Radical Impressionist: A Mathematician Paints

Santa Fe River Turbulence, with Dynamic Symmetry Armature, 10x16.  (golden ratio) Notes: (in progress: return again from time to time to see additions or modifications) 9/15/25: Edited a bit.  For example, the Wise Photo App Dynamic Symmetry Grid now works properly with a 2 to 3 proportion image (AR 1.5) since SLR cameras use this proportion. 5/15/23:    Added "Golden Rectangle" to the title to emphasize that this post is about the golden PHI proportion. Added an image with the Rule of Thirds superimposed over a PHI grid for the viewer to see the difference visually. Removed comment about other proportions since they are not represented in this post. Removed the Wise Photo App version of the Dynamic Symmetry grid applied to this golden proportioned painting since it distorted the size in its downloaded version.  In the App, it's best to use the PHI/Golden Triangle/Spiral 'grids' for golden rectangle proportions and the Dynamic Symmetry Grid for 3 to 4 or 2 to 3 ...

 Scroll down for construction steps for the basic dynamic symmetry. In ascending order of aspect ratio: Construction Example: Step 1: main diagonals and one reciprocal (perpendicular to the 'baroque' diagonal) Step 2: main diagonals and all 4 reciprocals. Note the red x indicating a possible focal point. Final Step: horizontals and verticals through the intersections of the diagonals and their reciprocals. See also: blog post on the Spiral:  https://karenhalbert.blogspot.com/2023/06/whirling-squares-spiral.html Whirling Squares Spiral: We superimpose the whirling squares' spiral onto the Dynamic Symmetry Armatures for PHI* rectangles, as indicated in the following diagram: *or near-PHI such as 10x16, 15x25, 5x8, 8x13

Construction of the Whirling Squares' Spiral The light grey squares are marked off within this rectangle. They form what are called the whirling squares. Note that in a golden rectangle with aspect ratio 1.618... a smaller golden rectangle is formed with the part left over after constructing the square.  And this continues as we 'whirl' around. https://en.wikipedia.org/wiki/Fibonacci_sequence : In mathematics, the  Fibonacci sequence  is a  sequence  in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as  Fibonacci numbers , commonly denoted  F n   . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Whirling Squares (with 'eye' at lower right) My paintings often have Fibonacci number dime...

Golden Ratio Piedra Lumbre Piedra Lumbre II, 15x24, oil The Piedra Lumbre (shining stone) painting has a proportion that is approximately the golden ratio - or the golden mean, an irrational number, approximately 1.610287...., meaning continuing without end.  As an irrational number it cannot be expressed as a ratio of two integers.   I’ve done many paintings, with 10x16 dimension and a few at 15x24, and this 15x24, still with a ratio of 10/16 or 5/8. Note that these are two number in the Fibonacci sequence.  1,1,2,3,5,8, .... The radio of two consecutive numbers gets closer and closer to the golden mean as we progress through the sequence.  Try calculating:  5/3, 8/5/ 13/8, 21/13,... The golden mean is related to the golden spiral, a form found in nature everywhere since it in fact represents how growth takes form (think sunflowers, ferns, tree branches, Nautilus Shell). The ratio is thought to be the most pleasing to the eye, perhaps because it is so perv...

The canyons of Ghost Ranch and Chama River are an hour’s drive from my Santa Fe home and I try to go there to paint as often as possible. This was the home of Georgia O’Keeffe. Sometimes I try to channel her style but more often I tend to want to use more color and also to use a Palette Knife (or knives).  I like the feel of working with a knife and you can judge for yourself but the results are often better for me. So, I have done a series of canyon color paintings recently primarily using palette knife. This is of the first large ones (18x24). I actually took some liberty with a nearby ‘canyon’, Diablo Canyon, just up the road from me. Its rock formation consists of tones of grey rather than the colors of the Ghost Ranch area. But I enhanced the color to reflect the early light of dawn to arrive at this: Diablo Canyon, Morning Glow, 18x24, oil. This sold at the Marigold Arts Gallery in Santa Fe. I then continued with he theme, but this time in my chosen venue of Cham...