The Mathematics of Beauty

Aaarrggghhh.
No no and thrice no.
If you search hard enough you can find mathematic patterns in everything.
Yes maths is interesting and beautiful in it own right, but all those constructed lines on works of art are hokum.
Maybe the golden ratio (or an approximation) does occur in all sorts of places but the maths is just a descrition of it not the reason.
A Nautilus shell grows in a log curve for reasons of physics rather than maths...or to expand my point to the limit, it just grows that way because the forces of nature cause it to do so, we just explain this with maths and physics.
An artist will seldom use maths to plan a work, unless maybe the scale is huge and work is required to scale up a model.
Can you help me down of this hobby horse now please?
Del
(PS For those who havn't seen any of my artwork, this blog shows some, and I promise no maths was involved, and no equations are harmed in the making of any of my artwork)

PPS, I s'pose I should add IMHO

DTC writes: "A Nautilus shell grows in a log curve for reasons of physics rather than maths...or to expand my point to the limit, it just grows that way because the forces of nature cause it to do so, we just explain this with maths and physics."

A nautilus shell is very beautiful, IMO, and surprisingly its geometry is often cited as a natural example of the Golden Ratio at work and/or of a Fibonacci spiral. A nautilus shell does exemplify a logarithmic spiral, but the spiral is not a function of the Golden Ratio / Fibonacci sequence as many have suggested (nor did you say it was, thankfully, because I would have impounded your cat bowl if you had ).

In perfect specimens the angle between the spiral's origin and an arbitrary point on the shell wall is a constant. This is also true of a wasp's trajectory as it comes in for a kill and of the trajectories of certain aquatic birds when surveying a landing spot in a marsh. In both cases the creature adjusts its trajectory in such a way that it conveniently sees its target at a constant angle to its flight-path whilst reducing the separation at a constant rate.

Below is a photograph (and its inverse) I took last year of a nautilus shell that I own. I collect natural objects like this shell that exemplify Nature's various ways of expressing mathematical beauty (you should see my collection of bronzed fractal broccoli sometime). I've included the video inverse to make taking various measurements easier for readers so inclined (hint: use PhotoShop or GIMP for this)...

Sadly, the resolution at which CR4 displays these images leaves much to be desired. I wish Mr. Gaulin would modify his code to temporarily save the uploaded image at full resolution and then scale it appropriately after the user has submitted his/her post (post, not just image). This would really help things, I think, especially when images contain text that must actually be legible to other readers in order to be useful.

The two pics above have a much higher resolution and are far more beautiful than can be seen here. <sigh>

That's nice that you don't use math to plan your work, but the great masters did. There are plenty of examples citing this information in art history books or classes if you've ever taken any. You really shouldn't be offended by people analyzing these older pieces, because of the time-frame of their creation and it's coinciding with Fibonacci and his explanations. You just have to understand how important the discovery of the golden ratio was. It was considered sacred knowledge and guarded for a very long time. 1.618 (the golden ratio) is so important, that Greece made their letter Phi = 1.618...it was important in all elements of design from statues to great buildings like the Parthenon.. It's very true that modern artists rarely use mathematics to formulate their design....but they should, because it would probably automatically be beautiful. As many others have already explained, we, as beings, are attracted to these proportions. Watch the video called "The golden ratio and the human body", it's very informative, and will help you understand why people are so fascinated with it, and it's application in art and sculpture

I would venture to say mathematics is the result of an instinctual capacity to create composition, colour and style in works of art....not the other way around.

BTW...if beauty is the hallmark of artsy maths......what in blazes is ugly?

(Hey Del, nice method with the light coming off the lamposts)

Ta,
Nice pic, I'm a big fan of Corvids
Del

Whaddaya mean Corvids.........that's me and the wife!

This is absolutely true. The fact is, most animals perceive beauty and usually it's based on symmetry and proportions. Why? Because symmetry and correct proportions reflect health, specifically robustness of cell division during early development. This is a very important factor for choosing a mate, therefore perception of beauty evolved early and is a very basic and fundamental instinct.

The reason why people find the golden ratio beautiful in buildings and paintings and everywhere else is because of it's origins in healthy body proportions. The reason it appears in healthy body proportions is related to cell division. The golden ratio is related to the Fibonacci Series

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

it is actually the ratio approached if you take the ratio of a term and the term before it:

1/1=1, 2/1=2, 3/2=1.5, 5/3=1.67., 8/5=1.6, 13/8=1.625, 21/13=1.615.,34/21=1.619..,

You see above the sequence approaches Phi=1.6180339887498948482… which is the golden ratio.

