Joe Bartholomew

Girih Polyhedra with Bow Tie Hexagon

2012-01-21T07:39:00.000-08:00

These girih polyhedra are heptakaidecahedrons, but they include bow tie concave hexagons where my first version had elongated hexagons, and the second version had rhombi.

Heptakaidecahedron with Girih Pattern

Heptakaidecahedron

Heptakaidecahedron Wireframe Girih Pattern

Girih Polyhedra with Rhombus

2012-01-19T12:17:00.000-08:00

These girih polyhedra are heptakaidecahedrons, but they include rhombi where my first version had elongated hexagons.

Heptakaidecahedron with Girih Pattern

Heptakaidecahedron

Heptakaidecahedron Wireframe Girih Pattern

Girih Polyhedra Pattern

2012-01-18T16:39:00.000-08:00

This pattern is from twenty heptakaidecahedrons decorated with girih tile strapwork, and aligned side-by-side. The heptakaidecahedron is a seventeen sided, geometric solid with two girih tile pentagonal faces, five girih tile elongated hexagon faces, and ten isosceles triangles. In this pattern, the three dimensional solids are in five rows of four each heptakaidecahedrons with each alternating row rotated by 180 degrees, and viewed in perspective.

These are the same heptakaidecahedrons at a different scale, and rotating.

Girih Polyhedra

2012-01-14T17:29:00.000-08:00

These images are stills from an animation of rotating girih polyhedra. The two polyhedra shown here are a dodecahedron and a heptakaidecahedron. Both of these include faces that are from the girih tile set. The heptakaidecahedron is a seventeen sided geometric solid with two girih tile pentagonal faces, five girih tile elongated hexagon faces, and ten isosceles triangles. The triangles are not original girih tiles. The dodecahedron is a platonic solid, with twelve sides of regular pentagons. The girih tile faces can be decorated with girih lines to create a three dimensional girih pattern.

In one important sense applying girih patterns to polyhedra defeats the purpose of the girih pattern system. That is, girih tiles are designed so that the interior decoration lines continue across boundaries, tangentially. When the lines cross boundaries that are not in the same plane, the continuity is interrupted.

Dodecahedron with Girih Pattern

Dodecahedron

Dodecahedron Wireframe Girih Pattern

Heptakaidecahedron with Girih Pattern

Heptakaidecahedron

Heptakaidecahedron Wireframe Girih Pattern

Five Polygons, Asymmetry

2011-11-07T13:55:00.000-08:00

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

2011-10-23T16:39:00.000-07:00

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

2011-10-16T16:18:00.000-07:00

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

2011-10-15T16:12:00.000-07:00

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

2011-10-14T11:36:00.000-07:00

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

2011-10-11T16:22:00.000-07:00

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

2011-10-05T09:56:00.000-07:00

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

2011-10-03T07:17:00.000-07:00

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

2011-09-13T19:22:00.000-07:00

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

2011-09-13T06:08:00.000-07:00

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

2011-09-12T17:14:00.000-07:00

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

2011-08-15T11:44:00.000-07:00

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

2011-08-09T15:47:00.000-07:00

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

2011-07-24T16:47:00.000-07:00

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

2011-04-11T14:37:00.000-07:00

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

2011-02-24T17:08:00.000-08:00

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

2011-02-18T06:57:00.000-08:00

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

2011-02-13T19:00:00.000-08:00

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

2011-02-12T16:20:00.000-08:00

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

2011-02-08T07:22:00.000-08:00

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

2011-02-06T18:57:00.000-08:00

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

2011-02-04T13:53:00.000-08:00

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

2011-02-02T14:26:00.000-08:00

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

2011-01-30T16:10:00.000-08:00

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

2011-01-06T07:12:00.000-08:00

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

2011-01-02T16:34:00.000-08:00

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.

Design for Muqarnas

2010-12-11T12:12:00.000-08:00

Here's a scaling girih tile design in the shape of muqarnas. First is the design with tile edges, then just the girih tile strapwork.

