Five Polygons, Asymmetry

In Peter Schjeldahl’s review of the Metropolitan Museum’s new Islamic wing (“Old and New”, The New Yorker, Nov. 7, 2011), he says, “To grasp Islamic aesthetics, Westerners must upend their sense of ornamentation as a minor art.”

The first image below is constructed from five polygons — two shapes at three scales. The symmetrical border encloses asymmetrical, crisscrossing paths of the polygons, aligned edge-to-edge, with gaps. The second image, based on the first, uses a patterning technique originally developed by Islamic artisans. The polygon set is not Islamic, but patterning through the interior decoration of polygons is. The main difference between these images and Islamic patterns, besides the actual polygon shapes, is the use of scaling polygons to introduce variation, especially in density of line. Scaling polygons and the Islamic patterning technique borrowed for the second image below make a geometrically precise patterning system.

These polygons, based on interior angles divisible by π/7, are more difficult to arrange edge-to-edge without gaps than a polygon set based on π/5. An asymmetrical pattern like this required gaps. Nevertheless, the pattern has merit given that seemingly random connections link opposite symmetrical borders. The polygons are all precisely edge-to-edge, with no adjustments necessary. Polygons in the center align through as many as fifty other tiles to both borders. As vectors, each polygon edge has one of three lengths and fourteen possible directions.

7-fold

I have this quote, by Max Bill in his introduction to the catalogue of the Zürich Exhibition in 1947: "the goal of concrete art is to develop objects for mental use, just like people created objects for material use." If anyone has access to the entire document, please let me know. I'd love to read the whole thing.

These large 7-fold or 14-fold patterns with the girih-like polygon set are difficult to generate without gaps.

14-fold Rotational Symmetry, Spiral

This spiral tiling with 14-fold symmetry was built from three equilateral polygons, including a heptagon. There are several ways to fill the center so that the tiling is a complete tessellation. A rhombus and the two polygons other than the heptagon can be used, though the complete tiling becomes 7-fold.

Edge-to-edge heptagons can be followed from the center outward. The tiling might extend infinitely(?)

Girih Process Art

A potentially infinite process controls the development of these images. The process is difficult but not impossible to describe. It could be extremely boring to list all the steps required, and insufficient to help visualize the final image. Just covering a few guidelines, imagine six equilateral polygons including a heptagon and tetradecagon, all constructed with internal angles divisible by π/7. Arrange them edge-to-edge with no gaps and no overlaps. The radial pattern should have seven (or fourteen) fold symmetry, and be extendable forever. In my solution, outer rows of regular heptagons alternate with rows of equilateral hexagons and a rhombus.

As a Design System

Girih Extended, Arabesque, and Seven are design systems similar to girih tiles, a medieval Islamic patterning technique. Like the original girih tiles, they facilitate the generation of complex patterns. Instead of constructing patterns line by line, we can design by tiling polygons. The systems are fast and accurate. The design below was generated in a few hours. The bottom diagram reflects the polygons that are selected from a menu and positioned edge-to-edge with another. The top and middle diagrams are generated automatically from the polygons.

These designs can be scaled from small to architectural applications. The final vector files are suitable for digital prints and processes.

Girih Extended, Arabesque, and Seven designs are reminiscent of Islamic art. The application of scaling, arabesque, and heptagon-based polygons to girih tiles shares much in common with, but extends girih pattern possibilities.

Girih Seven Examples

I’ve published a group of Girih Seven images. The final girih seven polygon set includes a heptagon, tetradecagon, elongated hexagon, rhombus, and bow tie. This follows the original girih tile example. I’ve added a second rhombus, another elongated hexagon, and several scaling tiles, including a trapezoid and kites. My bow tie has eight sides, instead of six, though two trapezoids make one six-sided bow tie. This larger tile set makes it possible to design without gaps, but it’s more difficult than using the pentagon based girih tiles. I haven’t made asymmetric patterns like those that are possible with girih extended. Otherwise, girih seven designs look like girih extended designs, and arabesque. The π/5 girih system and π/7 girih seven system complement each other.

2011 Nobel Prize for Chemistry

The blogger, SIRENDER, sent me this link to the press release on the 2011 Nobel prize for chemistry. It's cool that they mention Islamic mosaics. I don't think they had. It's my understanding that there's really no evidence that Islamic architects and artisans knew the significance of aperiodic tilings. Peter Lu and Paul Steinhardt wrote in 2007 that Darb-i Imam is the only known example of a perfect quasi-crystalline pattern in Islamic art. I wonder if the Alhambra, aperiodic example cited by the press release is a newer discovery. Still, I'm glad the press release from Sweden makes the connection.

Speaking of tilings, here's another girih seven pattern, this time with no gaps in the underlying structure. Since my last post I've added a tile that makes it easier to avoid gaps, though tilings with the new set of heptagon based tiles are still more difficult than pentagonal tilings.

