euclid (detail from the school of athens by...

Euclid (detail from The School of Athens by Raphael)

Enchanting the rectangle

Session #12 What if I told you there was a magic code -- let’s

say, a pair of numbers -- you could recite to put your picture under a spell? -- a spell that kept your picture always fresh and interesting to look at? Does it sound too good to be true? I know. It does to me, too. If it sounds too good to be true, it probably is.

Except sometimes in science and art, you’ll find magic. In the last few sessions we talked about how you control dimensional grids and ‘construction (orthogonal) lines’ in your picture. (You’re a space commander, remember?) So let’s introduce another sort of measuring and ordering of space that happens in the first seconds of your sketch. You often do it unconsciously. How you do it can make or break your painting. You’re already doing it right when you scribble off your gesture thumbnail in 30 seconds or less. You don’t have to plan too much or think too hard about design. It springs from you. It’s kinetic -- a spontaneous expression of movement.

The ‘movement’ can be wild, explosive action. Or it can be resting. An inert subject contains the potential for movement, and so there’s tension. (A transition from nothing to something is highly charged.) Your scribbled gesture contains this motion or this tension. You felt one or the other or both when you dashed it down.

So if you let the impulse draw for you, you already have your design. It’s staring back at you. You just have to see and appreciate it and recreate it in the gesture for your detailed drawing. So Stage One of the big drawing is redoing the gesture on your larger page. You must recreate the experience. It’s sensitive and subtle -- but not hard. There’s no hard thinking required in casting a spell. Children see the world as enchanted, anyway. So they expect no less from you. So reliving “the movement” is the first step. The next step is discovering an important (maybe mystical) grid that lies hidden under your picture space.

Yup, like a magic world behind a mirror. And finding a rhythm that could be the secret heartbeat of the universe. Wow, this is like being initiated into a secret society. So, is all of this sounding whacky enough for you? All right, then!

Let’s proceed… Let’s start with an old story dating back to the Roman Empire -- the Pax Romana under the rule of Emperor Caesar Augustus, to be precise. The Romans were having a hard time with their art and sculpture, that everyone admitted looked stolid, stiff and well, clunky compared to the exquisite art of the classical Greek era that preceded them. And they knew it. They weren’t hearing the music And it bothered them They were great at sensible, practical things. It was like in the Monty Python film The Life of Brian. (It’s been a long time since I’ve seen the movie -- and I don’t have the script in front of me, so I’m

making it up here. But in the scene, two men grumble about the Roman occupation of Judea.) “Those bloody Romans, what have they ever done for us.” “They gave us roads.” “Agh! Anyone can do roads. Name one other thing.”

“Bridges…” “Well, yes, but what else?” “Water and sewer systems?” “So name one other thing we got from them!” “Public schools…” But the Romans wanted to be good at more than just infrastructure. They wanted to do Art and Beauty, like the Greeks. So Caesar Augustus appointed Rome’s most able architect, Marcos Vitruvius Pollio – one of his former army engineers – to head a task force. They were to examine the art and architecture of ancient Greece, and discover the secret.

Vitruvius and his committee members poked around the Greek ruins and antiquities still lying around the empire. They camped in Greek libraries. And one day, while Vitruvius was reading Plato (the Greek philosopher) he found Plato’s description of a good composition.

Unity plus diversity.

Unity is about the One. Diversity is the many, the variety, the having ‘no two shapes the same’ in your drawing. Unity is the reigning order. Diversity upsets the order – just enough to keep it interesting. Your painting, statue, temple, is mostly about this (unity). But it’s also got that (diversity.) In good design, everything’s the same. But there’s a difference. It’s just like a great movie script has a mostly familiar plot, with something new and unexpected added. (Plato didn’t say that.) The figure stands this way, his weight on one foot. So he throws his body out this other way for balance. This was the Greek way.

Unity is sameness. Diversity is the graffiti scrawled over it. (I’m just sort of paraphrasing Plato, now.) Unity is often achieved through the design principle of dominance. Dominance, of course, is also a life principle.

For a structure, a system, tribe or a family to work, there must be a consensus. (It may come from the boss’s office, but as long as everyone agrees, you’ve got a going group.) Unity – everything under one rule -- keeps things simple – and symmetrical. It’s the conformity of a crystal. Diversity keeps things vital. It would be the alternative to order, the fringe, the specialties, opposing factions, the ‘peanut gallery’ with its critics, and other challenges to authority. It brings the tension that’s the signpost of life. Biologists talk about integration + complexity in kind of the same way. With integration, everything ties back to the base, the center, the One. But complexity starts to pulls away from the One.