Girih tiling based on a decagonal fractal

2010-12-09T15:15:00.000-08:00

My last entry described a girih tiling based on a pentagonal fractal. It's also possible to base a fractal tiling on decagons. This process is simply nesting or subdividing a decagon with five smaller decagons. Starting with a regular decagon, place five smaller regular decagons, edge-to-edge, within the original decagon. One vertex each of the smaller decagons should coincide with a vertex of the larger. Two sides each of the smaller decagons should be edge-to-edge with another small decagon. The lengths of the edges of the smaller decagon are 1/(2x(1+cosine(72))) of the larger. This is the same as calculating the sides of the five smaller decagons by dividing the larger decagon sides by the golden ratio, twice. That is, divide the length of a side of the original decagon by approximately 1.618, then divide the result again by approximately 1.618. The result is a measure for the side of a smaller decagon to nest and repeat inside a larger. The process of subdividing or nesting decagons with five smaller decagons may be repeated infinitely. Following are diagrams of nesting decagons, and a fractal border.

Here's a completed girih tiling.

Girih tiling based on a pentagonal fractal

2010-12-03T16:52:00.000-08:00

These girih tile drawing are loosely based on a fractal, a variant of the Koch snowflake, but pentagonal. The inside edges of the outer border of kite-shaped polygons follow a fractal. The original pentagon edges are broken three times to generate the final image. Pentagons may be used in place of the kites.

Each successive transformation of an edge into four edges was done by creating two new edges that are 72 degrees to the prior edge, and the lengths of the four new edges are 1/(2x(1+cosine(72))) of the prior edge.

Starting with a pentagon and using an angle of 72 degrees allows us to fill the interior with girih tiles. In the examples below I used a couple of scaling girih tiles.

Here is the girih line drawing I generated from the tiling. Yes, it has gaps.

Here are two more images following the same process.

More Scaling Girih Tiles

2010-11-28T13:38:00.000-08:00

My scaling girih tiles project is inspired by a remarkable Islamic art patterning system that originated by the year 1200, and was rediscovered by Peter J. Lu [1] in 2005. Islamic artisans used girih [2], from the Persian word for “knot”, to develop intricate patterns from just five tiles decorated with lines. The lines rather than the tile edges become the pattern. I’ve extended the system for design purposes to allow for scale and density variations. This innovation would have been inappropriate before computers. The remarkable complexity of design in Islamic girih tiles doesn’t suggest the need for extension by multiple levels of self-similarity. Nevertheless, by expanding the system I hope to create patterns with the look and spirit of one facet of Islamic art while providing new design possibilities. The key is the simple addition of scaling tiles and a self-similar tile set.

By taking the basic girih tile set and extending it to include scaling tiles I’ve added a level of complexity that’s easily managed with digital media. With up to 32 tiles in my tile set, this would have been impractical and unnecessary for Islamic artisans. I decorate these extra tiles with lines (girih strap work) like the five girih tiles, but unlike the girih tiles the scaling tiles are not equilateral. These tiles also differ in that they may lack girih lines on some sides, but otherwise I preserve the angles of girih tiles. I generate multiple sets of the girih tiles, scaled to match the two different sides of each scaling tile. With the complete set I can create fractal-like drawings in which tile patches at different scales are similar.

Since this is art, not math nor historical architecture, I’m free to add tiles to the origianl girih tile set. Sometimes I add a second rhombus that is not in the basic set of five girih tiles. My designs are not all always tessellations – gaps and boundaries are OK. I choose to work with tilings that are always edge-to-edge, meaning adjacent tiles always share full sides. I often create symmetrical designs, but infinitely changing asymmetrical patterns are also possible. I’m searching for interesting designs whether or not they fill the plane. I’m not attempting to stay true to the historical methods.

It’s interesting though not surprising that in some of the tile sets in this project the long and short sides of the scaling tile are in the ratio of 1.61803…, the golden ratio.

References

1. Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". http://peterlu.org/content/decagonal-and-quasicrystalline-tilings-medieval-islamic-architecture.

2. Wikipedia contributors, "Girih tiles," Wikipedia, The Free Encyclopedia,http://en.wikipedia.org/wiki/Girih_tiles(accessed November 11, 2010).

Asymmetrical Scaling Girih Tiling

2010-11-22T05:54:00.000-08:00

Below are three renderings of an asymmetrical scaling girih tiling. This tiling demonstrates that the scaling girih tiles might be extended to fill the plane, and that the scaling could continue as well. In this case I have intentionally ended the boundary with pentagons and decagons at the same scale to indicate the degree of control possible. Depending on requirements, a scaling girih tiling could be made to create controlled patterns, fill a variety of forms, or control a range of line-shape densities.

