Stochastic, Limiting Options

This drawing continues the work I've been doing with stochastic tilings. It's based on eight pentagons arranged like a flattened dodecahedron, but missing four sides. I limited the boundary tiles for each pentagon thinking that the options might be few. I may have reached a happy medium where enough random choices are still available to maintain variety and keep the process sufficiently unpredictable. Here's the tiling that determines the girih-like pattern in the drawing above.

Stochastic

Here's another diagram, below, continuing the asymmetrical girh tile drawings that I described in my last three posts (Asymmetry, More Asymmetry, and Systemic). It illustrates another aspect of this system that I think is in contrast to the order of math art. That is, I'm using a stochastic process, arbitrarily selecting from available options which then determines what must happen next. I design an overall shape with its boundary tiles, but within the boundary (or in this case, six patches) I more or less randomly select some of the inner tiles. To keep the tiling edge-to-edge, I am limited to several choices from the full tile set. Each selection often necessitates that subsequent options are limited or determined, but predicting the outcome is difficult to impossible. The initial choices determine the options available for the rest of the process — filling the tile patch.

This particular drawing, with a central asymmetrical decagon patch and five radiating congruent but otherwise differently patterned hexagonal areas emphasizes the random nature of the process. I started with three scaled sizes of the five girih tiles, a narrow rhombus, and two scaling tiles. As tiles are added it becomes clear that there will always be a solution for filling the patches without gaps, but each selection limits the possibilities. Some choices that preserve the edge-to-edge force gaps that can't be filled, necessitating back tracking. The variety seen in the five hexagonal areas is a visual mapping of the possibilities. The common motifs around the decagons is conversely a clue to the limits of choice.

A Systemic Capability

Here are two more diagrams, below, continuing the asymmetrical girh tile drawings that I described in my last two posts (Asymmetry, and More Asymmetry). I've stated that they defy the math art penchant for symmetry and quantification; and, that these drawings were the result of a search for ways to vary scale and density with tiles. I should also add that the math involved is relatively simple. I've needed little more than Euclidean geometry, basic trigonometry, and an extremely simple application of self-similarity. It's convenient in the programming to use matrix mathematics, but it's not essential. I regularly ignore practices that mathematicians would prefer, like designing tilings to fill the plane and eliminating gaps.

With this project I'm searching for knowledge, not so much refining objects. As the project is underway, I don't spend a lot of time adjusting or working with the minutia. I record the project as a web page, the de facto communication tool for our times. Eventually I'll turn the files over to production, in glass and possibly with computer numerical control. In one case the refinement will be accomplished by another artist, and in the other, a machine.

So, a key aspect of this project is that it's a system that could be reproduced by anyone with the capability to create slightly complex tile sets. I've suggested everything needed to generate endless variety. I happen to have borrowed from Islamic art the girih system. My contribution and evidence that something worthwhile is happening here is that I modified girih tiles to get the scale and density changes that I felt were needed. But we're not limited to girih style decorations. Given edge-to-edge tilings, with defined vertices, we could create other patterning methods. Besides being a search for knowledge, this can be an open-ended path for further investigation.

The two diagrams below, with their underlying tilings, emphasize the projects systemic capability with their asymmetry. The underlying tilings are based on the proposition that we could arrange tiles in endless ways, but still create planned, ordered overall shapes. If the system can do this with asymmetry, then it's flexible and forgiving.

More Asymmetry

The two diagrams below continue the work with asymmetrical girh tile diagrams that I described in my last post. In these two, I returned to symmetry for the overall shapes, but shuffled the tiles within. I developed these with edge-to-edge tilings, having no gaps. It may be possible to fill the plane using patches from the complete diagram. The first, which has an overall decagon shape, can be divided into tile patches of three elongated hexagons and one bow tie, each with girih tile angles. These three patches, with all the tiles that subdivide them, might be used to fill the plane, though I haven't proven that they could. I'm not convinced that the patches could be repeated to fill the plane if I assume that tiles must be edge-to-edge. The second diagram can be divided into a single central pentagon, and five rhombi. Rhombus patches could be added to fill the plane. That in short is the geometry of these two diagrams.

I'm not nearly as interested in the geometry as the density of line and changing scale of motifs. The geometry sets up the diagram, but the self-similar tile sets, and asymmetrical arrangement of tiles gives me the result I'm after. These then are about discovering how to create highly irregular, unpredictable line drawings from a scaling tile set. These drawings defy the math art penchant for symmetry, quantification, and categorization. I prefer the discovery process over object manufacturing.

Asymmetry

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 http://gallery.bridgesmathart.org/exhibiting-artists-2010.) 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.

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.

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.