Girih Seven

The images below are my first examples of a girih tile system based on the heptagon. Girih tiles have interior angles that are multiples of π/5. In the examples shown here I've applied the girih concept to tiles with angles that are multiples of π/7. Like girih tiles these have interior decoration (strapwork) that make a pattern without the tile edges. The midpoints of every tile edge have two lines coming from the edge, always at the same angle.

Girih tiles are equilateral, but I extended the tile set to include scaling tiles, allowing me to repeat the tiles at different scales while maintaining an edge-to-edge tiling. This means I can create girih patterns with variation in density of line. Girih seven continues this system. The tile set includes a heptagon, tetradecagon, elongated hexagon, rhombus, and bow tie. I've added a second rhombus, and I'm experimenting with several scaling tiles, including a trapezoid and five-sided kites.

Girih patterns are all straight lines, but starting with girih extended and now with girih seven I've added arabesque versions. Girih patterns are ideally suited for conversion to arabesque using Bézier curves. Since the patterns cross tile boundaries in straight lines, curves are tangent at the boundaries, creating a continuous flow throughout the pattern.

I've found that it's more difficult to create tilings with girih seven. Girih extended made it easy to develop patterns, without gaps. The girih seven example shown below includes a few small gaps, but I think they hardly detract from the finished designs. In at least one case I broke the rule requiring all pattern lines to meet tile edges at the same angle.

The Islamic Decorative Canon, 2 out of 3

Girih Arabesque is another extension of the Islamic patterning system, girih tiles. These designs were developed much like Girih Extended, but this time every line is curvilinear. So now I've addressed 2 of the 3 elements of the Islamic decorative canon -- the geometric and the arabesque.

Girih Arabesque, Described

Girih patterns are composed of straight lines. The girih lines continue uninterrupted across polygon boundaries, and terminate at intersections within the polygons. This means that they are ideally suited for a conversion to arabesque using Bézier curves.

Where arabesque girih lines cross polygon boundaries the curves are tangent, creating a continuous flow of line. Girih boundary points define quadratic Bézier curve end points, and girih interior points become Bézier curve control points.

This technique of converting girih patterns to arabesque could be applied to the basic five girih polygons. I’ve applied it to my extended girih designs. As far as I know, this is the first application of arabesque to girih tilings.

Girih Arabesque

I've added Girih Arabesque to my web site. Girih Arabesque is another extension of the Islamic patterning system, girih tiles. These designs were developed much like Girih Extended, but this time every line is curvilinear. So now I've addressed 2 of the 3 elements of the Islamic decorative canon – the geometric and the arabesque.

Girih Arabesque #2

This is my second example of girih arabesque. The underlying structure is from a finite tile set based on Girih Extended. I've converted the girih strapwork to Bézier curves. As in example #1 I'm using an extension of the standard girih tiles, including some scaling tiles that give me the ability to vary density. There are two versions of the pentagon strapwork in this example, adding to the variety of pattern. This arrangement of girih polygons is without gaps and overlaps. It could be repeated indefinitely.

Girih Arabesque #1

Girih patterns start with simple straight-line decorations on the five girih tiles. I've extended the system with scaling polygons to get variation in density of line. In July and August of 2011 I showed examples of Girih Extended at Chambers @ 916. The image below is the first example of another variation on girih patterns. I've converted the geometric, straight-line patterns to arabesque. This is more a working example than a finished design.

It's possible to apply this same technique to just the five girih polygons. The underlying structure of the example includes four of the five. As in my girih extended series I've added an additional rhombus, plus the scaling polygons: a kite and trapezoid. I've been using up to six scaled copies of the extended polygon set within a single design. The scaling polygons allow me to create edge-to-edge tilings or tile patches with considerable variation in scale and density of line. Sometimes I design with inner gaps, and sometimes not. Sometimes the designs are symmetric, sometimes asymmetric. Since my interest is art, not math, I'm not constrained by a requirement to develop tessellations that might fill the plane. The design above could be filled and extended to fill the plane, without gaps or overlaps.

As far as I know this is the first example of girih arabesque. The decoration or strapwork on the polygons has been converted to Bézier curves. At the polygon boundaries the curves meet tangentially, or nearly so. This gives the appearance of continuously flowing lines. There are numerous possible ways to elaborate the curves. In the example I show the curves were easily programmable, and more or less true to the original girih patterns.

The Theoretical Versus Trial-and-Error

When math and art overlap it seems that the role of and relative complexity of the math involved becomes exaggerated. It's as though the art can't be good enough without a complex theory. Witness the never ending false attribution of powers of the golden ratio to art.