Scaling Girih Tilings

2010-11-11T17:25:00.000-08:00

This series of drawings is based on the discovery by Peter J. Lu [1] of girih tiles [2], a set of decorated tiles used by Islamic architects for centuries. I’ve taken the basic girih tile set and extended it to include scaling tiles. I decorate these extra tiles with lines (girih strapwork) like the five girih tiles, but unlike the girih tiles, the scaling tiles are not equilateral. The scaling tiles lack girih lines on two sides, but otherwise preserve the angles of girih tiles.

The scaling tile sides match the sides of two sets of girih tiles. I use multiple scaled sets of the tiles to generate fractal-like drawings in which tile patches at different scales are similar.

Since this is art, not math nor archeology, I’m free to add tiles, ignore symmetry, and disregard normal restrictions on tessellations. I’m searching for interesting designs whether or not they fill the plane. I’m not attempting to stay true to the historical methods.

References
1. Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". http://peterlu.org/content/decagonal-and-quasicrystalline-tilings-medieval-islamic-architecture.
2. Wikipedia contributors, "Girih tiles," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Girih_tiles (accessed November 11, 2010).

Pentagons and Isosceles Trapezoids

2010-09-09T06:12:00.000-07:00

This dot pattern diagram from the series, Self-similar Boundaries, is based on pentagons and isosceles trapezoids. The trapezoids are the scaling polygons. They share non-parallel sides with a pentagon and parallel sides with scaled pentagons. The trapezoids generate one pentagon that scales down and one that scales up. The initial pentagon generates one trapezoid, with two scaled pentagons. Each subsequent iteration could double the number of pentagons, except that many are duplicates. In this example I follow three scaling iterations which might have produced 1 + 2 + 4 + 8 = 15 different pentagons, but since some are equal the pentagons produced are 10.

Pentagons and Scalene Triangles

2010-09-06T18:30:00.000-07:00

This dot pattern diagram from the series, Self-similar Boundaries, is based on pentagons and scaling scalene triangles. Neither the design of the tile set nor the arrangement of tiles makes any attempt to create symmetry or even a tiling that fills the plane. Nevertheless, the tile set and tiling are interesting in that they can be used to generate tendrils that extend indefinitely. In this case the tile set scales down seven times.

Scaling Triangles

2010-09-02T07:14:00.000-07:00

These two new dot patterns from the series, Self-similar Boundaries, are based on different tiles sets. They have in common triangles for scaling tiles. In the first example below the tile set is three rhombi and an isosceles triangle scaling tile. The second image tile set is a pentagon and two mirror image scalene triangle scaling tiles. I've included the tilings for the dot patterns.

Affine Transformations

2010-08-20T07:04:00.001-07:00

This entry is about a series of drawings from tilings. It has a few simple math terms, but it's about art, not math. These drawings are dot pattern diagrams I call Self-similar Boundaries.

These dot patterns, structured with tilings dependent on affine transformations, reveal characteristic grids and alignments. An affine transformation is in this case a rotation or scaling followed by a translation. I'm applying the term to the placement of individual tiles, as opposed to patches of tiles. Each dot pattern hints at the design of the tile set because the patterns reveal all possible affine transformations from each tile, with all possible scaling, rotations, and translations predetermined by the tile set. The tile sets are simple polygons. The tilings or tile patches may or may not be symmetrical. They usually have gaps but no overlaps, are always edge-to-edge, and never fill the plane. Each possible tiling from a tile set is a collection of scaling, rotation, and translation of tiles in the set. There is an ordered selection of tile from the tile set, and an affine transformation of each tile to a position edge-to-edge with another tile.

The tile sets always include one or two scaling tiles with a side that matches the length of a polygon in the tile set and a shorter side that determines the affine transformation scaling. The scaling tile may be the only tile in the tile set, or it may be one of several tiles.

These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem. Scaling tiles provide the mechanism.

The three boundary diagrams below are from the same tile set. They represent regular and irregular versions of tilings and dot patterns. The dot patterns reveal the fact that each tile set determines a limited number of possible affine transformations for each tile added. There are usually many possible ways to add a new tile, and the dots represent the vertices of all possible additions. But the design of the tile set limits the pattern of vertices to visible grids and alignments. These are just three examples of the dot patterns I generate from the tilings. You can view the tilings as well here.