The above Fibanacci Sequence represents healthy cell division, which is why the golden ratio emerges as the proportion associated with beauty.

http://www.plantsci.cam.ac.uk/Haseloff/teaching/PDFlists/2002_PDFs/Klar2002.pdf

There is less to this than meets the eye...........

RAOFPMSL
I'm assuming that diagram is meant to be a spoof.
Del

Dear GKardys, your post is very interesting, with my GA.

You said, "In prehistoric times, a caveman would look for a mate with the fattest face and body – to club and drag back to his cave crib. ".

Now how do you know that he choose the chubby ones for the reasons you mention? Maybe the chubby ones were just easier to catch. Or maybe he figured that if she had a little meat on her bones there's a good chance she's a good cook.

What happens if everyone meets the ideal Golden Ratio proportions?

Let's say in the future we genetically engineer people to precisely control facial and body features and everyone is made to the "norm", golden ratio or "ideal" proportions.

Will everyone appear beautiful or will the "ideal" become ordinary or plain (average)?

Will the people that were created with small subtle differences, epigenomic mistakes or "sequencing errors" in the genetic engineering process be considered more unique and then beautiful?

Something I have often pondered about beauty is the difference between plainness and beauty.

Some people consider conformance of features to desirable norms (waist size, height, etc.) to determine who or what is beautiful. In Beauty, an article on Psychology Today's blog, "…we will likely conform our sense of what is attractive to the cultural norm." What has puzzled me is the "norm" should be the most common. If the "norm" is the most common, then why would people or objects meeting the norm not be considered plain, average, non-distinctive or ordinary?

What is the difference between or what separates plainness or the non-distinctive from beautiful? Perhaps the mold on Marylyn Monroe cheek is her deviation that makes her beauty unique and stand out.

The perception of Beauty has long been known to vary with the number and volume of alcoholic beverages consumed per unit time. From this we can probably conclude that the perception of Beauty is, at least in part, a function of chemistry. By the sixth drink Phi has little to do with it, and don't even bother trying to pronounce "Fibonacci," even if it is her last name.

I agree with part of your comment. If you make a work of art too symmetrical, then it looks stiff and unreal. The older ancient Greek and many Egyptian sculptures exhibit too much bilateral symmetry and they look unnatural (see examples below).

The sculptures weren't unreal because they were too symmetrical, it was because they were done of postures that weren't natural. What the Greeks brought to sculpture was action (as well as accurate proportions).

Here's an excerpt from wikipedia:

"The Sculptor Polykleitos could be deemed as the creator of the Classical Period. He created works with a true naturalism and balance, unlike the rigid poses of the Archaic period. He was the first to use ideas of scale and mathematical proportions in order to create the perfectly proportioned figure, referred to as "The Polykleitan Canon of Proportion". Polykleitos was very influential in Greek Sculpture, his realistic proportions were recognised by other sculptors such as Skopas and Lysippos who successfully followed him using the ideas set out in his canon.

Lysippos was the successor of Polykleitos; he took the ideas used in creating perfect proportions and gave rise to the "Lysippan Canon of Proportion". Lysippos noted a greater realism could be captured in making the heads of his figures smaller as well as elongating the body, creating much more realistic sculptures; which was his primary aim. His scrutinising attention to detail emphasised this desire to make his sculptures as realistic as possible. This sense of realism brought about the transition into the Hellenistic Period where use of the Lysippan canon of proportion and Contrapposto created extreme realism.

Creating realistic proportions was not the only way used to create statues as life like as possible. A perfectly proportioned figure will still look unnatural if in a rigid and unrelaxed pose. In the late classical period a combination of Contrapposto and "in the round" compositions (intended to be seen from multiple angles) created more interesting and natural poses. This was sparked by the sculptor Praxiteles, with his creation of the "Praxitelean curve" or Contrapposto. His fundamental aim was to create fluidity within the pose by changing from the conventional parallels of the shoulders, hips and knees to sloping angles. These angles, as seen in figures such as "Kritios Boy"- Early Classical Sculpture- and "Venus Braschi" (the first female nude) were much more comparable to the anatomy in real life, further emphasising naturalism and movement. This was a major step towards the extreme realism of the Hellenistic period."

And as the Greeks in later periods delved into realism they deviated from bilateral symmetry. A much more difficult sculpture to make, but in the later periods their techniques improved. Even with the strive for realism, the figures were "idealized". Youth would not have such perfect skin as they depicted. I don't think real depiction of people showed up until the Renaissance when you can find painting of a person with a warty looking nose like in "An Old Man and His Grandson" by Ghirlandaio 1449-1494.

