Tuesday, December 23, 2008

Piero della Francesca, Symmetry

In several previous blogs (Haeckel, Calder, Jess, Merian, and Baer) I referred to artists with technical backgrounds, and to the connections between math and art. I have also quoted artists and mathematicians who sought to separate math from art: Robert Mangold, — "Abstraction is an idea. Geometry is not."; Mel Bochner, — "Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)."; and the mathematician, 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."

I received a Kindle for my birthday, and the first book I read was Ian Stewart's Why Beauty Is Truth: A History of Symmetry. This is a fascinating history of the idea of symmetry, how central it is to mathematics, and of the mathematicians that worked out what we currently know about the concept. (Much of the math is over my head.) In one short section Stewart explains how artists Filippo Brunelleschi and others formulated, Leonardo da Vinci applied, and Piero della Francesca perfected the mathematics of perspective, and how this fits into the continuous thread of discovery.
"In those days [during the Renaissance] mathematics and art were rather close; not just in architecture but in painting. The Renaissance artist discovered how to apply geometry to perspective. They found geometric rules for drawing images on paper that really looked like three-dimensional objects and scenes. In so doing, they invented a new and extremely beautiful kind of geometry."

Flagellation of Christ by Piero della Francesca
Galleria Nazionale delle Marche, Urbino, Italy

In the 17th century, Girard Desargues used this math to develop a new non-Euclidean, projective geometry. Desargues' contribution was a key step in the path that leads to the mathematics of higher dimensions.

So despite what Mangold, Bochner, and Hardy had to say, there is an important connection between math and art. That relation may have been strongest during the Renaissance, but continues in the math if not the art of today.

Here's one of my paintings from last year. It's one of several I did based on the idea that I could break the rules by combining multiple horizon lines in a single projective plane.

Sunday, December 14, 2008

There is more than one way to do it.

"There is more than one way to do it" is an expression associated with Perl, a programming language developed by Larry Wall.

These two images are both modifications of grids mapped to a parabolic spiral. One line of Flash Actionscript code separates the two.

Here's the completed drawing, after several more modifications, but 95% of the code is the same as the above two:

Wednesday, December 10, 2008

Transfiguration of the Common Eye

Transfiguration of the Commonplace, by Arthur Danto, was published in 1981. As Danto explained in 2007, it is:
". . . a contribution to the ontology of art in which two necessary conditions emerge as essential to a real definition of the art work: that an artwork must (a) have meaning and (b) must embody its meaning."
The meaning embodied by this image of eyes in the pattern of Fermat's spiral escapes me. So, though not art, I offer it as design potential. It's utility would need to be determined before I knew if the colors are wrong, or if there are too many/few eyes, etc. Without meaning, and without utility it's neither art nor design.

There are 232 eyes, the spiral grid corresponds to some natural disc phyllotaxis, and each eye is content for a single cell in the coordinate based grid.Therefore, at least it's an example of content presented in a non-Cartesian, coordinate based grid.

Tuesday, December 2, 2008

Developing the Grid

Grids in art of the last century were mostly the Cartesian coordinate system kind, with orthogonal axes and rectangular cells. They were also usually module-based, having cells with common corner vertices, rather than coordinate-based. [1]. They didn't admit the possibility of overlapping and gapping of cells. Grids of many kinds – logarithmic, curvilinear, polar, geodesic, and unstructured grids like those used in surface modeling – are possible, but seldom used in visual arts.

A recent example of a grid-based image with a cubic function vertical axis.

Digital art is a recent exception. When artists have access to computer aided design systems they can use unstructured grids without having to calculate the data structure. Geometric primitives make up surfaces in the earliest stage of the graphics pipeline. Thanks to CAD, unstructured grids are commonly employed in architectural design, industrial design, and digital imaging. However, the fact that there is an underlying grid structure is usually lost in the final stages of rendering.

Victor Vasarely and Bridget Riley painted grids with quadrilateral cells. A few artists have been influenced by CAD imagery to create complex grid-like paintings without using a computer. Michael Knutson of Portland, Oregon, has painted grids that resemble polygon meshes or unstructured grids. They're based on the quasi regular rhombic tiling, but are mapped to freeform spirals that he lays out by hand. (See his paintings from 2007-2008 at the Blackfish Gallery in Portland, and this interview in Geoform.) The surfaces in the paintings of Vasarely, Riley, and Knutson undulate but never overlap or allow for gapping. They represent a development of the grid from flat surface to an optical bas-relief.