Irregular Boundary Diagrams

2010-08-15T06:55:00.000-07:00

This boundary diagram is based on a pentagon and three triangles. Two of the triangles are scalene, and the tile structure underneath is irregular, with gaps. The dot pattern transitions from regular to highly irregular. Nevertheless, from the regular bottom of the image to the irregular top the pattern retains some common horizontal and angular alignments.

This is one of several boundary diagrams that I've done combining tiles that would easily be regular tessellations with a transition triangle or two. Previously, I have designed self-similar scaling tile sets that suggest a regular if not gap-less tiling. In this case, the scalene triangles make the transition of pentagons and isosceles triangles from one scale to the next.

Here's a similar example, but with three rhombi and an isosceles triangle making the transition.

Boundary Diagrams

2010-07-28T10:34:00.000-07:00

I've created a series of Self-Similar Boundary diagrams. I'm using original self-similar tile sets as the structure for these dot pattern diagrams. I begin the process of creating each diagram by designing a self-similar tile set. Then I create a tiling that emphasizes the transition from the largest tile to a potentially infinite boundary. Finally I generate vertices and extrinsic vertices for the tiling. The vertices or dot pattern recedes from open to closed at the boundary. I've included images of the dot pattern and the tiling structure for the vertices.

Examples of Self-Similar Tile Sets (Part 3)

2010-06-22T07:15:00.000-07:00

These are examples of tilings from a variety of polygons. My purpose for making these is art, not math. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem. See Part 1 and Part 2 for detailed descriptions of two of these examples.

The tile set I describe in Part 1 is closely related to, and can be derived from tilings described by Robert W. Fathauer. Fathauer found two families of self-similar tilings based on segments of regular polygons [See the references below]. He went on to discover a great variety of fractal tilings that, as tilings, are more interesting than most of those I describe here. Unlike Fathauer’s discoveries the tilings below are either trivial, or they require more than one prototile. My method for developing a tiling is not necessarily straight forward or fixed. My purpose is not to generate interesting fractal diagrams, but to develop often asymmetrical structures for vertex pattern diagrams.

I've imposed a few restrictions on these tilings. That is, they should be edge-to-edge, with no overlaps. Tilings may not fill the plane, but should be infinitely scalable at the boundaries, as in a fractal. In some examples there might be gaps, but these gaps might be bounded by infinitely scaleable tiles forming singularities. These are actually tile patches, not tessellations. In most of these examples the tile set could be extended by adding other polygons, especially triangles.

1: A square and a triangle. The triangle is the s=4 case described by Fathauer, and I have retained the square for which the triangle is a segment. In most of these examples I'm showing one possible tiling that might extend indefinitely in a similar way.

2: A pentagon and a triangle. The triangle is the s=10 case described by Fathauer, and I have retained the pentagon that remains after removing five s=10 tiles from a regular decagon. This is the tile set I describe in Part 1.

3: A regular hexagon and two triangles. The isosceles triangle subdivides the equilateral triangle, which subdivides the hexagon. Therefore, a similar tiling could be accomplished with just an isosceles triangle.

4: Two trapezoids and two triangles. Other tilings could be made with just the two trapezoids or one trapezoid and a triangle.

5: A dart and a triangle. In this example the triangle is necessary to continue the scaling when the darts fold in on themselves. Darts alone without the triangles would overlap.

6: Another dart. Unlike the previous example, no triangle is necessary. When the darts fold in on themselves the gap can be bounded by infinitely scalable darts.

7: Another dart and triangle. A right isosceles triangle or square is necessary to fill a gap when the darts fold in on themselves.

8. A hexagon. This example is trivial, but the hexagon prototile can be combined with triangles that subdivide the hexagon to create interesting asymmetries. The hexagon fits inside a regular pentagon. This is the tile set I describe in Part 2.

9: A pentagon and two triangles. The pentagon fits inside a regular hexagon.

10: Golden Rectangles. The lengths of the two longest sides of each L-shaped hexagon are in the golden ratio.

References

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).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Self-similarity in Architecture

2010-06-10T20:36:00.000-07:00

Most tessellations, especially in architecture, limit the available diversity within the chosen system. Part-to-whole relationships are singular when they could be scalable. Self-similarity solves the problem by allowing a potentially infinite range of parts within one whole. It's possible to design chaotic part-to-whole relationships that are flexible but impractical. Self-similarity restores the practicality while maintaining acceptable flexibility.

