The Laws of Nature

Tuesday, November 17, 2009


Before I discuss today's topic, I should explain this painting. It's watercolor and was painted many years ago. The title is Measured Life at 51 and it's about a time in my life when I underwent chemotherapy for the first time. I began to think of all the ways my life was being measured: dosage of pills, timing of dosages, losing weight, measuring everything I ate, etc. Glad those days are behind me! It's not a very masterful painting, but it expresses how I felt at the time.

In the last post, I left you with Arnheim's idea that The great works of art are complex, but we also praise them for having simplicity, by which we mean that they organize a wealth of meaning and form in an overall structure that clearly defines the place and function of every detail in the whole. This way of organizing a needed structure in the simplest possible way may be called its orderliness.

Over the past year I made it a point to study composition, and specifically Renaissance designs. Talk about making the complex simple! Although I haven't yet applied too much of what I learned, I plan to start a whole new series of paintings in 2010 that employs the mathematical division of space developed by the Ancients. The idea that a work of art will be more beautiful if it is made according to numerical relationships began in mid-fifth century BCE. Renaissance artists utilized that math, as do a growing number of contemporary artists. Everything old is new again!

For those of you who are brave enough to follow this line of reasoning, I'll briefly depart from Arnheim's book to discuss how the Masters placed shapes to create "orderliness." It's not a comprehensive analysis; just some tidbits.

I'll start with a very simple concept. You want to paint on a rectangular canvas or paper. According to Renaissance artists, the length and width of that rectangle should be a specific proportion to yield the best results. These are called root rectangles. Here's how it works:

1. Start with a square of any dimension (example - each side is 3")

2. Measure its diagonal (for a 3" square, that would be 4.25")

3. Calculate the square root of the diagonal (the square root of 4.25" is 2.06")

4. Add that number to the length of the base of the square (3" + 2.06" = 5.06")

5. Draw a line that defines the length of the base of the new rectangle

6. Draw the other three sides of the rectangle. (your rectangle has a height of 3" and length of 5.06")

This is a root 2 rectangle.

You can keep going, like the figure below, by drawing the diagonal of the new rectangle, taking its square root, and drawing the next larger rectangle. That would be a root 3 rectangle, etc.

Figure from Hambidge, 1920

If you use pre-made canvases/papers they're already sized according to these proportions!

OK, so you've decided on the size of your rectangular canvas/paper. Now, you must create a balanced (harmonic) composition to create "orderliness." For now, I'll forego a discussion of the size and color of the shapes. How did the "ancients" figure this out? Pythagoras, a Greek mathematician/philospher who lived around 500 BCE, discovered harmony through proportions, or ratios. This is the foundation of our modern musical harmonic scale, and also the foundation for Roman architecture and Renaissance art.

Eventually, Pythagorean theory on harmonics made its way to the Renaissance, when artists wanted to create a less complicated way to organize the elaborate designs of their predecessors, the medieval artists. These designs, or ways of creating orderliness, were eventually identified as the armature of the rectangle by Charles Bouleau in his book The Painter's Secret Geometry. Here's a complex armature:

Notice that the diagonal lines are drawn from either the corners or middle of each side. The verticals are calculated by halves, thirds, fifths, (sevenths?). Nodes occur where lines cross, and it's on those some of these nodes that the artists would place focal points.
Here's the armature used by Johannes Vermeer in his painting, The Astronomer. Notice that he ony used a few of the diagonals, verticals, and horizontals from the more complex armature in the diagram above. Also, notice the central nodes that define the placement of the globe, wrist, and shoulder in the middle of the composition. The left hand is immediately below the right wrist on another node, and the toe of the left shoe is below that. The angle of the light follows the two diagonals drawn from the upper left to the bottom right. And so on. (Figure taken from Classical Painting Atelier by Juliette Aristides, 2008).