Simple programming, nothing as complex as a CAD system, can be used to generate grids other than unstructured and Cartesian grids. It's possible to create new designs using grids with nonorthogonal axes, and when coordinate-based, designs can include overlapping or gapping. Spirals can be used to map regularly spaced coordinates for grids resembling disc phyllotaxis. Coordinate or module-based grids can be generated from polynomial functions – sine waves, or cubic functions for example. This means that the grid no longer resists development. It's also neither flattened, nor antinatural. [2]

Grid scale also offers the possibility for development. Artists have just begun to use GPS systems. GPS systems use trilateration (not triangulation) to determine positions. If you think of the position of a receiver as a coordinate, the resolution or error in the system means there is a natural cell or area for each numerical position. Depending on the type of system, the accuracy may be from 10 meters to as little as a few centimeters. Appropriately larger cells of any size and shape could be calculated. The key concept though is that practical grids can now cover huge areas, such as an entire city. Alyssa Wright effectively used the entire city of Boston to map coordinates for "Cherry Blossoms". The coordinates make either an unstructured grid with significant coordinates as vertices, or a grid with all possible coordinates locating cells sized by the resolution of the system. For grid-like structures at the microscopic scale, look at diatoms and the drawings of Ernst Haeckel.

Another obvious area for development of grid-based art is in cell content. Until recently, grid cells were often modular, without gapping or overlapping, and filled with color (Ellsworth Kelly), texture (Nevelson), images (Warhol), or space (some Sol Lewitt structures). The natural world, from the atomic scale and up suggests that coordinate-based systems can position a variety of phenomena in grids. Disc phyllotaxis and social networks are two examples.

The more we consider algorithms, scale, or content (biological, social, political...) as means to develop the grid, the more we discover and invent underlying structures, the more possibilities the grid should offer artists.

Here are some graphic examples of new grids:

1. Jack H. Williamson. “The Grid: History, Use, and Meaning.” Design Issues, Vol. 3, No. 2. (Autumn, 1986), pp. 15-30.
2. Rosalind Krauss . "Grids." October, (Summer 1979), pp: 50-64. Reprinted in The Originality of the Avant-Garde and Other Modernist Myths. Cambridge, Massachusetts: MIT Press, 1985.

Saturday, November 29, 2008

Photography, Science and Jazz

In several previous blogs (Haeckel, Calder, Jess, Merian, and Baer) I referred to the occasional overlapping of art and technology.

The recent Scientific American magazine has an article on the 2008 BioScapes Photo Competition: story and photos. Entrants were allowed to use computers to enhance the images, turning scientific study into computer photo art. If you're still not sufficiently impressed, see the Nikon Small World Photomicrography competition.

For a completely different take on revealing the art in science, see the December 1 issue of The New Yorker magazine, which has an article on the collaboration between a musician and a biologist: Swing Science.

My latest series of digital prints is about grids other than the Cartesian. So far, I've covered curvilinear, parabolic spiral, sinusoidal, and cubic functions. Here's the last print, a grid following a cubic function in one axis:

Friday, November 21, 2008

Cubic Function Grid

Here's my first example of a cubic function grid. The vertical axis of the grid is a polynomial of degree 3.

Wednesday, November 19, 2008

A Sinusoidal Grid

Sinusoidal Grids: It's possible to create coordinate-based grids based on functions. In the example below I'm using a sine wave. I can simulate the module-based grid, with common corner vertices, or use coordinates to position cells without common vertices, overlapping and gapping the cells.

This image was inspired by columnar basalt.

Thursday, November 13, 2008

Plant Senescence, Sample, Remix

Here's a new print, "Plant Senescence, Pixelated Sample, Vector Remix":

Monday, October 20, 2008

Pierre de Fermat

Malcolm Gladwell writes in the October 20, 2008, New Yorker, that in our accounting of creativity we have forgotten to make sense of late bloomers. We expect poets, artists, and mathematicians to do their best work before middle age. We tend to accept the conventional wisdom that age is the enemy of creativity.

