Painting on the Left Side of the Brain

(Originally published Jan, 2019) 

Everyone refers to Betty Edwards' book, Drawing on the Right side of the Brain.  And I have to admit that it was an important book for me when I began to draw and paint seriously, in fact, so much so that it contributed to my efforts NOT to use the left side of my brain.

But I was a Mathematician first and throughout my painting career I kept being bombarded with left-side ideas or logic. I couldn't escape it even if I tried. But I kept pursuing my need Not to allow logic to affect my painting.

Finally this year I came to grips with the fact that I am driven by logic; I am an analyst first. And in fact I realized that for me, it's important to accept the logical part in order to grow as a painter.  So I determined that I would not discourage any left-brain ideas popping up while I paint.  And I would marry both sides of my brain perhaps to establish my brand as a painter-mathematician (or mathematician-painter).

We all know that mathematics is at the fundamental core of understanding our universe.  This includes not only theorems and proofs and formulas but the underlying concepts of what makes up everything in nature.  At first glance something like fractals or infinity might be considered ideas that arise out of the right side of the brain. However, ultimately, all the important fundamental ideas of the universe can be boiled down to formulas.  Ah, but you argue, what about the unknowable; how do we explain that.  It turns out that science is getting closer and closer to explaining that which cannot be proven - with theorems and proofs impossible for the layman to follow along with formulas.  It has become the dream of every physicist to discover the underlying essence of the universe with the core theorems describing it.

A century ago a woman mathematician, Emma Noerther, discovered breakthrough theories that contributed to or in fact drove the development and discovery of major physical concepts.  The book by my friend, Jack Leibowitz,  Hidden Harmony - the Connected Worlds of Physics and Art, describes these so that the layman can appreciate the idea of "Harmony" in our universe and how the understanding of art contributes to this.
(Note added, 1/16/2020): Another friend, Joel Spruck, Mathematician at Johns Hopkins, recently solved an important theorem, https://hub.jhu.edu/2019/11/07/joel-spruck-mathematics-cartan-hadamard-manifolds-proof-science/ solving Dido's problem. The article is described as "The Proof of Life". (The mathematics is way beyond my knowledge or experience.)
But I have deviated from my original thoughts on this matter.  My Mathematical side wants to understand the fundamental theorems of the universe, just as I studied -and taught - the fundamental theorem of calculus.  Let's not forget also that Mathematics lies at the core of Physics.

But back to my original thoughts about painting with the left side of the brain. What does this mean?  To try to clarify this for myself, I am going to list, stream a consciously, any mathematical ideas as I perceive them that I have come across while studying and doing painting - from the simplest concepts to the more complex.

But before I leave you with this list, please watch this blog for new posts on my thoughts on mathematics, painting and composition, where some of these topics will be discussed..

Karen

shapes - geometry
composition - balance, unequal measures
informal subdivision
golden rectangle and beauty
calipers
golden ratios with the golden mean as the limit
calculus
aerial perspective, single point etc.
volume
2- and 3-dimensional shapes in the landscape
light and volume
3-d to 2-d
ruler, verticals,  horizon line,

fractals and Mandelbrot - in the landscape

chaos theory, infinite series
self-similarities
groups and their graphs
infinite combinatorial group theory
symplectic groups (subject of my PhD thesis)

transformations of the plane
tessellations of the plane
4-color mapping theorem
repeated patterns

tools: rulers, calipers, divisions of a rectangle into interesting geometric shapes

And now to another logical area of painting:

color and color theory:

red, blue and yellow and white
CYMB - cyan, yellow, magenta and black
complements:  yellow/purple, red/green, blue/orange

color wheel, harmonic paintings

limited palette, expanded palette

color charts

cooler/warmer; lighter/darker

value-scale (4 values, 10 values)

formulas, integrals, derivatives: fundamental theorem of calculus and how integrals lead to derivatives and vice versa. Meaning of 2nd and 3rd derivatives and how do this relate to painting and art?: Curves, maxima and minima. Mountain shapes, saddles, parabolas, asymptotes.


I have almost forgotten all the mathematics I knew. I hope that the fundamentals are still with me so that if I needed to, I could construct proofs of theorems for example. Or solve differential equations.

So I look at a mountain range and I see a "saddle". And I ask myself, what is its formula (answer later).  I might see a parabola. Well I know this: y = x squared is an example. hyperbola?  circle (y - pi times radius squared, where pi is approximately 3.141257...., not a rational number), hypotenuse of a triangle (x squared plus y squared = hypotenuse squared)
irrational numbers

golden mean ~ 1.618....... not a rational number Use a calculator to find phi. 

In another post we will show that phi = (1 + sq rt of 5)/2 a remarkable finding or in fact this is how phi is defined. But then its numerical properties turn out to be remarkable.

So phi = 1.618033988749895... on my calculator if use it to calculate sq rt of 5 + 1 all divided by 2..
fibonacci series (with base of 1: 1, 2, 3, 5, 8, 13, 21, 34)  So a series of rectangles with sides the successive numbers of this series might lead to rectangle that are pleasing to the eye:
3x5 or 6x10;
5x8 or 10x16 or 15x24 or 20x32,;
8x13 or 16x26;
13x21 or 26x42., etcetera.

 So I will use these proportions for my paintings (not counting the frame).

Or if I have a 3 inch wide frame perhaps the panel could be 20x36 for example so that the total framed piece is 26x42 (with aspect ratio, 1.615...).

The ratios of the successive fibonacci numbers get closer and closer to the golden mean:
3/2, 5/3, 8/5, 13/8, 21/13, 34/21,...

It's clear from this that we can use any of these ratios to approximate the golden ratio to a degree that we are not able to see any distinction.

So I decided to concentrate initially  on the 5x8 proportion (1.6) , which is the same as 10x16, 15x24 or 20x32.

But I like the idea of using a proportion as close to the golden mean 0.618.... as possible. Well I don't want to work on a panel/canvas as large as 21x34 and I don't want to work in a fractional width or height. And the ratios from 13:21 on all involve fractions or large numbers.

But the 13x21 ratio is very close since 13/21 - 0.61905.  The 8x13 is a little small.  So I start thinking about which ratios are closest to the 1.618 with integral sides that are manageable for me as a painter.

Assume that the largest size canvas I want to use will be something like 36 or 40, which is really not my favorite sizes.   I might try a 21 x 34.
The following seem to be reasonable sizes

a	b	a/b
5	8	0.625	y
7	11	0.636
8	13	0.615	y
10	16	0.625	y
11	18	0.611
13	21	0.619	y
15	24	0.625
16	26	0.615
20	32	0.625
20	33	0.606
21	34	0.618	closest to 0.618
22	36	0.611

So i will order or make panels that are:
standard fibonacci series: 5x8, 8x13, 13x21, 21x34
and
(0.625):  10x16, 15x24, 20x32 approximating the golden mean of 0.618033988......
(0.61538):  16x26, 8x13


Next: proportions involving radicals; e.g., square root of two, three or five........

Watch this blog for more posts on these ideas, especially as they pertain to Composition.