There are many variations on this basic armature, and certainly any artist can invent one. They may be very simple, or include all the lines found in the figure above. Some artists included circles and arcs in the armatures. What do they all have in common? Harmonic symmetry.

Does an artist have to use these geometric principles? No, not at all. As I said earlier, I plan to experiment with it in 2010 to see if I can correct many of the design flaws I've been noticing in my work. This gives me a great opportunity to marry my two "selves": the nerd and the aesthete!

So, for those of you who made it this far - thank you! and here's a great quote as a reward meant to provide a counterweight to all that precedes it:

There are some things in painting which cannot be explained, and that something is essential. - Pierre August Renoir


Mike said...

Amazing! This stuff really gets into the psyche, (doesn't it?), Kathy!

Fibanacci series, golden mean and all this stuff completely amazes me. I simply cannot fathom how any person, much less an ancient, could come up with this stuff . . . .a mathematical harmony!

I simply MUST sit down for an afternoon with a note pad and read carefully (study) this blog. The content is wonderful!

Kathy said...

Thanks so much, Mike. I love numbers, equations, the whole biz - so it's fun to take it all apart incrementally to examine. Fibonacci is one of my favorites. It took me a few weeks to fully understand that numerical system and apply it, but now I see it as a simple elegant solution.

Sheila said...

I swear Kathy, you need to write your own book... you have a group of followers who will be buyers and promoters already.

Kathy said...

Hi Sheila,
It's so nice of you to support my efforts by suggesting a book, but I won't be writing one in the near future. However, I am teaching workshops all over the country for the next few years and if anyone wants to learn more, just sign up. The schedule and locations are listed on my website. Thanks!

PAMO said...

WOW!! I had no idea. Awesome stuff as always. Thanks Kathy!

hwfarber said...

I have seen this Vermeer example before and always wonder why that globe is not centered on the node.
Thanks for including Renoir's quote--I like counterweights.

Kathy said...

Thanks, PAMO, for reading through all this!

Hi HW - Neither the globe nor the shoulder are centered on the nodes. I don't think they need to be, they just need to be proximal and, in this case, they're balanced. Interesting to note, though.

Tonya Vollertsen said...

Hi Kathy, great article! I'm enjoying this series. I love to read all the theory on the math of harmony in art but I must say I never actually work that way before hand,( I know, that explains a lot in my work, right?) LOL! I do use it to evaluate after the fact though but usually use the simple division of dividing the paper in thirds each direction and consider the points where the lines cross as the "sweet" spots. I guess I do tend toward more simple compositions though. Not sure that's valid. I can't even imagine trying to plan out your eggshell paintings! How do you do those? Do you use photo references or life or both? They are amazing. It only just occurred to me to wonder how they might be planned out.

Kathy said...

Hi Tonya,
Oh ... it takes me forever to plan the eggshell paintings and I'm still not happy with them. I must work on better arrangements. I start with a pile of eggshells that I have in my studio. I've done a lot of sketches and also photographs. I look at the sketches and photos together to make decisions, but I haven't yet applied Renaissance math to this, so my designs are more intuitive. After I get an idea, I do a large sketch, making lots of changes as I go, and transfer it to watercolor paper (usually a full sheet). I select a palette (three to five hues) and decide which areas should be illuminated. Those I block while I apply washes and paint the other areas. Typically, before I finish I photograph the piece, download it onto my computer, import it into PhotoShop Pro and change it to greyscale. There, I can make it smaller and evaluate the distribution of values to see if I'm on track. I also flip it in all different directions to check for balance. Then, I make a print out of the greyscale image and decide on corrections. After that, I finish the painting. It takes a long time to complete one of these, but I don't mind. Thanks for asking!

Margaret Ryall said...

I've read some of the mathematical ideas behind composition but I can't say I took much of it in. I start to glaze over when I'm told there's math involved.After I had a successful career as a student in mathematics I felt I had done my part. It's a good thing we are all different. Thanks for including the Renoir quote. It gives me hope. Orderliness I get.