Forget the dudes, just look at that Italian landscape out of the window...it's almost abstract , hints of Japanese...I wanna be there, cos the weather is pants here.
Havn't had a holliday all damn year_{...<exit monitor left mumbling>}
Del

I see what you mean...

Bravo!

I think Marilyn's beauty is in her expression of 'one-up' playfulness - as a great deal of Nefertiti's beauty is in her self-possessed serenity. Expression isn't quantifiable and is the essence of beauty. Imagine either of these ladies depressed and downtrodden - would they attract?

Beauty comes from spirit. Although I'm not an adherent of "intelligent design", fractals were conceived of abstractly, long before anyone noticed that they in fact described growth patterns. I find something very mysterious and beautiful in this convergence.

Re: dovy's remark that natural objects are beautiful - well, I can think of some that aren't (deer season upon us - the dog will fill the yard with carrion soon). But essentially, the natural world is alive and filled with spirit. This includes crystals, mountains, etc. in that they grow and change, are filled with light and the substance of life.

I'm including animal architecture in this conception, and many human constructions as well. Why then, are so many human constructions appallingly ugly? They are utilitarian - can it be that their uses are ultimately harmful, unhealthy to us and the biosphere?

I don't think that was mold on her cheek, more like a mole.

So if the air had been slightly thicker or thinner and gravity had been slightly higher or lower when the first organic compounds sparked to life would those ratios and number sequences still be the same in representing many life's proportions as well?

Or would they be different? The thing I see is that we logical rational humans tend to naturally seek out some explanation for every thing and then at times falsely conclude that there is a connection between things because we think we found a similarity. Some times there are in fact similarities other times there are just random coincidences.

The tires on my vehicle are round. The tires on your vehicle are round also. So in mathematical conclusion all circular objects are tires? The numbers add up so it must be true right?

Life has no more a corner on Φ than do tires on the Essence of Roundness (just ask Φ's ubiquitous cousin, Π, and tell 'em e sent you... )

e

I disagree with your analogy. I would argue that the tires of your vehicle are round, just as the tires of my vehicle are round for a specific functional reason. Round wheels show up on cars because cars are supposed to move. It is reasonable to extrapolate that most cars have round wheels because of this inherent function of the car (to move).

In the same way, all multicellular organisms develop through cell division. Is it really so surprising that the process involved in cell division, which is the same species to species, would have largescale consequences?

We look at this world of life with all it's diversity and sometimes forget how similar most of us species are to each other. We all start out small, incubate through the beginning stages of cell division (either in a womb, or an egg), then after a period of time we hatch or are born and continue to develop and grow until we reach maturity. Our robustness coupled with environmental factors effect the symmetry and proportions developed during these fast developmental stages. Beauty is simply a measurement of how close we are to a human developed under perfect conditions (who would have perfect symmetry and proportions).

This is not to say that there aren't different face types and body types, but ask and artist, and they will tell you that the faces may change, the proportions rarely do. Go ahead and hold your hand up to your face. You'll see that your hand is as long as your face from the hairline to the chin.

And of course we find inanimate objects that reflect these symmetries and proportions beautiful. That's because the idea of beauty originates with us, not with them. That's just our consciousness meshing with our instincts. We call it aesthetics.

The point is, to think that our perception of beauty, with the incredibly important role it plays in mating, which for all intents and purposes is the point of life, isn't somehow an evolved trait to effectively determine the viability of a mate quickly, but rather some universal thing that exists "out there" that we perceive, just seems insane.

And yes, I know I'm losing romanticism points here, but I can't force myself not to see the obvious truth. It's not so bad if you accept it. You can understand how a car works and still enjoy the ride.

I might also add that the Golden Ratio shows up in places and in ways having nothing at all to do with the Human concept of 'beauty.' For example, I seriously doubt that most humans would find population-growth patterns in nascent honeybee colonies as something worthy of a sonnet.

Or how about the spines in those criss-cross patterns you see on a pineapple? The number of spines in each row going one direction and the number of spines in each row going the other direction are adjacent Fibonacci numbers - but only for healthy pineapples. Would the lowly pineapple be any less lovely if they weren't adjacent Fibonacci numbers?

Perhaps in addition to its obvious role in mating behavior, might beauty also imply a degree of safety?

Well quite an amusing and entertaining discussion....Beauty is, in the end, in the eye/ear/touch/taste/mind of the beholder...Patterns of nature approach a mathematical sequence..perhaps even the stars/galaxies etc..have a similar findable number but at the end of the day beauty is personal opinion....and hence a subjective,non measureable entity.

