Sunday, January 30, 2011


Mel Bochner: "Art in our culture never wants to be viewed as a pursuit of knowledge, but as a manufacture of objects. We don't want to deal with any artist as a thinker. That way Malevich's art is converted into social history. But his art continues to make trouble. Look at the negative reviews his recent retrospective at the Guggenheim got. Art has a way of proceeding, like any other theoretical endeavor, by accruing knowledge about itself. Later some artists might understand—within their own contexts—and proceed with that knowledge. That's why it's not dead-end business." ["Mel Bochner on Malevich, An Interview with John Coplans", June 1974, Artforum, reprinted in "Mel Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007", Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4, page 114 ]

Lately, I find it helpful to acknowledge the distance between art and math art. Reading Bochner inspired me to think about art as an attempt to accrue knowledge. I have a couple of new drawings that I hope illustrate the difference between math art and art, and I think they got that way through a little knowledge accretion.

The two drawings below extend the application of scaling girih diagrams into the asymmetric. Up to now, I've worked with diagrams that have usually been at least symmetrical. These two differ radically. They also happen to use none of the original girih tiles. Instead, I used a single rhombus and scaling kite, at five scales. In the first drawing below, the rhombus is not scaled. The underlying tiles are still edge-to-edge, but gaps are everywhere, and of course filling the plane is impossible.

Given that I start with a self-similar tile set, based on girih tiles, these drawings defy mathematical approaches to patterning. I've ignored the rich tradition of ordered, quantifiable, and predictable mathematical preference for symmetry and categorization. I retain only a semblance of the usual practice. I rely on a precisely designed tile set, and I retain a preference for edge-to-edge tile placement. The two tiles include girih-like tile decoration that becomes the drawing, and this decoration is in the spirit of girih tiles. Otherwise, these patterns are not geometrical.

This is not math art. I use a bit of math to design and define tile sets. I'm aware of some of the concepts governing symmetry, self-similarity, and tilings. But I'm grateful that asymmetry and self-similarity confuse the quantification of relationships in these drawings. Using girih strap work as opposed to a zellige style completes the separation of the completed pattern from any underlying math. Obvious symmetry in most of the drawings in this series makes apparent that there is some system at work. These two drawings remove the possibility that their order might be easily deduced.

Math art is about math. Mathematicians and recreational mathematicians create beautiful math informed art. (For example, see But math art is usually about using knowledge, and seldom about accruing knowledge. The two drawings below are the result of a search for knowledge despite math. These drawings use asymmetry, obfuscation, and design. These were the critical directives for this work. The result would have been impossible with just math. Math is a useful tool, but it doesn't explain the art.

I've posted this before. In the article, "Serial Art Systems, Solipsism", Mel Bochner said, "Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)." ["Mel Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007", Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4, page 42]

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

Thursday, January 6, 2011

Scaling Girih, Seventh Scaling Tile

This drawing from a series of scaling girih tilings was created with a new kite scaling tile. It differs from earlier drawings in the series in that the girih lines and the tile edges for the kite share a common vertex. Traditionally, and in all other tiles in the series, girih lines or strapping have endpoints on the midpoint of tile edges or within the tile. In this case, the girih lines share one vertex with the tile edges.

This scaling tile also lacks a girih line ending on either of the two shorter edges. Consequently, this means that all the girih lines at a common scale connect, but are discontinuous with the lines at other scales. I choose to limit these drawings to only edge-to-edge tilings, so this disconnect is complete and only at the transitions.

The scaling factor for this unique kite is the golden ratio, which is also the scaling factor for the trapezoid scaling tile. Both the trapezoid and kite could be used in the same drawing to transition tiles, and the trapezoid would connect girih lines from one scale to the next.

I have used two other scaling tiles that lack girih lines along the shorter edges. There are no such tiles in traditional girih tilings.

Sunday, January 2, 2011

Scaling Girih Tile, Arabesque

Some of these scaling girih tile drawings are crystalline. Some are fractal. All are geometric. This one, below, is biomorphic, and nearly arabesque. It's my attempt to use the strongly geometric system of girih tiles, with the addition of scaling, to create arabesque from straight line. I've added a couple of tiles to the girih tiles, and extended the set with scaling. However, the extra tiles include girih strapping in the spirit of the original girih tiles. Also, I'm able to get the almost arabesque form with the girih lines only, rather than filling in geometrically delineated areas with arabesque.

The design above was generated from the tiling below.