Randomness

As in my previous post I'll borrow a term from math and science to describe what is entirely art. The term, randomness, is an interesting way to consider asymmetrical tilings like some of the scaling girih tilings. If I'm clear that this is neither math nor science I think I can suggest that aspects of this art are random.

I'll use the example of the diagram, below, with its eight large pentagons each divided by a tessellation of tiles from a common tile set. The large pentagons are identical in their boundary tiles, but their interiors appear randomly ordered. The pentagons can be filled with tiles, no gaps, edge-to-edge, in a number of asymmetrical and random-like patterns. The process of arranging the interior tiles is almost entirely non-random. There are initial random choices in tile selection for each, but the rest of the process of filling in the pentagon is driven by those choices and is a tedious trial and error attempt to find tiles that fit edge-to-edge leaving no gaps.

Putting aside for a moment the fact that the tile patches within each pentagonal boundary were carefully selected, I can say that a little randomness is enough. Following the fixed arrangement of boundaries, tile selection began with an arbitrary choice. The tiles do appear somewhat randomly arranged within. The limit of their scale relative to the boundary suggests that there are few such solutions, though how many is hard to say. So, we have eight examples of a closed system in a fixed state that at least started with some randomness, analogous to a system with very small but not zero entropy.

This highly systematic process appears free form because of the asymmetry. It's mostly not. Trial and error are necessary to complete any one pentagonal tile patch because I choose not to catalog all possible arrangements. The final diagram is thoroughly asymmetrical. In this case there may be as few as eight possible states. I could only know how many possible arrangements might exist by somehow proving that I've diagrammed all of them. I think there must be more than eight, but probably a small and finite number.

Now consider that the scaling tile set I've used could be extended infinitely. Some of the tiles could be replaced by smaller tiles, and still maintain the tessellation with no gaps and no overlaps. The elongated hexagons can be replaced, but not by infinitely scaling patches. The decagons can be replaced by three elongated hexagons and a bow tie. There are other possibilities, as well. Most close the process. However, the decagons can be replaced by rings of scaling trapezoids and closed finally with a small decagon. Also, the decagons can become scaling kites indefinitely, and with the rhombus the center gap can be closed at any point, preventing a singularity gap. As long as the tile set is infinite, and a decagon is included, the number of possible tilings is infinite. When to stop and close the final gap could be randomly determined. This potential for large entropy applies if the initial shape is simply a decagon instead of a pentagon, and the tile set is only decagons with scaling trapezoids and/or kites plus a small rhombus.

Singularities

If I'm careful not to imply that this is mathematics, and clear that I don't use these terms in a mathematical sense, then I can borrow terms such as singularity and cusp to describe an aspect of the drawings that I'm calling scaling girih tilings. In these tilings it's interesting that there may be singularities — areas in which there either seem to be no solutions or an infinitley scaling point. Scaling tilings, or self-similar tile sets lend themselves to creating infinitely expanding, spiraling, and converging designs. But, they also require either trial and error or careful design to avoid dead end solutions.

In most of these drawings I use the scaling tile sets to vary density. That, and overall shape drive the design. As a design transitions from low to high density I can manage the tile selection through trial and error to avoid dead ends, or treat them as singularities. As smaller and smaller tiles are added, tile patches may fold in on themselves. It's possible to design tile sets and tilings that form infinitely converging singularities. [1] These are well behaved in the sense that we can add scaling tiles infinitely. If my diagrams were math then I would select only well behaved solutions, discarding those that can't be continued infinitely. Since this isn't math I'm just as likely to investigate dead ends, limits, or ill behaved singularities. Variety and options are preferred. Not that I'm mimicking nature, but natural examples of fractals are only approximately fractal, and limited when things fold in, come together, and can't overlap. For instance, ice crystals on a window pane grow until they collide.

So the diagrams below are a subset of the scaling girih tilings, and with these I've demonstrated a few singularities. I've found that with a certain tile set that includes a dart scaling tile, placing two decagons forces limiting cusps. Scaling continues everywhere there are pairs of equal sides at 216 degrees, as with the decagon. Two of these on a decagon accept three scaling darts, creating six more pairs of sides for the next scaled darts. But at some point between the original two decagons, I'm faced with a single side, and no possible solution. I can't add scaling tiles indefinitely.

Another type of singularity in these drawings is the type that is well behaved and infinite. As adjacent pairs of three scaling tiles converge around a cusp eventually they close, creating a bounded gap, maintaining the edge-to-edge property of the tiling, and creating new inner and outer edges where scaling tiles can be added infinitely. These are the singularities sought by mathematicians, and I encourage them here because they provide areas of smooth, symmetrical density change.

I also want to break up designs with ill behaved singularities. I can introduce seed tiles (imperfections, like a dopant) that create new singularities, some well behaved, and some not. I can also connect two decagons with other girih tiles, and generate singularities.

What if a scaling tiling can be extended indefinitely even as it generates singularities indefinitely? Is some original choice, such as two decagons connected by scaling tiles, necessarily going to generate both singularities and infinite boundaries? Or, will one or the other, infinitely scaling, or finally singular close the boundary?

Reference
1. Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).

Filling the Plane

The drawing below has a orderly border that could be extended to fill the plane. It requires just three girih tiles at the same scale to continue the pattern up-down and left-right. Yet, this pattern converges in the center to asymmetrically arranged girih tiles (plus three) scaled down, twice. This is the closest I've come to demonstrating that the scaling girih tiles are capable of filling the plane asymmetrically. The only method I know of to create such a drawing is by trial and error. The possibility that any of these patterns might fill the plane defies math — you'd have to complete an infinity of trials and corrections for error to accomplish the task. There's no simple or recursive definition at work.

It's interesting that this system of scaling girih tiles can support a creative system of trial and error. It doesn't depend on math nor symmetry, but allows for arbitrary choice without precluding resolution. This drawing began with ten decagons surrounded by common patterns, like seed crystals, but arranged by whim. The only limitations were the fixed tile set of eight tiles times three scales, and the requirement for the result to be edge-to-edge, no gaps, no overlaps — a tessellation. These ten tile patches were surrounded by two levels of scaling tiles then girih tiles to the grid-like border decagons — the solution.

It's mildly interesting that the three scales of girih tiles in this drawing have sides in the ratio of 1.6180339. . . , the golden ratio. This of course falls out from the fact that we're dealing with pentagons.

I'm more interested in what I can create with a system of scaling tiles than the math, which is fairly simple. Still, it occurs to me that there should be some mathematical interest in such a system that accomplishes a task like filling the plane, but requires endless trial and error. If it's extended to three dimensions is it a model of crystal discontinuity? Are there any natural systems that continue even infinitely through a constant correction like trial and error?

Trial and Error

The diagram below brings me to another reason why this is not math art. In my previous four posts I've described several reasons why math is not emphasized. Now I can offer the deciding reason — that these are drawings are by trial and error, the antithesis of math art. I've designed a system for drawing with tiles, but selecting which tiles and even designing the tile set is largely trial and error.

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.