It’s the leaving, the differentiation, the pushing out from the center and moving into the unknown. Integration is the fat trunk. Complexity is in the roots, fruits, twigs, high branches and leaves so necessary to the tree… You need both.

You play them against each other like you did the complementary colors on the color wheel. If your drawing is about thin, tentative lines, you enliven it (quite a lot) by adding a few bold, sure, thick lines. Or if your drawing features round soft cuddly shapes, you can improve it by adding a few hard angular shapes.

The love of visual contrast seems to be hardwired into us. It’s a “feel good” mechanism, probably tied to survival. Probably it goes way back to the time of our forerunners, the trilobites -- the first creatures on the planet with eyes. I say “eyes.” I mean photon receptors that could distinguish light from dark. “Eyes” gave the tiny trilobites a tremendous advantage over their fellow bugs, worms and other wrigglers. They could see trouble invading their field of vision (such as it was) and respond accordingly.

They could see their next meal coming, too, if it was a little darker or lighter than the background. Their lives depended on their discernment of change (contrast) in their environment.

Visual contrast still grabs our attention today. It’s fraught with significance, metaphorical and neurological.

We don’t want too much, because contrast triggers stress. It’s hard to process a message when we’re always bracing for changes. Our brains crave the order, calm and (we hope) security of the status quo.

So Plato’s formula for beauty is more like the law of the jungle. But Unity with Diversity is also the stuff of drama. (The best comedy has touches of pathos. See: Charlie Chaplin.) And it’s the stuff of great cooking recipes. (Add the salt, but don’t forget the sugar.) Vitruvius took the idea back to Rome with him, to build his canon of Roman aesthetics on. Nobody said things better than Plato. It’s a subtle formula. Look carefully at the words. Unity with Diversity can’t be an equal split. It does not mean 50-50 fusion between the two contrasts. (If it was 50-50, there would be no unity, you see.) So beauty is order + something that threatens the

order, but mostly it’s about the order. So your design must be more about the rule than the exception. How much more? A little more.

How much a little more? Well, more than half. How much than half, exactly? Well, you’re getting all quantitative on me. OK, actually, there is a way to state Plato’s idea with numerical precision. Plato may not have known it. But Greek mathematicians like Pythagorus and Euclid knew it. (Sigh, there’s just no getting away from math, it seems.) It’s .6. Actually it’s 1.62. The ratio of unity to diversity that we seem to want in our ordered universe.

Euclid expressed the relationship as a measure on a line: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”

8 is to 5 as 5 is to 3 This proportion binds the parts to the whole. And that’s a beautiful relationship. The relationship can be expressed as a ratio, 3 to 5 Or a fraction, 3/5 Or as a decimal number. .6.

Or if we’re using 5 to 8 (5/8), the decimal would be .62. But we painters we can use the simpler .6. It’s still considered the same proportion. Every time you multiply a length of line by that decimal number, you create the bond between the two parts and the whole that Euclid was talking about. Actually the decimal (1.6180339887…) goes on forever because this is one very irrational number. Some say it’s a magic number, or even a ‘sacred’ number. It has fascinated mathematicians, architects, musicians, astronomers, artists and mystics for centuries. It’s harmony found in a decimal. The ratio of opposites implicit in Plato’s formula:

Unity + Variety, Diversity, Contrast Dorminance Order Sameness 60% to 40%

The pleasing proportion.

It’s a number important enough to have its own Greek letter associated with it -- phi (pronounced “fee”)

A 20th century American mathematician named the special number for Phideas, the great sculptor who may (or may not) have used the ratio in his designs for the Parthenon. Phi is often confused with another number special enough to have a Greek letter name, Pi (Pronounced: pie)

Pi (3.14 -- the ratio of a circle’s radius to its circumference) is also used in art and seems to play role in nature. Both numbers have their cults.

The Phi proportion is called the golden proportion, the golden ratio, or the golden section. It has nothing to do with the golden mean. (That’s Aristotle’s phrase to express his philosophy of living a balanced life: moderation in all things.) It has nothing to do with the golden rule (He who has all the gold makes all the rules. That’s the golden rule of the children’s book industry.) OK. Are you ready for this? The math hounds of history have tied the golden ratio to an interesting series of numbers called the Fibonacci sequence (named for an Italian mathematician who lived in the Dark Ages. But number crunchers of very early India may have caught on to it first.)

