Pentamerism

This is the only non-periodic tiling I've done that is pentaradial. It has five polygons, but the square and rectangle (which has an aspect ratio of 1:1.618033989) can only be used along the five rows extending from the center pentagon. If I attempt to use them anywhere else, a gap is created.

Camouflage

In 2007, the Portland Art Museum exhibited Camouflage, an exhibition of eight paintings that explored artists’ use of pattern. The exhibit included a large camouflage pattern painting from a series that Andy Warhol did around 1986.

In his 1940 essay, A Mathematician's Apology, G. H. Hardy (full text here) discusses, among other things, pure versus applied mathematics with one difference being the application of mathematics to war. He stated that "a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged".

In Arthur Danto's The Transfiguration of the Commonplace: A Philosophy of Art, and 25 years later in his 2008 reply to his critics in CA online, Danto uses Warhol's Brillo box to discuss what makes an object an art work. (See Ontology, Criticism, and the Riddle of Art Versus Non-Art in The Transfiguration of the Commonplace. Contemporary Aesthetics, Volume 6, 2008.)

The Warhol camouflage series is interesting in that it turns a pattern designed for war back into art. Hardy said "A painting may embody [an] 'idea', but the idea is usually commonplace and unimportant." I wonder what he would have thought about the Warhol painting's ability to wrap all these concepts — beauty, seriousness, the ontology of art — in one art work.

Here's an unrelated, gratuitous Bezier curve image from a series:

Here are a few close-up pictures of a painting (a polyptych) just under way. It will resemble the digital print, Slip, but will undoubtedly have a completely different color scheme. I shot these as close as I could to show what I look at all day while I'm laying down these outlines in white on white oil paint. All the curves are paint applied with a small brush along pencil lines. The pencil lines are plotted/interpolated Bezier curves. That is, I plot about ten points per curve, and interpolate with ships curves or other French curves. The plots follow a spreadsheet of grid points that I exported from the original digital print program data.

Erosion

First test drawing of new program about erosion and alluvial fans:

Beziér Curve Drawing

The developer-artist has the ability to collaborate with a client, and enhance understanding of the project in the process. The client can propose a requirements list or request permutations based on variables built into a program. Architects, engineers, and designers work with developers to extend or modify software, achieving through collaboration a client-directed variation. The artist-developer creates a program which reveals more through the interjection of client requirements than an artwork rendered from just the artist's requirements.

This new "Beziér Curve" drawing was created with a Flash Air program written for managing and creating modules and whole drawings within this project. It's meant to explicitly demonstrate modularity open to client requirements. The client can dictate requirements for the arrangement of squares, the selection of curves within, the color and thickness of line, overall scale, or as yet to be determined requirements.

New Sequence

I finally decided to include continued fractions in my project on rectangles and spirals. While doing the research I read that "numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients." My previous rectangles and spirals were all based on irrational solutions of quadratic equations. So, this got me thinking that all I needed to do was start with a periodic continued fraction, and I might find another series of rectangle and spirals similar to the golden rectangle. The continued fractions for my second sequence are:
[1; 3,1,1,3,1,1,3,1,1, …]
[1; 7,1,1,7,1,1,7,1,1, …]
[1; 15,1,1,15,1,1,15,1,1, …]
[1; 31,1,1,31,1,1,31,1,1, …]
etc. . . .

So, I decided to start with a similar sequence. The continued fractions I came up with are:
[1; 2,1,1,2,1,1,2,1,1, …]
[1; 4,1,1,4,1,1,4,1,1, …]
[1; 6,1,1,6,1,1,6,1,1, …]
[1; 8,1,1,8,1,1,8,1,1, …]
etc. . . .

And, the corresponding ratios look like:
b/(a–2*b/3) = a/b
b/(a–2*b/5) = a/b
b/(a–2*b/7) = a/b
b/(a–2*b/9) = a/b

I haven't drawn the spirals, but I think they look a lot like my second sequence.

Hackers and Painters

Hackers and Painters, the essay: http://www.paulgraham.com/hp.html

Hackers and Painters, the book: http://www.paulgraham.com/hackpaint.html

Gratuitous tile set image made with a Flash Air program written by a painter: