

  5. Rajini Rao says:

    Let’s define the parameters of inner beauty, John B We can probably deduce a plausible mathematical formula from there:)

  6. Marian Wirth says:

    Rajini Rao When will I learn not to tease you? I have two Twitter buddies already who always win the ensuing fight. Do I really need that on G+, too? sigh

    But yes, let’s define the parameters of inner beauty. You first, Rajini Rao John B

  7. Larry Mayer says:

    Perhaps the solution to a “mathematical formula” for beauty would not converge on a single solution, or even better, the solutions would all be imaginary numbers.

  8. Rajini Rao says:

    I tease you because you are so gracious, Marian Wirth 🙂

    How about starting with generosity , material and spiritual?

    I also find intelligence to be irresistibly attractive. Some of these could be quantified.

  9. I dunno. I find a few crooked teeth behind a genuine smile rather endearing.

  10. Rajini Rao says:

    Perhaps the inverse of (ego), XiXi Zhai ?

  11. Marian Wirth says:

    Related: sapiosexual

    I want an incisive, inquisitive, insightful, irreverent mind. I want someone for whom philosophical discussion is foreplay. I want someone who sometimes makes me go ouch due to their wit and evil sense of humor.

  12. Rajini Rao says:

    My sapient sense discerns an aberration in the last sentence of that definition, Marian Wirth ! (“I want someone I can cuddle with.”) There must a virtual version of that, no? Edit : Aha, I see you edited out that aberrant sentence! Our brainwaves must be in synchrony 😉

  13. Rajini Rao I saw a Simon Singh talk he gave about hidden messages and predictions in the bible and that if you look hard enough you can indeed find vertical and diagonal words about Newton and Gravity etc. He then demonstrated the same could be done using Moby Dick as it is statistically very likely to find these patters.

    Isn’t that what is going on here? For example, the width of the base of a nose to its tip – where exactly on the tip do you stop measuring? As I say, if you look hard enough, you will find anything you like.

    However, I agree these proportions are indeed very easy on the eye…

  14. Rajini Rao says:

    Mike McLoughlin , I don’t see why measurements of facial features cannot be precise and based on anatomy. Within a millimeter or two, the ratios either hold or don’t. The Marquardt mask can be placed on any face (see Flash interactive) and one can see symmetry (or lack thereof) for oneself. Of course, beauty lies in the eyes of the beholder, so all this is just so much theory, isn’t it? 🙂

  15. Marian Wirth says:

    Rajini Rao Our brainwaves must be in synchrony 😉 I can totally fathom addressing my brainwaves like “Prof. Rao said you are in synchrony with hers! Would you please do me the favor and act accordlingly? Thank you in advance.” #selfconvo

  16. Kapil Ranade says:

    I wonder how this applies to simha-kati gaja-gamin . . .

  17. Rajini Rao says:

    You lost me there, Kapil Ranade !

  18. Rajini Rao, as Mike McLoughlin noted, it’s probably an artifact. If you measure enough ratios you’re going to find numerous instances of the golden ratio. Especially when you’re measuring biological quantities, which vary continuously.

    A whole bunch of other ratios he tried, bet you, came out close to exp(1) and to pi and to 1/sqrt(2) and to pretty much every other number on the number line, but he ignored them because he was just looking for the golden ratio. It’s not impressive marksmanship if he ignores his misses.

  19. Rajini Rao says:

    David Ratnasabapathy , I’m guessing you are going with your gut feeling on how this came about, rather than on some inside information or sites that you’ve researched on this topic? If so, that’s a bit unfair don’t you think? I’m not defending him, but Marquardt being a maxillofacial surgeon who reconstructed severely damaged faces, earns him a bit of respect in my book. Now, moving to facts: the key is this Golden Mask which is “entirely constructed from the “golden decagon matrix”configuration.This is why all of the “phi elements” appear in the face.” According to his observations, beautiful faces through different cultures and history fit this mask..that’s all he is saying. Oversimplification? You bet. But an intriguing hypothesis, not one to be summarily dismissed because of bad measurements or failure to report alternative ratios.

    Check out this video that takes an ordinary face and photoshops it to fit the Mask. Formula for Beauty — anyone can look hot with the golden ratio

    Take a look at how he comes up with this “mask”:

    As I said before, beauty is in the eye (phi?) of the beholder, we are just having a little fun here. 🙂

  20. Rajini Rao no, I’m going by the fact that he’s doing it wrong. If he has a theory to guide him, that explains why he measured those particular ratios, then fine, he’s found something interesting. Instead what he’s doing is measuring every conceivable ratio in sight and picking the ones that are close to the golden ratio!

    Look at one of those examples: the ratio of the width of the two front teeth to their height. Why two front teeth? Why not one? Why not the width of all the visible teeth to their max height? — because none of those gave him the ratio he was looking for. He’s doing numerology, not science.

  21. Rajini Rao says:

    I’m do you know that he is measuring every conceivable ratio in sight and picking the ones that are close to the golden ratio? He is fitting faces (front or side view, close mouthed or smiling) to a template that embodies the golden ratio! It’s as simple as that! Because the template is derived from the golden decagon, it embodies phi in its dimensions. He did not go looking for it. I would have thought the controversy in this post would be more along the lines of faces fitting the mask (i.e. with major features in fixed proportion) are not particularly more beautiful than ones that do not!

