A tetracontakaidigon (42-gon)

Here's my first attempt at an arc-shaped tesselation of triangular designs. I'm exploring the possibility of plane symmetry groups composed of triangles instead of the usual rectangular pattern. This could carry over into space groups since a mosaic of triangles can approximate a 3-dimensional curved shape. I am inspired by physicists like Renate Loll, who connect triangles to simulate multi-dimensional universes.

Ridges of California at 33,000 feet

My series of Plane Symmetry Groups was inspired by the view from flights between Portland and Los Angeles. Thanks to Google Earth, I can revisit the scenery.

Plane Symmetry Group, Cityscape

The Hills Above Los Angeles. While trying for a drawing inspired by the California hills outside of San Francisco, I created this image that looks more like a cityscape.

Artist-Scientist Maria Sibylla Merian at the Getty Center

In Los Angeles see the Getty Center exhibition of Maria Sibylla Merian (1647-1717). Merian was an artist and entomologist, with a special interest in the metamorphosis of butterflies.

Here's a vaguely botanical plane symmetry group of mine, with an animated version. Click on the image to enlarge it.

Jess and Julian Voss-Andreae

I've been emphasizing artists like Jess who had prior training in math or science. There are two new exhibits in New York that relate. "Jess, Paintings and Paste-Ups" is at the Tibor de Nagy. (In Portland, you can still catch Jess: To and From the Printed Page at Reed College's Cooley Art Gallery.) Also see this listing in the NY Times.

The second exhibit pertaining to my theme on artists with training in math or science, is To Infinity and Beyond: Mathematics in Contemporary Art, at the Heckscher Museum of Art on Long Island. I expected to find a few artists in the show who trained and worked in something related to math. Of the 33 or so artists, as far as I can tell, there is one with some kind of rigorous training — Julian Voss-Andreae. Most if not all of the others are artists with a peripheral interest in math. Voss-Andreae, now of Portland, Oregon, studied physics at the universities of Berlin and Edinburgh. He did graduate research in quantum physics. He then went on to earn a BFA in sculpture from the Pacific Northwest College of Art. Unlike most of the other artists in the Heckscher exhibit, his work isn't based on a superficial understanding of math or science. As elegant as many of the other artists works are, they are generally based on trivial or simple math concepts. Voss-Andreae on the other hand has actually developed something new from a scientific source.

Dynamic plane symmetry groups

I have two running themes. One is to identify artists who were trained in some other field, particularly math or science. The other theme has been a gratuitous display of my arc shaped digital prints.

My artists list now includes: the chemist Jess Collins, the biologist Jo Baer, the mechanical engineer Alexander Calder, the inventor Hans Hofmann, and the lawyer and professor Wassily Kandinsky.

My image below, from my latest project, represents a dynamic, arc shaped plane symmetry (wallpaper) group. The arc shape for the grid was arbitrarily chosen. It has no particular significance though it slightly complicates the math. It's a pleasing shape. The cells are arranged like discs in sectors and tracks around a center point, with axes dividing sectors and concentric circles dividing tracks. The animations of images are done by one of two transformations. When an image moves to an adjacent sector it amounts to a mathematical reflection about an axis. When a image moves to an adjacent track it represents an oblique or scaling reflection.

The cell images are designed to be rotated to one of four orientations. An animation may contain multiple related images which are rotated in quarter-turns to three other orientations. The image selection is random, but the initial orientation is calculated using the greatest common divisor method described above.

Plane symmetry groups show up naturally, in mathematics, in architecture, and art. The recent Portland Art Museum presentation, Every Picture Tells a Story: Persian Narrative Painting, includes several fine examples.

Gratuitous still image from the animated GCD (dynamic plane symmetry group) project:

Jo Baer's "Mach Bands"

In an earlier blog on Jess Collins, I remarked that Jess began his career in chemistry. I have a self-serving interest in identifying artists with science or math training. I recently found the web version of Aspen, a multimedia magazine of the arts published from 1965 to 1971. In issue 8 from 1970-71, is an article by Jo Baer, "Mach Bands". (Also see Ernst Mach and Mach Bands.) Baer's writing on color is much more technical than say Walter Darby Bannard's "Color, Paint and Present-Day Painting", and I think more factual than Interaction of Color, by Josef Albers. Checking Baer's background, I see that she majored in biology at the University of Washington. She also did graduate work in physiological psychology at the New School for Social Research in New York. That puts her at the top of my list of artists with rigorous training, along with Jess.

I've no way to tie the content of this particular blog to one of my images, so here's just a couple of gratuitous images from my on-going animation series.

Abstractions and Tangibilities

A few years before Mel Bochner linked the best art with the clarity and rigor of mathematical thinking, he had something else to say about math and art. In his article, "Serial Art Systems:Solipsism", from Arts Magazine, Summer 1967, [reprinted in Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007, Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4, page 39-43], Bochner wrote about a Sol LeWitt structure, in which a square within a larger cube (ratio 1:9) becomes a small cube within the larger cube, then a box the height of the larger cube:

"There are no mathematics involved in operations such as these. Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)."

Contrast that last quote with this one from G. H. Hardy:

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." G. H. Hardy (1877 – 1947)

This quote is from Hardy's essay, A Mathematician's Apology. The full text of that essay is here.

Here's my latest digital tangibility and animation.

Not the Antithesis of Artistic Thinking

I just completed a new animated grid, and I'm thinking of the following quote from Mel Bochner — in his “ICA Lecture”, 1971. [Reprinted in Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007, Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4. Page 90-92.] he wrote:

"Any sort of information can be diverted by a set of externally maintained constants. Concentration on these constants, rather than on the information itself, results in the surfacing of a structure. This means a shift in focus from an object to the order.

When order is focused upon it, it reveals that there are no inherent necessities that define what form a work of art should take. This mode of thinking immediately links us to certain areas of mathematical thought. Mathematical thinking is generally considered the antithesis of artistic thinking, but it is not. The two aspects of mathematical thinking that interest me are its clarity and rigor. These are also the characteristics of the best art."

The new grid is from a program that should free me from some of the tedious reprogramming I've been doing to create these animations. I've distilled the process down to a set of easily changed points, lines, and curves so I can try more variations, quickly.