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.

Ditching Perspective Rules

How much do you think about the rules of linear perspective when you draw or paint?

One-point, two-point, three-point—just establishing a horizon line and finding vanishing points can be such a chore that many artists find it too stifling to deal with. But there’s a way to find the correct angles of perspective for any object without having to memorize a single rule, and it’s something that artists have been using for centuries: the angle finder.

A brush handle, pencil, finger or even a stick can be turned into an angle finder. All it takes is holding your angle finder at arm’s length, closing one eye, and aligning it with the edge of the object you’re planning on drawing.

Locating the angle of the an oak tree trunk by using an brush handle as an angle finder.

Finding the angle is simple enough, but sometimes there’s a disconnect that occurs somewhere between reading the angle and translating it to a drawing or painting.  Years ago I recognized this problem among a number of my college students.  I discovered that when students were unsure about the tilt of an angle they were looking at, they could see it better if they could label it.  (An odd thing, labeling: something we humans seem to need.)  To enable an accurate labeling system, I came up with a method to teach angle finding by equating each angle to a number on a clock face.  

It works like this:  stand in front of a clock face holding your angle finder at arm’s length. Position it so that it becomes like another hand on the clock as if connected to the center. Looking with one eye closed,  align the angle finder toward a number on the clock by rotating it either clockwise or counter clockwise to find 7 o’clock, then 9,  12 and 2.  You have just read four perspective angles.

Now try this:  Lay a book on a table, pick up a pencil to use as an angle finder,  stand back three or four feet, hold the pencil on the eraser end, straighten your arm so your elbow won't bend, close one eye, then rotate your pencil until it aligns with the edge of the book's cover.  Look at where the pencil is pointing, then name the angle according to its clock number.  Try this again along the front edge of the book cover.

Keep your attention on the pencil so that it doesn't flop forward or  point backwards.  It  must move only clockwise  or counter clockwise, else it will mislead you.

The photo on the left shows the pencil pointing to a 10 o'clock angle while the one on the right points between 2 and 3.   Labeling these angles according to where we find them on a clock can enable us to draw these angles, by-passing perspective rules .

Try doing a simple line drawing of the book, labeling the angle of each line like I've done in the drawing below.

Practicing the exercise several times can give you a idea of how the system works.  After feeling secure, try the same exercise using a building as your image.

Just like any skill, the more you practice this the more adroit you’ll become at using it. Eventually you will have the clock positions firmly engraved in your memory and you’ll no longer need the clock diagram to help you out.

You will also discover that you’re recognizing angles in many more places, and that it’s much easier to draw them accurately—without ever being concerned about the actual rules of perspective again.

(Disclosure:  this tutorial is an adaptation of a tutorial I wrote for Empty Easel in October, 2008.)

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See Dianne's latest work on One Artist's Journey

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Ten Thousand Hours

We listen to music, we look at paintings.  We play a musical instrument, we compose paintings.  But what's behind it all?

Referring to the Compose tutorials of the past two weeks,  Ann Feldman commented:  "Malcolm Gladwell's book, Outliers...hypothesizes that mastery can only be accomplished after 10,000 hours of dedicated practice in any field, including art, music, and sports."

Ann continued, "Makes me think that an artist can only relax and allow the painting to be discovered after that artist has spent many hours learning the underlying techniques of art. Until an artist feels comfortable in his or her understanding, that relaxation is elusive! Do you agree?"

Gesture drawings by Rodney Grainger
following the Nicolaides approach

I cut my teeth on The Natural Way to Draw by Kimon Nicolaides. I did not realize as a young art student that Nicolaides' approach was the key that would unlocked my journey as an artist and as a teacher.  Nicolaides emphasizes process over product, a concept I believe underlies all artistic success.  So to answer Ann's question, I believe an emerging artist can become comfortable within every layer of the process.  In Gladwell's hypothesis, I believe "practice" is the emphasis.

There is always a degree of awkwardness with anything new. But that awkwardness dissipates as we  practice.  With practice the newness wears off and what was once new now is familiar. And the more familiar it becomes, the more comfortable we become until it what was once new is now second nature.

Thinking about Sharon Isbin's mastery of guitar playing, I think back to when I was age nine taking my first guitar lesson.  It was all new--how to hold the guitar, how to place my left fingers, how to use my right fingers, how to read the music score, and how to strike the strings. Add to this that those tiny young fingers were too tender and lacking in muscle dexterity to hold the strings tight to the guitar's neck.  A year later,  my fingertips were toughened, my finger muscles were toned, I held the guitar correctly without a thought, both right and left hands moved automatically and the score was like written words on a page. I could play "Danny Boy" without thinking.  I was ready for my first recital.