Jack Foster said...

Great blog you have here Kathy. Very informative and interesting at the same time. (thats hard to do :0)
You ARE an artist for sure. Glad I stumbled upon your stuff! Love it!

Kathy said...

Hi Margaret - I used to be "allergic" to math, but eventually saw the beauty of this universal language. It applies to everything. Glad you liked the quote!

Hi Jack - Thanks so much! Hope you'll join in the conversation in the future.

-Don said...

OK, the math formulas are interesting and very thought provoking, but my focus in this comment is going to be on your painting that you included.

First of all, I wish you wouldn't have put it down as "not a very masterful painting". What does that really mean? To me it's a wonderfully intense piece that tells an emotional story.

I always look at your work before I read about it so I can have my own reaction to it. My first gut reaction was a feeling of anxiety and fear in the midst of chaotic energy. My eyes followed the tape measure to your reflection with the sunken, deer-in-the-headlight eyes which let me know right away that everything was not right with the world. I then allowed my eyes to more slowly drive along that ribbon of numbers leading me thru the medication decanters, the scales, the measuring devices, and the clock. Chemotherapy did not come to mind for me, but a sense of foreboding and uncertainty gripped me. My initial literal response was self-induced, excessive, obsessive dieting and exercise creating an unhealthy situation.

Of course, my eyes noticed the 51's and wondered at their meaning. But, in a self-preserving move that any guy with even half a brain would understand, I will refrain from listing any of my guesses as to the nature of the numbers.

Upon reading the title I then revisited the painting and my reactions to it. I don't think I was very far off from your original symbolic intent. I just misinterpreted the literal story.

I will wrap up this dissertation with a tip of my hat to a survivor who continues to find joy in her chosen profession as she shares of herself and her work in such selfless and thought-provoking ways. Kudos.


Kathy said...

Don - I'm speechless! (rare for me, I know). Your analysis of my painting is wonderful and I feel humbled that you'd even take the time to "read" it so well and write about it. Thank you for your encouragement. It means a great deal to me!

vanidor said...
This comment has been removed by the author.
vanidor said...

Madam Katherine,

Greetings in the Name of St. Thomas,

I have come across your painting at your blog specifically referring to your experience of undergoing chemotherapy... it has captured my imagination.... the painting so vividly captures the feelings of undergoing chemotherapy... By the way, i am Rudolf Martinez, a registered nurse from the Philippines, a graduate of University of Santo Tomas College of Nursing... It has been always my passion to help those children afflicted with cancer... As part of my continuing effort to help them, i have taken up Masters in Nursing and I'm at the moment finishing it... I am currently finishing my thesis tackling about the experiences of undergoing chemotherapy in the eyes of an adolescents hoping that in the end, their views and experiences will further help us nurses understand the thing they are going through and tailor fit the care rendered to them... With these regards, may i ask your permission to allow me to use your painting as a basis for reflection of my co-researchers (the adolescents, my participants) regarding their experiences... i firmly believe that the arts reveals otherwise hidden meanings in the experiences... as i have been having a hard time finding paintings to which they may reflect... and yours is absolutely beautiful and full of symbolism... I promise that i will only use your precious paintings in this educational pursuit and nothing more... i hope this would merit your kind consideration...
God bless you and your family... Happy Holidays!


Rudolf Cymorr Kirby P. Martinez
Complementary & Alternative Therapy
Nurse Practitioner
Pain Management Clinician
(Specialized in Pediatrics)
University of the Philippines Diliman, AKKAP Healing Arts
Asian Social Institute,
Kahingalay Center
School Nurse, Tomas Morato
Elementary School

Deborah Elmquist said...

I too have been reading and studying about armatures and dynamic symmetry. Please check your measurements for pre-made canvases. Only the 1.5 rectangle is a regular canvas size. Root 2, root 3, root 4, and root 5 don't. A great resource is a website for accurate information:
Happy designing.