Mathematician Pierre de Fermat (1601–1665) had completed a manuscript for his pioneering work in analytic geometry by the time he was 35, but he also helped lay the groundwork for probability theory when he was 53. Fermat's example contradicts what G. H. Hardy said in his essay, A Mathematician's Apology (full text here):
"No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." G. H. Hardy (1877 – 1947)
Hardy may have been right that original mathematics is the most difficult discipline to continue into middle age and beyond, but Gladwell says something else is going on. Gladwell's article covers the work of University of Chicago economist, David Galenson, who showed that there are two different life cycles of artistic creativity — the conceptual and the experimental. Gladwell writes:
"The Cézannes of the world bloom late not as a result of some defect in character, or distraction, or lack of ambition, but because the kind of creativity that proceeds through trial and error necessarily takes a long time to come to fruition."
Fermat was a lawyer first, and though devoted to mathematics, he never felt the need to publish. His work survives through correspondence and notes he made, rather than finished writings. It seems to me that he chose to be the experimentalist, and wasn't distracted by his own early success. Like Cézanne, Fermat was more interested in the process of discovery, and less distracted by his own success.

I usually end a post with a gratuitous link to some new artwork of mine. In this case, I'm connecting Fermat's spiral, which he discussed in 1636, to the first drawing for a new series — Fermat's Spiral. (More to come. . .)

Sunday, October 19, 2008

Ernst Haeckel

In previous blogs (More. . ., and Artist-Scientist Maria Sibylla Merian. . .) I referred to the artist/naturalist. Here's a link to 100 images by Ernst Haeckel, whose work survives alongside the prints of Maria Sibylla Merian. Here's a link to an exhibit of natural history books by ten authors, including Merian. Here's a link to the work of Richard Hertwig, a scholar of Ernst Haeckel.

Here's just one image of radiolarians from Haeckel's Kunstformen der Natur (Artforms of nature), followed by my own interpretation, "Radiolarian Skeleton."

Friday, October 17, 2008

Alexander Calder

Continuing my search for artists with science or math training, while reading about the Whitney Museum of America Art exhibit "Alexander Calder: The Paris Years, 1926-1933", I discovered that Calder trained as an engineer. Previously, I identified Jess Collins, Jo Baer, Portland artists Julian Voss-Andreae and Stephan Soihl, and the artist and entomologist, Maria Sibylla Merian (1647-1717). Calder went to the Stevens Institute of Technology in New Jersey, and worked for a short time as a hydraulics engineer and a draughtsman for the New York Edison Company.

Surely the other end of the art spectrum from an exhibit at the Whitney would be a gallery show that accepts all entrants and charges a fee. For the time being, I'll have to be content with the later — Snap to Grid 2008, at the Los Angeles Center for Digital Art includes one of my "Canopy" prints.

Wednesday, October 15, 2008


The digital art at joebartholomew.com is grouped by project. Many of these projects share a common purpose — to extend a graphics system in some new way and thereby provide a tool for creating new imagery. The Rectangles and Spirals project extends features of the golden rectangle to other rectangles by generalizing the golden ratio formula and the Fibonacci number sequence. The Small Programs and Plane Symmetry Groups projects relate to plane symmetry or wallpaper groups, but differ in ways that disqualify them from, while extending the mathematical form.

The Coprimes and Greatest Common Divisor (GCD) projects plot the GCD of integers as colored cells within a grid. Patterns are generated by extending the integers provided to the GCD calculation beyond the grid column and row numbers. These numbers figure in a pair of formulas to generate different GCDs.

I based the Perspectives project on a simple two-point perspective drawing function. Instead of using the tool to create visually accurate perspective drawings I mix several viewpoints within the same drawings.

The Cubes and Cabtaxi Fleet project is based on the recreational math idea of taxicab and cabtaxi numbers. It suggests possibilities for representing the numbers in sculpture.

Here's my latest Small Programs drawing, "Apse". The apse, a frequently vaulted recess at the sanctuary end, may extend the exterior of a church.

