Mathematics and Art

I confess I surprised myself when I check out from our local library a video course on mathematics, but the title Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas caught my attention.  And I was totally intrigued with every lesson.  To my delight, the course explores how math can be used as a tool to explore aesthetics and the mysteries of nature.

Some folks think it is not logical that artists could be fascinated by mathematics.  Somewhere in the mix of professional mandates, artists are told that logic and art don’t mix, to use logic is to stifle creativity.  Art is suppose to come altogether from the intuition, they say.  Leave logic to the mathematicians. As a result of this kind of thinking, art and math get pigeon-holed as poles apart when, in fact, there is a lot of intuition in mathematics, and artists and architects have been depending upon mathematical proportions for centuries.

One obvious way artists use mathematical proportion is when we locate a rectangle’s "sweet spots" (eyes of the rectangle) for placing our centers of interest.  We have discovered that those four areas midway a canvas' center and each of its corners are ideal spots for placing points of emphasis.

I've shown these with the green dots in the diagram below.

Qiang Huang has used the lower right "sweet spot" for placing the orange in his painting, "Out of Shadow."

Carolyn Anderson uses the same location for placing the hands in her painting, "Boy Reading."

I admit that if we artists had to do the math to come up with these "sweet spots," we'd refuse to deal with it, but the beauty of it is that hundred of years ago mathematicians figured out where these are: they discovered the Golden Rectangle on which the "sweet spots" are based.  

Fibonacci who lived in the 1200's AD discovered a sequence of numbers that show the logic of the Golden Rectangle.  It begins with 0  followed by 1 and here is where the sequence begins. The next number in the sequence, 1,  is the results of the first two added together (0 + 1).  That 1 is added to the first 1 to get 2, then the 2 is added to this 1 to get the next number 3. Then 3 is added back to the previous number (2) to equal 5, then 5+3=8, then 8+3=11 and on and on so that the sequence looks like this:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 and on and on: there is no end to it.  This sequence is exciting because it represents a ratio pattern that is found throughout nature, whether spirals in seashells, leaf arrangement on plants, scales of a pineapple, or all living cells.  (For more on this, go HERE.)

Here's how it sets up the Golden Rectangle:

Slight disclaimer:  the squares here are not exact, but illustrate  the point.

The formation begins with 2 squares the same size (1 and 1 in the illustration above) .  Add another square (2) whose edges are the same length as the combination of the first two together. Add a third square whose edges are the length of 2 + 1, a fourth whose edges are the length of  3 + 2, another whose edges are the same as 3 + 5 and on and on.  This rectangle could continue until it circles the earth a zillion times.  When built on this sequence, it is an infinite rectangle, the Golden Rectangle. Perhaps its perfection is in its infinitude.

I often wonder whether there is a strong relationship between intuition and infinity.  Whether there is or not, it's all fun to think about.  Perhaps logic and intuition are two sides of the same coin. But that's a discussion for another day.  Meanwhile, I hope you enjoyed this little mental exercise.

The Zen of Composing

Don't let the noise of other's opinions

drown out your inner voice. -  Steve Jobs

Composing is about making decisions, but decision-making need not be limited to intellectual choices.  In fact, some of the best artistic decisions we make are intuitive.  After all, it's within our intuition where we find our inner voice.

Our intellect plays an important role in our learning the mechanics of how composing works and we need that; otherwise, our work would suffer.  In fact, those who would argue that learning composition principles is limiting are both misinformed and naive.  But we are not unlike Olympic champions who learn their craft, but preform their best when they depend upon their intuitions to exercise that craft.

And it is within our intuition where we make discoveries.  Our intellectual knowledge enables us to recognize, understand and communicate what we have discovered.  And it enables us to grow out of ignorance,  but without  intuition there would be no such thing as creativity.  There would be no scientific discoveries, no new inventions.  I would argue that the finest art grows out of both working together in balance.

But we need to acknowledge that too often artists fall into a trap of blocking intuition when we have to make judgments.  With our eyes tracking backwards while we're working, it's easy to deem what we have done inadequate.  Artist-author Frederick Franck has what I think is an approach towards breaking through this potential trap.

Franck's  Zen Seeing, Zen Drawing guides the artist past the self-critical phase into an attitude of mind where we respond to what we see without hanging ourselves up on whether we're getting it right.  He guides the reader through simple exercises that when practiced daily becomes a journey of seeing rather than just looking at our subjects.

The value of being able to do this, as I see it, is the freedom we can gain  from trying to make our work acceptable.  We lose that tendency of judging ourselves inadequate and therefore grow more confidence--confidence in our painting skills, confidence in the way we use them in composing, but most important of all, confidence in allowing access to our inner voice.
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As as a sidebar:  With this week's One Artist's Journey post, I'm introducing my new association with Daily PaintWorks, an on-line sales gallery and auction website owned and managed by David Marine, husband of artist Carol Marine.  I'm excited about this new opportunity to make my paintings available for sale on a website of such integrity.