Other self-similar tile sets can be based on regular polygons including squares or hexagons. Here’s an animation using three scaled sets of three prototiles: http://joebartholomew.com/aniVertices_HSS9.html

Self-Similar Vertices, Part 2

2010-06-10T19:46:00.000-07:00

This is the second of two methods that I use to create dot patterns with self-similar tile sets. My interest in tilings is relative to art, not math. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem.

The tile set described in Part 1 is closely related to, and can be derived from tilings described by Robert W. Fathauer. Fathauer found two families of self-similar tilings based on segments of regular polygons (see Reference below). One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. The tile set I describe here in Part 2 is also self-similar, but doesn’t include Fathauer’s prototiles.

The method I describe here is based on an infinitely self-similar tile set consisting of a partially concave hexagon and two isosceles triangle prototiles. The hexagon is congruent to a regular pentagon except one side of the pentagon has been made concave with two smaller sides forming interior angles of 72, 252, and 72 degrees in addition to three 108 degree interior angles. The triangles are 72-72-36 and 36-36-108 isosceles triangles.

The initial hexagon and triangle prototiles are sized so the two long sides of the 72-72-36 triangle are equal to the long sides of the hexagon. The single long side of the 36-36-108 triangle is also equal to the long sides of the hexagon. The 36-36-108 triangle fits exactly in the concave area of the hexagon, and together the 36-36-108 triangle and the hexagon form a regular pentagon. The tile set in Part 1 was infinitely scalable pentagon-triangle pairs by setting the next pentagon side equal to the short side of the previous triangle. The tile set describe here is infinitely scalable by setting the next hexagon long sides equal to a short side of the previous pentagon.

Tilings should be edge-to-edge, with no overlaps. Gaps are always fillable. Singularities are not necessary. Tilings may not fill the plane, but could be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations. A radially symmetrical tiling of just hexagons looks like the following figure.

The tilings are a structure for the finished diagram of vertices and what I call extrinsic vertices. This demonstrates the advantage of using a tiling with properties of self-similarity. The changing density of tiles translates to a greater variation in vertex density.

It's possible to create numerous symmetrical tilings with these tiles, but I often choose to create asymmetrical diagrams. The processes, lattices, and patterns I use are not math. I'm influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. These diagrams have no practical use or purpose other than art.

Reference

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).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Self-Similar Vertices

2010-05-30T13:59:00.000-07:00

I've been using the process described here to create tilings from a pentagon and triangle. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem.
My interest in tilings is more art than math. The tile set I’m using is closely related to, and can be derived from tilings described by Robert W. Fathauer (see Reference below). Fathauer found two families of self-similar tilings based on segments of regular polygons. One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. Other hexagonal or square prototiles can be combined with triangles to make self-similar tile sets.

So, this method is based on an infinitely self-similar tile set consisting of pentagons and 18-18-144 triangle prototiles. Tilings should be edge-to-edge, with no overlaps. Gaps are inevitable, but they should allow lining with infinitely scaled tiles. Tilings will not fill the plane but should be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations.

The initial pentagon and triangle prototiles are sized so the long side of the 18-18-144 triangle is equal to the pentagon side. Each subsequent pentagon-triangle pair is scaled so that the next pentagon side is equal to the short side of the previous triangle. Using this scheme, as the boundaries of the tiling grow outwards they form singularities, or gaps surrounded by tiles. The inside edges of these gaps can be continuously and infinitely lined with scaling pentagons and 18-18-144 triangles, or just 18-18-144 triangles.

An interesting feature of these prototiles is that the ratio of the areas of each pentagon to the next smaller pentagon (or triangle to triangle) is always 3.618... or 2 plus Phi.

It's possible to create numerous symmetrical tilings with these tiles, but I often choose to create asymmetrical diagrams. The processes, lattices, and patterns I use are not math. I'm influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. These diagrams have no practical use or purpose other than art.

Reference:

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).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Extrinsic Vertices from mixed n-gon tilings.

2010-03-04T15:36:00.000-08:00

Plots of the extrinsic vertices (my term) for tilings from mixes of 4-, 5-, and 6-gon regular polygons with 3-gon tiles are generally more interesting than those with just triangles and one of the 4-, 5-, or 6-sided regular polygons. Less variegated plots come from tile sets that include only polygons with angles that are multiples of 18 degrees (pentagons and triangles that subdivide them), or multiples of 30 degrees (hexagons and their triangles), or multiples of 45 degrees (rectangles and their triangles).