Monday, October 13, 2008

Four Similar

Here are four new images from my Small Programs series. They were developed with mostly common code. As usual, I generate the unit cell with a small program. The program randomly varies the resulting pattern. I like to consider this technique an extension of plane symmetry or wallpaper groups in that instead of translating, rotating, or reflecting the primitive cell, the small program regenerates a varying image each cell. This gives me control over the total image. For example, I did a series of three entitled "Lessness", I, II and III, with clumped images in the center, in a grid, and along diagonal rows. In the first image below the program generates sparse cells toward the center.

Friday, October 10, 2008

Lessness III

Here's the final drawing of three entitled "Lessness", I, II and III. "Lessness" is a short story by Samuel Beckett in which he randomly ordered the sentences. The Lessness drawings are from my series, Small Programs, which relate to plane symmetry or wallpaper groups except I generate the unit cell with a small program. The program randomly varies the resulting pattern.

Saturday, October 4, 2008

More Art and Science

Last June we went to see the Getty Center exhibition, "Maria Sibylla Merian & Daughters: Women of Art and Science". Merian was an artist and entomologist, with a special talent for natural history illustration and printmaking. I have an interest in artists with scientific or technical backgrounds such as Jess and Jo Baer, or artists that understand how to use technology as well as Larry Bell does. Last night I saw Susan Murrell's exhibit at galleryHomeland, in Portland. Murrell recreates the effect of scientific illustration and museum display without the actual science. She fabricates displays and specimen boxes of fantastic faux creatures. She paints multi-layered abstracts on transparent acetate that hint at purposeful geological renderings and maps. The whole presentation is tied together by call out label leader lines, installation style with tape applied directly to the wall. It's similar to installations and sculpture that reference without being informed by architecture. Eric Zimmerman is one of many young artists taking architecture and doing installations and drawings similar to what Murrell does with biology or geology. (See www.ezimmerman.org and Art Lies, Issue 58.)

This forces me to think about my own pursuit of art grounded in math. In my work I use programming, simple math, elementary algebra, trigonometry, and Euclidean geometry. Yet, the closest I've come to actually practicing math was with my project on a generalization of the golden ratio formula. So, what's the point in mimicking, paralleling, or otherwise borrowing from math, science, and architecture to create art? I've noticed that many artists want to separate themselves from the technical. They blatantly use technical imagery while distancing themselves from the technology. Both Robert Mangold and Mel Bochner de-emphasized the math in their art. Susan Murrell takes this much further. Joel Courtney said of Murrell in the 6/20/2008 Las Cruces Bulletin, "Even though science is a large influence, Murrell goes out of her way to avoid studying the sciences further so as not to cloud over the messages of her artwork with the particulars of the technical world." Murrell was quoted, "My ignorance (of natural science) is part of the equation in a way. The abstraction does dumb it down a bit, but it allows my work to explain it to others."

In contrast, I prefer the technical illustration of Merian to the parody. This inspires me to renew my efforts toward taking a stand opposite to mimicry and parody. Merian used artistic talent to advance science. Can we use math or computer science to advance art without trivializing the technical aspect? Mel Bochner said,
"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." 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.]
So Bochner seems to say that art should be precise and valid like mathematical thinking, but I think there's much more opportunity here than Bochner's statement implies. I think twentieth century art opened up possibilities for artists to use materials, skills, and techniques from all professions. I hate to see artists stuck within the limitations of pre-twentieth century art simply mimicking other professions when they could employ the techniques and knowledge of those professions to make a new art.

I'm sure there are other artists with a technical background and a portfolio of work that advances rather than parodies science in art. In Portland we have Julian Voss-Andreae who studied physics at the universities of Berlin and Edinburgh and did graduate research in quantum physics. Also, there's Stephan Soihl (B.A. in Physics) of Mt. Hood Community College and Blackfish Gallery. Coincidentally, Soihl also does botanically accurate watercolors.

Here's a second version of a drawing from my Small Programs project.

Friday, October 3, 2008

Magnetic Field

Here's another drawing from my Small Programs project. This image depends heavily on the program's ability to concentrate images toward the center of the grid.

Wednesday, October 1, 2008


An object or a system is called chiral if it differs from its mirror image, that is it cannot be superimposed on its mirror image.