I'm aware of a couple of controversies surrounding Islamic patterns and math. There is the general problem of how big a role mathematicians played in the development of patterns. Also, there's the small controversy over whether or not an Islamic pattern predicted the Penrose quasi-crystal pattern. (See Peter Lu and Paul Steinhardt's answer to Emil Makovicky here. [1])

Previously, I referred to the fact that my Girih Extended drawings are developed largely through trial and error. In A.J. Lee's 1987 paper, "Islamic Star Patterns", he addresses the part played by mathematicians in the development of Islamic patterns, and shows how many of the patterns could be developed simply. So simply, in fact, that little more than trial and error was needed. Lee states:

"It may be an advantage for a modern author to develop a systematic analysis of Islamic patterns in purely mathematical terms, but a knowledge of pure mathematics or geometry is unnecessary for those who wish merely to draw Islamic patterns or invent new ones. A theoretical background will often allow the artist to see a number of combinatorial possibilities more quickly than the use of trial-and-error methods, but it forms no substitute for true creativity." [2]

Trial and error can take you a long way. My Girih Extended patterns are not truly Islamic. They only resemble some of the historical patterns. However, they are new art, not a mathematical analysis, cataloging, or repetition of Islamic patterns.

I should add that there’s no simple or recursive definition at work in the Girih Extended designs. After settling on a general goal, I reach a graphic solution through trial and error. Though digital tools were used and custom programs were developed, the designs are not a programmatic approach to drawing. It is still drawing with polygons.

Reza Sarhangi's paper presented to the Bridges 2004 conference provides the best insight into the possible role of the mathematician in Islamic design. He cites a a treatise written by Buzjani in the 10th century. Sarhangi writes:

"Buzjani wrote in On Those Parts of Geometry Needed by Craftsmen that he participated in meetings between artisans and mathematicians. 'At some sessions, mathematicians gave instructions on certain principles and practices of geometry. At others, they worked on geometric constructions of two- or three- dimensional ornamental patterns or gave advice on the application of geometry to architectural construction.'" [3]

So Islamic mathematicians instructed artisans on basic geometry. Did a 15th century Islamic mathematician understand principles behind quasicrystals? (See "The Tiles of Infinity" by Sebastian Prange. [4]) Or, is it possible that the singular example, the Darb-i Imam shrine in Isfahan, Iran, was created by chance, that is by trial and error?

References:
1. Lu, Peter J., et al. Response to Comment on "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture". Science 318, 1383 (2007). [The full comment is available at www.sciencemag.org. here.]

2. Lee, A.J. 1987. "Islamic Star Patterns. In Muqarnas IV: An Annual on Islamic Art and Architecture". Oleg Grabar (ed.) Leiden: E.J. Brill. [The full document is available from ArchNet.]

3. Sarhangi, Reza. "Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations". Presented at the Bridges 2004 conference. [The full document is available here.]

4. Prange, Sebastian. (September/October 2009). "The Tiles of Infinity". Saudi Aramco World: 24–31. [The full document is available here.]

Art as Catalyst for Change

The following exchange is from "Richard Serra: The Coagula Interview", by Mark Simmons, from Coagula, Issue #36 (1998). (See: http://coagula.com/richard-serra/.)

MS: What would you hope that the people who assist in the production of your work would get from having experienced working on Richard Serra's vision, from idea through fruition? And I'm talking about the people who do the computer thing and the steel workers, and the riggers.

RS: I think this, I think basically I'm not interested in people following my work or making work like my work. But what does interest me is the notion that if you do a lot of work it means there's a potential for other people to understand that a lot of things are possible with a sustained effort and that the broadening of experiences is possible and I think that's all art can be. A little catalyst for change. It's not going to change the world. But it can be a catalyst for thought and thought can change and how people think about what's possible can change and I think that if the work has any value at all on its interpretive level I can't get into how people are going to experience it but if it has any value at all I think it stands for one person understanding that the potential for change is in all of us.

The image below represents my latest attempt to broaden experience, seconding Serra's proposal. I've been generating patterns with polygons and code. Last year, I realized I wanted to add density and scale changes within the patterns. Up to that point my diagrams were like many patterns, overall consistent and therefore a bit boring. Scale and density changes outside of patterning are usually simple because you can control graphite, ink, paint in any way you choose to get the full range of densities possible within a medium. But with polygons as your drawing unit and programming as your tool, it's different. Then too, I imposed a not so arbitrary restriction that the polygons had to be from a finite set, placed edge-to-edge, with no overlaps, with boundaries, but not necessarily gaps. So I encountered the density and scale problem, and found a solution that changed the way I make patterns.

Now, I wouldn't expect anyone to repeat this same process, or get into all the particulars of how to generate girih patterns with scale changes. This is my own solution, and not likely of interest to another artist. On the other hand, patterning is frequently subject to innovation by artists and designers. What might be of interest to other artists is Serra's idea about the potential for change. Rather than simply personalize existing techniques, we should promote change.


Here's the last exchange from the Coagula interview:

MS: One last question. Do you have any advice for sculptors and artists?

RS: Work out of your work. Don't work out of anybody else's work.

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.