Learning to paint is exactly the same process, just different materials. Perhaps the most important similarity in the two is that neither can be done without practice.  But what does practice mean?  Doctors practice, but they already have their degrees. (Even so, we hope they don't stop learning.)

Practice means to exercise and perform repeatedly to improve and/or maintain what one has gained.  When we study techniques, we practice; when we're making a painting, we practice.  When we do exercises, we practice.  Practice is essential to the process no matter what we're doing.

Leonardo da Vinci practicing

Leonardo da Vinci practicing some more

Whether it's painting or guitar playing or figure skating, our practice undergirds the process that enables us to master new levels. Within each level once the awkwardness has given way to relaxation, we can discover.  I don't think we have to wait until we've put in those 10,000 hours; I think the painting can be discovered at any level of competence.  It's a matter of attending to the process rather than focusing on the end product, finding our comfort zone in each of those 10,000 hours, all along the way.  If we do this, mastery after 10,000 is virtually a guarantee.

Bring Life Into A Still Life

I've always thought the term "still life" a bit antithetical:  what life has ever been still?  But somewhere back in history, powers-that-be deemed still life a genre and left it up to future generations to define.  Still life painting has a long history, going back as far as the tombs of ancient Egypt, proliferated in the Middle Ages and holds strong today as a genre.  These days, any painting of a human-made arrangement of objects is called still life.

Left, Henri Masisse, 1912     Top right, Qiang Huang, 2010,
Bottom right, Roman wall painting, 70 a.d.

The question Judy Warner asked is how to add life into a still life composition. More specifically, Judy added "...how to add life so they don't look dead."   It is true that some still life paintings carry with them a ho-hum feeling, but so do some landscapes and even some figure paintings.  Is there something about the still life genre that puts it in danger of becoming a lifeless work?

I'm not convinced that any genre is more vulnerable than any other to resulting in a lifeless painting.  But sometimes while setting up a still life, the artist can get a bit fussy during the process of arranging objects and that disposition gets transferred into the work itself, defeating the success of the painting before picking up the first brush.

I think Charles Reid is a prime example of an artist who brings life into his still life paintings.

Photo by Sandi Hester                      Still life demo by Charles Reid
Click on the image for a larger view.

Those of us who have watched Charles set up a still life know how nonchalant he goes about putting an arrangement together. Often the objects he chooses are randomly selected.  But in the end, we see that he has placed a number of actors on a stage, an advantage given still life painting unique to its genre. No other genre offers the artist total control over the subject matter.

Charles then approaches the painting from a viewpoint of discovering and responding:  rather than trying to copy the setup, he begins with a contour drawing to discover what he is looking at.  Sometimes he will change the setup midstream--take something out, move something to a different place or add another piece.  

Here's where begins something important to bringing life into a still life painting:  the artist's attitude toward our process. When the intention is to discover what the eyes are looking at rather than that of trying to copy stuff, the artist stays alert during the process because we don't know what's going to happen.  If, on the other hand, our attitude is one of trying to get it right or trying to force some preconceived notion, we run the risk of suffocating the work.

Charles Reid preliminary drawing for a painting different from examples above.
Photo by Mick Carney
Click on image for a larger view

Once his drawing is done, Reid approaches the painting like a kid in a candy store.  He simply responds and keeps moving forward with confidence.   He doesn't labor the piece.  When asked whether he has a game plan, he always answers "No."  He says he likes to approach the painting as if he's never done it before.

(During one of Charles' workshops, Mick Carney recorded the progress of his demo.  Go HERE then cursor down a bit to see this progression.)

I think the difference between a tired-looking still life painting and one that's vibrant and full of life is a matter of the artist's attitude and confidence.  Any painting that is labored over will most likely look tired and lack vibrancy.  But when the artists moves forward, confident with a child-like approach of discovering what's there and responding with whatever degree of available skill, the end results has a better chance of having a life of its own. 

I take issue with those who approach painting from a formulaic attitude. Keeping a painting fresh and alive is not a matter of following a set of rules--including intentionally trying to loosen up--nor is it slinging paint willy-nilly.  Rather, bringing life into a painting comes from an inner attitude of wondering what one will discover next and allowing the painting to move forward within that intention.  It is in the laboring over a painting that we steal from ourselves and consequently, the painting itself,  freshness, spontaneity and wholeness that yield life.

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SHAMELESS PLUG

Check out Dianne's new book,  In Praise of Mountain Waters:  Paintings of Rivers, Waterfalls and Streams in Northeast Georgia.   Available at Amazon.com