These images show that it is not the complexity of the tiling, but the tile set itself that determines the patterning of the vertices and extrinsic vertices. The first image is from an animation. The second image is the tiling that is the underlying structure for the first and last images. The last image of the extrinsic vertices reveals its regularity. The tile set includes a pentagon and two triangles whose angles are multiples of 36 degrees. The tiling is highly irregular, including gaps that can't be filled with the tile set. It is an edge-to-edge tiling though.

Compare these images to those from my last post where the tile sets are generally mixed 4-, 5-, and 6-gon regular polygons with 3-gon tiles.

Extrinsic Vertices Animations

2010-02-19T17:29:00.000-08:00

Here are the final frames of a few new animations. These are four of the most complex images from the series, "Extrinsic Vertices". These animations are controlled by tiling structures that preserve in spirit if not in fact preferences for tessellations from small tile sets that fill the plane, edge-to-edge, with no overlaps.

Tiling Generation

2010-02-08T18:04:00.000-08:00

It’s difficult to design a tiling that fills the plane, is edge-to-edge, with no gaps and no overlaps, if you start mixing pentagons with squares and hexagons. This particular tiling is just VERY STRANGE in that it has two seemingly incompatible lattices in connection. So, from my point of view, it’s very jarring – even irritating in that it forces two systems into one, almost. It’s intentionally irksome.

Embodied Cognition

2010-02-02T07:08:00.000-08:00

Compare this idea, that the body takes abstract thoughts literally (see Abstract Thoughts? The Body Takes Them Literally, by Natalie Angier) with this idea — that the integration of the human body into performance is a fundamental problem with electronic music (see "Human Bodies, Computer Music", by Bob Ostertag). In the first case, the field of embodied cognition shows us how the body reacts to abstract ideas. In the second case, the lack of body is seen as limiting to the art form. Perhaps the study of embodied cognition might hold some clues as to why electronic music and computer art in general are cold to our "senses". Though we can grasp the abstract idea of computer art we have no experience connecting the idea of the art to our bodies. We have all seen live performances of music in which the bowing of the violin, the strumming of the guitar, the blowing of a horn, or the beating of a drum were all accompanied by corresponding volume, pitch, intensity and the emotions of the performers. With a recorded performance, we still react in empathy with performers we have seen. The abstract idea of a song is felt with our bodies partly because we have been trained by performance artists to feel what they emote. Correspondingly, if we sense that the body was removed from the performance, with electronics, we react less with our bodies. You get into a hard rock guitar solo as though you were the guitarist, but computer code and electronics tamp down all those emotions. We can't imagine a programmer strutting, thrashing, and banging out code.

Code and Art

2010-01-17T18:47:00.001-08:00

Programming is simultaneously the problem with and the source of digital art's potential. With culture the abundance of digital solutions is directly related to the fact that coding is so intensely creative. Those who can love to code, so they flood the world with every conceivable image, animation, architecture, or other art form built with code. Programming is a thoroughly rewarding creative process. It's unstoppable because it's so completely satisfying to produce. It sustains innovation. The problem for digital artists is that though we can feel the emotion that goes into a song, painting, film, or novel, we can't empathize with the act of programming. Apparently devoid of emotion, the digital visual product is cold. The coded digital image, at home on the Internet, appears out of place in museums and galleries.

Another problem is with the attribution of the art to the artist. Writers, musicians, painters, directors, and architects don't have this problem. Digital art is more difficult. You never know how much credit to give to the hardware, the compiler, the application, the Internet, and the thousands of engineers that contributed to making it possible. Even if you program you're not apt to know quite how much credit to give the artist for an interesting image.

Coding is incomprehensible to those who don't, so a programmed image blocks potential empathy for the real creativity behind the image. Looking at a digital print, we're no more interested in the creativity of the coding than we are in the coding behind our browser app. We're satisfied if our browser works as well as other applications, and we'd be satisfied if a digital image could hold up against all other images, digital or otherwise. Again, even if you program you're not apt to credit an artist for their code.

A solution that has been tried is a programmatically arranged massive aggregate of unitized elements. Fractal art is an example. I’ve tried this approach, repeatedly, and I think it’s insufficient. I’ve also tried the creative use of math. This works no better than code with little or widely known math. I doubt there’s anything less appreciated in the Euclidean arts than Euclidean geometry.