Regards...Marty Wolf

You Wrote:"beauty is personal opinion....and hence a subjective,non measureable entity."

Sure, a personal opinion driven strongly by instincts.

You have to draw the line somewhere!

If beauty is mathematically expressed by Phi, I wonder if there's a mathematical expression for 'has character'... You know, some things have "a lot of character" and some have "a bit of character" so... there's math there somewhere, just waiting for a brilliant mind to provide the definition..

Asymmetric hairstyles don't look good on everyone, right? Why do you suppose that is?

In my photographic career I have discovered that people love pictures of pigs.

So what is it about pigs?

Content wise it is one thing. How the image of the pig is made, and composed hardly seems to have any bearing.

Let us apply the rules to the face of a pig, and see what mathematical precepts apply?

One theory of art is dependent on the frame, and the concept that perfection can be achieved only within the limitations imposed by the frame.

It would be interesting to note when art was first commonly presented in a frame.

I suspect that architecture, and the door created the concept of a Frame.

Cave paintings and cave wall paintings were not framed.

Sculpture is not framed.

The history of art and sculpture in particular illustrates something odd about mankind, for there was a time when the sculpture was actually worshiped as if it were God.

I have a poet friend who is fastened on the idea that if God exists, he must be in one place somewhere. My friend wrote a book about a character who finds God fleeing into a door at the end of a dark alley.

I like my friend but feel that his expectation that there is one address for God, odd, and not particularly sensible.

The worship of a sculpture would illustrate the desire for the art to actually be the thing, similar to my friends desire to find by detective work where Gods office is.

So then we have the 2 dimensional and three dimensional arts both seeking perfection.

The arts and sciences both seek what we ascribe to only God, perfection.

In a way it as if the pig becomes beautiful and perfect because it is imperfect, and expected to be ugly.

"Why is that pig smiling?"

I have a painter friend who paints scenes within tire shaped "framing" on the canvas.

So I consider her a brilliant artist by freeing herself from the "frame" and being able to swim the images together within the space of the circle.

(Lynda Curry is her name, and you may find a site with her stuff on it.)

Photographs do illustrate something innate within us for we instinctively know a great photograph from one of little distinction.

I myself am not so impressed by resolution and detail of a photograph, as I am by composition and content.

The work must have "heart".

Perfect work out of me has ideally Strength, Wisdom, Beauty, and Humor, in balance. Give me a mathematical formula for that please, and I'll try and apply it.

Oh to be a Priest and give funny sermons on Sunday!

When I got married I had permission for a Special Service from the Bishop, to throw and read along with the Readings for the Day, the I Ching.

So there I am and I throw for the Reading and it comes back, Time of Decrease on the issue and event of my marriage.

Good thing I got counseling.

At any rate I myself try for conformance to rules of art, as if there is some hope for perfection, knowing that there is conflict between the finite and the infinite. Russell

I like your philosophical look at the topic.

One thing you said sort of struck me as odd though.

"I have a painter friend who paints scenes within tire shaped "framing" on the canvas.

So I consider her a brilliant artist by freeing herself from the "frame" and being able to swim the images together within the space of the circle"

If she is using tire shaped framing, isn't she still confined by the "frame", just not a rectangular frame?

Possibly I ought to amend the sentence to read: freeing herself from the rectangular frame within the frame, or something like that.

She is not completely free of the frame, but frees the work within the frame, by reframing in the round.

The size of the circle, and where it is put on the Rectangular canvas conforms from afar. Then the painting is in the painting within the circular painted in frame.

The question becomes really could there be more than one mathematical rule determining beauty?

I do not know the answer to that question. I am caused to ask it due to my involvement with you all in the discussion.

I am also caused to examine where I come from primarily as a visual artist.

I was primarily trained as a photographer. I primarily worked with motion picture cameras and lighting. In motion picture photography you are limited to a frame that is nearly always horizontal. Further your subjects are actually often supposed to move and so to maintain visual integrity it becomes demanded that you move the camera as your subject moves.

It is labor intensive.

It may have as much to do with rules of sculpture, as it does with rules of composition as applicable to the still, or the painting.

Motion pictures are a distinct and "new art" that combine all arts and sciences.

You may argue that it is just captured theater, but this does not obviate its unique experience and technology that determines a specific "Frame", or way of seeing the art.

It is a unique artistic experience, and we love films that conform visually to the rule of thirds even when we have never heard of it.

I feel that there is a distortion in how we see things and what we will accept as visually pleasing, or correct caused by the historical television tube experience which is a terrible aspect ratio to work with, causing many compromises, and hopefully to be overcome as HD screens and the new TVs more and more allow for the perfect frame for the artist to work in.