The Small Programs are digital drawings that I program instead of using traditional artist's media. They grew out of my Plane Symmetry Groups project. These drawings relate to plane symmetry or wallpaper groups, but they differ in ways that disqualify them from, while extending the mathematical form. The primitive cell in each of these patterns is generated by a small program. Like plane symmetry groups, I can transform, rotate, or reflect the cell. However, the drawings differ from the mathematical form in that I generate the unit pattern programmatically. I have replaced the static cell with a small program which may randomly vary the resulting unit pattern. The patterns are isosceles trapezoids arranged radially in tracks around a center point. The patterns grow in width as the tracks radiate out.

I explore small programs in somewhat the same way I would use traditional media. I have a general idea of the result I am attempting to achieve before I start coding. I work through various challenges during the process. I selectively discard or keep elements as I approach the final rendering.

The programs are Flash ActionScript. I often use random selection within the program, so the result varies each time an image is rendered.

Here are two new closely related prints in the Small Programs series.

"Prebiotic Earth"

"Mass Ascension"

Sunday, September 28, 2008

Devil's Claw & The Far Shore

Here are two new digital prints, continuing a theme of images inspired by the natural world. First is "Devil's Claw":

"The Far Shore":

Sunday, September 21, 2008

Hollerith Acacias

A punch card or Hollerith card is a piece of stiff paper that contains digital information represented by the presence or absence of holes in predefined positions. When I began programming, I used punch cards to record each line of code.

Friday, September 19, 2008

Almost Independence Day

"I can hear Them calling, 'way from Oregon" – Van Morrison

Wednesday, September 17, 2008

Ice Core

This image from my Plane Symmetry Groups series reminded me of ice cores.

Wednesday, September 10, 2008

Tuesday, September 9, 2008


Here's a detail of my new painting layout. This is how we used to do technical illustrations — with pencils, compasses, dividers, and French curves:

Sunday, August 31, 2008

Coronal Mass Ejections

Solar storms like one that occurred in 1859 could disrupt communications worldwide.

Here's a drawing entitled "Coronal Mass Ejection" from my Plane Symmetry Connects series. (It's only an interpretation, not a scientifically accurate diagram.) :

Saturday, August 23, 2008

Standing Waves

The title, "Standing Waves", is not meant to imply that this drawing actually shows a standing wave. It was created with Flash Actionscript and Bézier curves. Also see Schumann resonances, and A Short History of Standing Waves: Part 1.

Sunday, August 17, 2008


On 8/11/08, the Cassini spacecraft flew by Saturn's moon Enceladus, passing within 50 kilometers. Because Enceladus and Cassini were moving relatively fast with respect to each, a remarkable technique was developed to get photographs of the surface. Cassini was first positioned ahead of Enceladus. The spacecraft with camera was then spun as fast as possible in the direction of Enceladus' predicted path. As Enceladus overtook Cassini, the spinning craft matched Enceladus' motion across the sky, snapping photos as it flew by. The new "close-ups" pinpoint where icy jets erupt from the moon.

The surface of Enceladus is marked with craters, fissures, and corrugated terrain. Images like the one below, from a prior flyby, inspired me to attempt my own version of a Enceladus moonscape, following.

Saturday, August 9, 2008

Larry Bell's Studio

I was in Taos a couple of weeks ago so I went by Larry Bell's studio at 233 Ranchitos Road. An assistant was kind enough to give me a quick tour. I was thrilled to see a large number of glass boxes under construction, and the various equipment they use to create his signature pieces. I got a peek at the large vacuum chamber, purchased from the Los Alamos National Laboratory, that Bell uses to coat the glass plates by thin film deposition.

The assistant also showed me prints and explained how Bell is creating what looked like iridescent areas in the prints — colors that reflect light like oil on water. Of course, her explanation went in one ear and out the other. God, I wish I could remember and comprehend how she explained he does it. They were like no other prints I've ever seen.

Though I don't think Bell had training in math or science he seems to effortlessly master whatever skill he needs to accomplish his goal. I have previously focused on artists who had prior training in math or science. Bell is different. I don't think he studied or practiced in a technical field prior to becoming an artist. Yet, he masters and uses science or technology in his work.