Now let’s look at something, here. From Wikipedia:

In mathematics, the Fibonacci numbers are the following sequence of numbers:

The first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two:

The Fibonacci sequence is 0,1,1,2,3,5,8,13,21,34,55,89…and it keeps going, up, up, up into outer space. Each number in the sequence is the sum of the previous two numbers. Now let’s find phi, the golden proportion here, somewhere.

Take any number in the sequence, and divide it by the number that comes after it in the sequence…

It does take a while to get rolling and for the magic to kick in. But let’s try it. Go ahead. Get a calculator. (I’ve got my HP12C out.) Here’s the sequence again: 0,1,1,2,3,5,8,13,21,34,55,89… Let’s go: 0 divided by 1 = 0. 1 divided by 1 = 1. 1 divided by 2 = .50 (50 percent.) 2 divided by 3 = .666666667 (Hmmm. No wonder the cults look at this stuff…) 3 divided by 5 = .60000 (OK. We’re in Golden Ratio territory.) 5 divided by 8 = .62500000… (It went high, but

we’re still kind of in the ball park, I’d say.) 8 divided by 13 = .615384615… (Back in the zone.) 13 divided by 21 = .619047619… 21 divided by 34 = .617647059… 34 divided by 55 =.618181818… 55 divided by 89 = .617977528… Look at that.

Let’s do one more. We’ll add 55 to 89 and get 144 (the next number in the Fibonacci sequence.) 89 divided by 144 = .618055556…. All the numbers in the Fibonacci sequence after 8/13 are irrational, which mean they never resolve themselves. They keep going. But they all round to .62. We’ll get .62 from here out as we go up the sequence to infinity. .62 -- the 1.618 or .618 phi relationship -- keeps showing up in geometry. It’s found in plants. Wikipedia: Fibonacci sequences appear in biological settings,[34] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[35] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone

It shows up in human anatomy proportions (like the ratio between the top of the head to the floor and the belly button to the floor, or so Leonardo DaVinci believed.)

It appears in animal population patterns ( like the breeding rates of rabbits, some claim.)

It repeats in crystals, fractals and the patterns on moth wings

It repeats in the twisting chambers of the Conch shell and the spiral arms of galaxies.

The universe can’t seem to get enough of 1.62. It’s just plain weird. Maybe it is truly a building block of Nature. Or maybe our brains just want to see it everywhere. Members in good standing in the various Phi cults do see it everywhere. They’re like conspiracy theorists. (Da Vinci Code, anyone?) You can spend the next eight months of your life reading about Phi on the Web and you will have fun.

Here’s my freehand drawing of a phi spiral. Oh, you could tell I did it freehand? I kept multiplying the rectangle by its little golden section cousin. The arc of the spiral starts and stops at diagonal corners of each box.

So what does this have to do with illustrating a children’s book? I was wondering when you’d me ask that. Well, phi is a spatial harmony. It’s the proportion between any adjacent Fibonacci numbers after 3 (3/5, 5/8, 8/13 etc.) It’s also the unity plus diversity formula expressed as a fraction. With phi, you can divide the space in your picture in a way that never gets boring or repeats itself. And weirdly enough, the decimal you get from dividing 5 by 8 (and all the remaining Fibonacci pairs) is an algebraic irrational number that never quite resolves itself and keeps calculating, out to an infinite number of decimal places.

So the number never repeats or gets boring either. As low tech artists, we get to round phi down to .6 . We get to use the simplest phi ratio of 3 to 5.

So when you’re thinking of how to lay out your illustration, you can, if you want, multiply each side of your picture by .6 or 60 percent to find your composition sweet spot.

Well, it’s roughly 60 percent here. You don’t actually need a ruler for this. You can eyeball 60 percent. It’s just a bit over half, right?

The intersection of these golden sections will make the ideal center of interest– the place where your eye is most glad to look. It will also give you four unequal spaces. Not too unequal -- just enough to be interesting. The spaces are “divinely proportioned.” They’re barely asymmetrical. And one space dominates in size. That’s unity with diversity. And it’s perfect.