    P.S. As far as the teeth are concerned, the second two are also in the same ratio as the first two, according to Youtube videos on this topic.

  22. Rajini Rao says:

    Thank you to TECHNICS ► for finding this intriguing video on Da Vinci’s use of proportions in Mona Lisa: Mona Lisa — Da Vinci’s Use of Sacred Geometry
    It’s a great watch!

  23. Rajini Rao, the Discover Magazine article at the link you provided describes him as actively looking for the Golden Ratio.

    “…So Marquardt went it alone. He collected photographs of faces the world deemed beautiful and began measuring their dimensions… As his data set expanded, Marquardt found indeed that the perfect face was lousy with golden ratios.”

    The article also mentions his prior infatuation with the ratio. What he’s doing is no different from those pyramidologists who find the fundamental constants of nature encoded in the ratios of various measurements of the Great Pyramid of Giza.

    Re: the teeth, my point is that the ratio of the width of the visible teeth to their height wouldn’t give the Golden Ratio. The ratio of the width of eyes to the distance from the top of the nose to its bottom wouldn’t give the Golden Ratio. Bet you, I could fiddle his numbers and find pi all over those faces. Heck, I bet I could find a whole bunch of ratios all equalling square root 2. It’s easy! Finding interesting numbers isn’t significant because there’s an infinity of ways to find them. A theory which told him which ratios to pick, a theory which predicted what he’d find — that is the hallmark of science.

    Him, he’s doing pseudoscience.

  24. Rajini Rao says:

    David Ratnasabapathy , I think we are talking past each other because we are both fixated on different things. You, on how he got to his theory (a theory which I take with a big grain of salt and not at all seriously). Me, on his final product which is highly symmetrical, so it is intriguing that beautiful faces fit into it. Agreed?

  25. Rajini Rao, I tried their flash program. Beautiful faces don’t necessarily fit, and an ugly face did fit!

    Picture 1’s eyes don’t fit into the mask properly; nor does the smiling woman’s chin in picture 2. The cheekbones of the man in picture 3 don’t fit the mask, once his eyes and chin are lined up, which immediately raises a suspicion as to why we can’t see the top of the head of man in picture 1. Picture 5 was a beautiful face which did match the golden mask well; but picture 6 following was an ugly face that fitted the mask just as well. It’s all highly subjective! A properly blinded test would just show the key points the mask is supposed to fit — the eyes, the chin, the mouth etc and erase the rest.

    It’s perfectly possible that we come pre-programmed with inflexible standards for facial beauty. It’s perfectly possible that these standards involve symmetry and geometrical ratios. These are interesting ideas, well worth testing. But these guys aren’t doing that. They make a specific claim: that the Golden Ratio determines beauty. What evidence, what theory suggested this? None at all. They just pulled it out of thin air! And given that their own mask fails to fit their own supplied faces, their claim fails their test.

    One day we will learn why it is that we enjoy viewing some faces but not others; one day science will indeed reduce beauty to a network of numbers. On that day we’ll feel pleasure at the sight of beauty, and pleasure squared at understanding why we appreciate it.

    But these guys? Alchemists pretending to do chemistry.

  26. Our mind and bodies are extremely complex and are shaped by our evolutionary journey. I am just an average Joe and evolutionary biologists in this thread should be able to add more scientific basis to this subject. Our brains have to make snap judgements in a nano second. Is that lunch or am I going to be lunch? Friend or foe, us vs. them.. All these decisions are life or death situations. Your gene pool make its way to the next generation or not. Its binary. To make these really fast calls, we have to have a large templates to quickly narrow it down. Face is a the central identity for us as a species and gives us a lots of clues. I think what appeals to us a a beautiful face (symmetry, ratios) are clues about sexual health and compatibility for reproductive success.

    I am sure there is some research out there that says the same faces appears non appealing when sexual reproduction is not possible at that instant. Examples: pregnancy or non ovulation durations.

    If you have ever observed kids when they encounter people with one arm missing, one eye missing, they cant but stare. I see parents being ashamed and drag the kids away. The reason I believe is the kid, had a certain model and this is an exception. The brain system is throwing exceptions from the unconscious and the kids conscious brain is forced to make sense of this situation.

    If we did not lose our hair in our evolutionary journey, we would find this photo very beautiful.

  27. Jeremy Teman says:

    You don’t want too much symmetry, as to go into the uncanny valley:

  28. Rajini Rao says:

    Wow, fascinating insight into human (and primate) psychology. Thanks, Jeremy Teman !

  29. Lillian Ríane Kelly, the Fibonacci sequence has no ending.  There are infinitely many numbers in the sequence.

  30. Rajini Rao says:

    Thanks, David Ratnasabapathy . I was planning on making that point and time got away from me. How have you been? 🙂

  31. Hi Rajini Rao!  I am fine, am enjoying the end of the school year.  Hope you are well?

  32. Rajini Rao says:

    David Ratnasabapathy Same here, summer means no pointless meetings and more time to focus on research (trying to get those papers out). Please post some science/tech stuff for us with #sciencesunday , okay? Anything that interests you.

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