The solution might be to animate. I’m confident that this works. Given the immediacy of motion, images come alive with potential for feeling and emotion. I’m concerned that it’s acceptable because it’s film, and not considered programming.

There may be a solution other than animation, but boy do I not know what it is, yet. Digital art can be cold, but coding is anything but. We should reject cold art, but it's an error to deny the process. New media is begging for a solution to this problem. Writers, musicians, cinematographers, and photographers are adapting to digital technology without appearing to abandon their art form. Visual artists alone are stuck with a preference for the hand made, and a prejudice against the machined. MP3s, 3D CG, and Giclée photo prints are acceptable uses of technology. The programmed digital print is not.

See "Human Bodies, Computer Music", by Bob Ostertag.

Onward through the fog. . .

Robert Motherwell on Math and Abstraction

2010-01-17T14:38:00.000-08:00

Continuing a thread on quotes about math and art, I found these from Robert Motherwell:

"I have often quoted Alfred North Whitehead in what I think is one of the crucial statements on abstraction, that 'the higher the degree of abstraction, the lower the degree of complexity.' In that sense, mathematical formulae are (ironically) by nature of a lower degree of complexity than a painted surface with three lines, even if it's an Einsteinian equation."

"Advanced mathematicians say that when there are two mathematical solutions to the same problem that are equally valid, mathematicians will often reject one of the two solutions as less beautiful than the other. Even in something seemingly as cool and remote as mathematics, there is an element of the aesthetic involved."

Both of these quotes are from a lecture Motherwell gave on 2/7/1970, at St. Paul’s School, Concord, New Hampshire. I picked them up from The Writings of Robert Motherwell, edited by Dore Ashton with Joan Banach, 2007, Berkeley, CA: University of California Press, p. 250 ; from the article, "On the Humanism of Abstraction, the Artist Speaks", 1970, Robert Motherwell at St. Paul's School, exhibition catalogue, and reprinted in Tracks: A Journal of Artist Writings, vol. 1, no. 1, 1974.

I purposely selected these quotes for their reference to math, but the article they come from is not much about math. The quotes are out of context, and I recommend reading the entire selection, if not the book. Nevertheless, I take exception to the idea that a math formula is less complex than a painted surface with three lines. Applied mathematics, being the language we use to describe natural concepts (as in e=mc²), is not necessarily abstract. The formulas of pure mathematics are often as not the language of a larger process or proof, so though abstract are still complex. Sometimes three lines are equally as complex &/or abstract as a formula, as in three lines making a right triangle and the formula, a²+b²=c².

Here's a gratuitous design, Samurai.

Contrapuntal

2010-01-15T07:28:00.000-08:00

Carol Yoon on taxonomy: "Reviving the Lost Art of Naming the World".

Carol Yoon on "Avatar" and the order among living things: "Luminous 3-D Jungle Is a Biologist's Dream".

Inching away from symmetry

2010-01-11T15:01:00.001-08:00

This system I call "Extrinsic Vertices" is great for generating symmetry, but slightly more difficult to manage with asymmetry. This image began as a relatively complex tile patch, with seven tiles. I'm fairly certain the tile patch couldn't be extended to fill the plane, without some significant rearranging. The biomorphic look I'm going for is one with symmetry changing to asymmetry and back.

Bio Blaster

2010-01-11T10:05:00.000-08:00

A mathematicians mind is not necessarily required to appreciate the aesthetics of math. Some math may be made accessible and aesthetically beautiful through geometry and graphics. Fractal art and domain coloring (the graphical representation of functions of a complex variable) come to mind.

I suspect or intuit that all math exists independently of my discovery. At most, I discover a mathematical concept that someone else described, and I use, extend, or elaborate it. There’s a bit of Euclid in every grid, but there’s also a lot of nature.

Number or geometry based biomorphism then brings representation full circle, from a posteriori math through some system to a representation of nature.

Extrinsic Vertices Lattice

2010-01-09T18:35:00.000-08:00

The first image below is one more from my Extrinsic Vertices project. The Extrinsic Vertices diagrams are an effort to create biomorphism through geometry. Sometimes the lattice of vertices is more interesting than the tiling it's based on. This diagram is from a radially symmetrical tiling based on a pentagon, an isosceles triangle, a square, a rectangle, and an isosceles trapezoid. The triangle subdivides a pentagon, but not the pentagon of the tiling. The lattice may reflect the 54, 72, 90, and 108 degree angles of the tiles.