Monday, July 21, 2008


I just finished reading Flatland, by Edwin A. Abbott. First published in 1884, this story of a two-dimensional world provides a subtle introduction to the concept of worlds with four or more dimensions. Beings in Flatland are lines, polygons, and spheres. I don't recall reading anything about the natural Flatland world. If Flatland could have Bézier curves the landscape might look like this —

Saturday, July 19, 2008

Observer Effect

An image created with a Flash ActionScript program is different each time it's created. It almost demonstrates an observer effect. Not really, but that's the idea.

Monday, July 14, 2008

Jeffry Mitchell talks with Denise Patry Leidy

Last Sunday at the Portland Art Museum, Denise Patry Leidy, Curator of Chinese Art at the Metropolitan Museum of Art, spoke on perfection and form in Asian art. As she explained, it's a huge topic covering a vast part of art history both in time and space.

After her talk, she and Jeffry Mitchell presented an equally interesting one-on-one presentation of Mitchell's three contributions to the museum's Contemporary Northwest Art Awards exhibit. Mitchell and Leidy complement each other because Mitchell has an abiding interest in Asian art. He often draws on images and concepts with strong Asian influences. I still have difficulty with some of Mitchell's work. I disparaged it in a previous blog. I don't connect with what I read as his storybook imagery. However, two of the three Mitchell pieces in this exhibit interest me.

Of the three pieces Mitchell has in the show, "Sphinx" has been described by critics almost to the exclusion of the other two pieces, "From a Muddy Pond: Like an Elephant in a (Plexi) Box" and the black-on-black painting, "Black Star". "Sphinx" shares some of the same imagery that I criticized when Mitchell showed at Pulliam Deffenbaugh. I prefer "From a Muddy Pond" and the black-on-black painting — they're atypical for Mitchell. There's an aspect of "From a Muddy Pond" that I can relate to. Mitchell created a sort of wallpaper group from lithograph prints on transparent paper. Because the paper is transparent, he can achieve the reflection of a plane symmetry or wallpaper group by flipping the image over.

In the black-on-black painting Mitchell uses a highly geometric process to develop a remarkable image. In the center of the painting is a pentagon. Toward the edges are five circles. Concentric pentagonal shapes radiate from the center and morph into concentric circles radiating from the five outer shapes. This is all accomplished with black drawn on black oil paint. The entire painting is meant to be the image his "Sphinx" (self-portrait) contemplates from across the room, and therefore seems to me to be the real focus of his exhibit, not the "Sphinx".

Monday, July 7, 2008

Math Ideas, Art Ideas

I've posted several quotes (and here) on the relationship of art to math. I found a good one in the excellent monograph, Robert Mangold. Mangold ended his "Statement for a Panel on Abstract Art, 'The Geometric Tradition in America Art 1930-1990', Whitney Museum of American Art, New York, 1993" with the following:
"Whatever role geometry plays in my work I see as incidental. I have used circles, squares, ellipses, and all manner of four- and many-sided forms and combine forms. I see no difference between this and the way a writer or poet would use words and made-up words to express an idea: the key is to express an idea.

Abstraction is an idea. Geometry is not." [Robert Mangold by Richard Shiff, Robert Starr, Arthur C. Danto, and Nancy Princenthal. New York, Phaidon Press Inc., ISBN 0 7148 4448 9. page 164]
There's disagreement among artists about whether art or math is based on ideas. Compare Mangold's statement with this one from Mel Bochner's:
"Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)." [Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007, Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4, page 39-43]

Once again, here's my favorite quote from mathematician 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)
Below, I include one of the latest prints from my Plane Symmetry series. Disclosure: to make this print I used programming, simple math, elementary algebra, the number PI, trigonometry, Bezier curves, and Euclidean geometry.

Sunday, June 29, 2008

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.

Monday, June 23, 2008

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.

Friday, June 20, 2008

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.

Sunday, June 15, 2008

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.

Sunday, June 8, 2008

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.

Saturday, June 7, 2008

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:

Wednesday, June 4, 2008

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.

Tuesday, June 3, 2008

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.

Sunday, June 1, 2008

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.