From a simple visual experience standpoint illustrating the importance of framing to the art and its impact, and feeling of correctness, I suggest one watch a letterboxed version, and a TV format version of Saving Private Ryan. You simply are not seeing the same movie.

While my brother is a classical sculptor in that he puts things together permanently, my own forays into sculpture have been installations, or assemblages. Titles have been Modern Industrial Disaster, Bicycle Disaster, and The Raft. If I get around to making another Assemblage I've determined that part of the art would to covertly film the insured destruction of the work.

It is part of the human artistic experience to either destroy, or see, or have the art they made destroyed.

Three Soldiers by Dos Passos about WWI is interesting for our era of war and art and technology.

If you can find a way to see the film Light by Jordan Belson made back in the 70s, or maybe look at La Strada by Fellini, or photos by Cartier Bresson, and then Dutch Masters, and then go to Seurat we end up seeing revolutionary conformance.

Breaks away towards the banal as come to us in Photo Realism rebelling against proper composition depend so much on technical execution and shear size as to often make you wonder if a rich artist is as arrogant as any venal banker, and to be avoided in the same way.

Andy Warhol always played by the rules in a subversive way.

He made a number of pictures of Marylyn Monroe, who has been mentioned as a Beauty in this discussion.

Marylyn Monroe made herself into a living sculpture, a face, when she had a nose job.

Any way the question that arises in me caused by this discussion, is what errant mathematics are opposed but partner with to guide great art, and where might that diverge from or support pure mathematical realities.

In other words I have seen the wrong be right in art, and wonder if in science the wrong can also be right, or if once there is no frame, there is only one right, and confusion reigns as much for that, as the other.

Perhaps a less controversial and more fruitful but rather similar question would be: "Why do we consider so many phenomena within 'nature' to be beautiful?"

Finding ugly clouds, sunsets, mountains, trees, flowers, crystals, animals is rather difficult.

As for Phi, perhaps some quantitative analysis would clarify the issue. Do some natural things contain Phi? A few? All? It is not at all surprising that natural things grow according to varying mixtures of regular and chaotic descriptive rules. Extrapolating from Nautilus to, say, a race horse, a sunset or the late, lamented, Nefretity seems somewhat overenthousiastic.

Well, it seems that many things (process states) or, presumably, the processes which partially took part in producing them, depend on the 2 former states thus manifesting Phi.

Is there an analogous well-investigated process where a state depends on the 3 former states? On some other n former states?

Some (not all) things are considered beautiful, some (not all) things look regular, that does not mean of course that anything regular will be considered beautiful, nor does it mean that regularity can be found in anything considered beautiful.

dovy asks: "Is there an analogous well-investigated process where a state depends on the 3 former states? On some other n former states?"

In the case where "a state depends on n former states" (which covers both 'cases' above because n can assume any value, including 3) could be taken as, "The current state (however realized) depends on visiting each of the n former states in-turn."

A student learning mathematics is an example of this. She doesn't study calculus before learning algebra, and certainly not before learning her numbers. A largely linear sequential process.

The current state could possibly be arrived at by more than one route. For example, a traveling salesman may visit n cities via one of m possible routes (taking him through other cities) before arriving at his current location, yet he arrives at his current location nonetheless. His 'state' is assumed here to be his location. Unlike the study of mathematics, a non-unique sequence of states leading him to the same final state.

See Graph Theory.

Forcing Wikipedia to find extensions of Fibonacci numbers leads to Google which mentions Tribonacci and Tetranacci numbers which, in turn, are treated by Wolfram - with nothing exciting showing up.

Speaking of graphs, here are several representations of The Internet showing backbones, major IPs, average load vs route and so forth...

... and a mapping of a social network ...

Enjoy!

-e

Thanks, I have.

I'm getting more & more convinced, that apart from being 'a joy for ever' and residing in 'the beholders eye' beauty has very little to do with regularity (at least known, logical and/or mathematical regularities...).

One has to wonder what works harder....the hamster in Europeums head or the processor in his computer.

The fact that he has just pointed out some fascinating examples of artsmath he also points out the possibility that any work of art can be judged according to a preset value. This approaches tyranical (AND AMAZING) proportions.

Imagine if Clement VII told Mike that his Fibonacci #'s were off. It would've taken another decade to finish the Sistine.

On the other hand, and if I understand Europeum correctly, what if the human mind was predisposed to an instinctual logic ???? Would math therefore only be a complement to an otherwise evolving pattern of human perception.....hmmmmm

I am attempting to address this for I agree with myself, Del, you, RP and Europeum.

I get left with an answer, and a question.