Every time you scribble in your sketchbook or make a splash on your watercolor sheet, you’re setting up invisible graph paper. The asymmetrical grid. With your first mark, you start dividing your sheet into “parts.” How you carve up the space determines if your finished painting will “work.” You don’t have to think too hard about this. Phi practically renders itself (though you may not always notice or heed it.)

Have you ever drawn a scene that just bothers you at some pre-verbal, primordial level? You can’t accept it, yet you can’t figure out why? The drawing is good. The figures are well-realized. The perspective makes sense. The line quality is fun.

But something in the composition doesn’t add up. You hate it. Usually it means you’ve run roughshod over a divinely asymmetrical proportion.

If you re-arrange the elements a little, crop the picture differently so that three fifths (or 5/8 or any other Fibonacci pair -- the phi ratio) clearly happens in the frame

You’ll fix it.

Just nudge the dominating shape or subject over a bit -- to about, let’s just say the 60 percent mark... Your composition will work fine It won’t bother you any more. Phi is potent medicine.

It’s the photographer’s great joy when he’s looking through his camera viewfinder. He hopes to “find his picture.” (The old single lens reflex (SLR) camera viewfinders with their ground glass lenses were so great for this.) When plein air painters hold up their little viewfinders to an actual landscape, they’re looking for this pleasing space division. The key elements in the view –- the barn, the tree, the branch, the rusted tractor -- determine those division points. When artists throw a mat around a watercolor, or play with the crop tool in Photoshop, they’re hunting for the phi proportion. They may not know it’s what they’re doing. The instant you put a mark on your blank sheet, you wake up the “hidden grid.” You’re invoking natural laws and secret forces.

You’re triggering them.

You saw the astronomy photo. The forces are out there -- happy, poised to play and ready to work with you (or against you) as you pick up your pencil. Look out. It reminds me of a story from early 20th century history. Leon Trotsky was talking to a civic leader at the time when the Bolsheviks were organizing in

Tsarist Russia. The community official told Trotsky flat out, “But we’re not interested in revolution.” “That may be so, comrade,” Trotsky responded. “But the revolution is interested in you.” Yikes. (I would have taken it as a threat.) Phi is interested in you whether you agree with it or not. My message, though, is that phi can only help your pictures. It’s the big hedgehog idea of design. It’s about more than dividing the picture space. It’s about the proportions you’ll select in Color and all the other “design elements” that you remember from middle school: Shape, Size, Line, Texture, Direction and Value.

It’s about phi all around you.

Check out this video by music educator Rebeca Mauleon, in which she talks about the “hidden rhythm” of Latin music http://www.youtube.com/watch?v=dafWseVsJCY

(Slow…slow…slow… Quick-quick …) Is the exciting beat of Salsa from the Fibonacci sequence?

Your home fun: El cuarto de Tula Buena Vista Social Club http://bit.ly/eBoNK 1.) Use a calculator to divide some of the adjacent numbers in the Fibonacci sequence to get phi.

(The first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two.

To get phi divide any number 3 or above by the number that

immediately follows it.)

If you don’t have a calculator handy, use this online one. http://www.math.com/students/calculators/source/basic.htm

(It’s a very basic one but it does the job. The divide key is the one with “/”.)

It’s an experience when you see for yourself phi coming up over and over. 2.) Make your own cool “enchanted” viewfinder. Cut a rectangle with the dimensions 1.5 inches by 2.5 inches out of an index card.

Use it to isolate imagery – real or reproduced. You’ll be looking through a magic frame that has phi proportions 3 to 5. Move it around until whatever catches your eye falls into the “golden section” of your view. 4.) Go to the opera. In this scene of Pirates of Penance, Frederic learns that he has a longer contract with the pirates than he realized. http://www.youtube.com/watch?v=XXhJKzI1u48

http://math.boisestate.edu/gas/pirates/web_op/pirates18.html

http://math.boisestate.edu/gas/pirates/web_op/pirates18d.html


http://www.youtube.com/watch?v=rQGrQPZMLK8&feature=related

http://www.youtube.com/watch?v=JGAxcjhV1EQ


The Pirate King and crew vs. the Keystone Cops (Only in Gilbert & Sullivan…)

http://www.youtube.com/watch?v=gzSTTTCOHr4&feature=PlayList&p

=49D375CEE15882E3&index=0


http://math.boisestate.edu/gas/pirates/web_op/pirates23d.html



Make Your Splashes; Make Your Marks!: The power course on creating great drawings for books, magazines and other media for children

Content © Copyright 2009 by Mark Mitchell

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