Here's another:

Tessellations, Vertices, Lattices, and Fibrils

2010-01-07T16:11:00.000-08:00

These diagrams are a development of the Extrinsic Vertices project, with an emphasis on radial symmetry and biomorphism. The Extrinsic Vertices diagrams are an effort to create biomorphism through geometry. Natural systems are often unitized and geometrical, so geometrical patterns can be an efficacious means of building biomorphism from simple curvilinear units.

Extrinsic Vertices

2009-12-19T13:27:00.000-08:00

Extrinsic Vertices are lattices and patterns developed from tilings of polygons. I create edge-to-edge tilings or tile patches, and then plot the extrinsic vertices of potential tiles, creating a lattice structure. From these lattices I germinate patterns. The pattern lines extend out from the vertices of tiles and potential tiles.

The processes, lattices, and patterns in this project are not math. I’m influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. The patterns have no practical use or purpose beyond this project.

Extrinsic vertices and potential tiles are terms that describe objects unique to this project. They are elements I designed to create lattices and patterns, but are not recognized elsewhere. I begin with tile sets of two or more regular polygons, rectangles, isosceles triangles, isosceles trapezoids, or rhombuses. The selection and arrangement of tiles is not necessarily predetermined as in a periodic or symmetrical pattern. At any step in building a tiling I can make multiple selections from the tile set, each with potential vertices. The vertices of potential tiles are extrinsic to the tiling. These vertices lie in characteristic dot patterns or lattices depending on the angular properties of the tile set.

I almost always borrow from math a preference for edge-to-edge tilings, simple polygons, and filling the plane with no gaps or overlaps. In contrast to typically symmetrical tilings like those in Islamic architecture, I often opt to create nonperiodic tilings, to select asymmetry. Ultimately I obscure the tilings with overlapping patterns.

These designs begin with tile patches, not in fact tilings or tessellations. They are tile patches – a finite number of tiles from some tiling. All of them could be extended, and most if not all might fill the plane if extended. However, it’s not necessarily clear how they would be extended or what a tiling extending any patch might look like.

My tilings based on a pentagon tend to include polygons with angles that are multiples of 18 degrees. Those based on a hexagon tend to include polygons with angles that are multiples of 30 degrees. Tilings based on a square tend to include polygons with angles that are multiples of 45 degrees. Tile sets with just one of these three angle groups have vertices and extrinsic vertices that lie in characteristic patterns depending on the angles used. Other mixed tile sets have their own characteristic vertex lattices. As tilings become complex with mixed tile sets the lattices reveal new patterns characteristic of the tile set, often repeating rosettes.

For reference, I refer you to the dot patterns described in Tilings and Patterns, by Grunbaum and Shephard (p. 238-246). Some of my simplest lattices correspond to Bravais lattices in 2 dimensions. The more complex lattices are possibly overlain combinations of dot patterns or lattices. I emphasize that the images and techniques in this project are not math.

I advocate using complex technology to create art that mimics natural beauty. If you know the technology and use it assertively, resolutely, and creatively you can produce something new and interesting. If it’s a bit innovative it might also be instructive, at least for other interested artists. Maybe it opens a path with potential. You can work with the natural beauty of math, though no one is likely see it in your work. You can also mimic or parallel nature. I do this kind of art because I think I’m particularly good at it: connecting, organizing, coding, elaborating, extending systems, following narrow paths that haven’t been pursued.

Tiling, Bravais Lattice, Squiggles

2009-10-21T08:41:00.000-07:00

This is the first successful image in a new project that grew out of my tilings project. I created the image in three steps. First, I made a non-periodic tiling. Then I generated a lattice of vertices including all potential vertices around each tile. Then I plotted a shape or lines (in this case, a squiggle) at each lattice point. The plot of vertices is somewhat like a Brazais lattice, though I'm extending this math concept for my own purposes.

Tilings of the plane using tile sets of regular polygons, rectangles, isosceles triangles, trapezoids, and parallelograms, have vertices that lie in characteristic patterns or lattices depending on the properties of the polygons — its angles and sides. The Bravias lattice system categorizes these patterns. The five Bravais lattices for two dimensions approximate the arrangement of vertices of simple periodic tilings. I'm applying this concept to complex, non-periodic tilings with large tile sets.

Storm, Sepia Approach (Hexamerism Lost)

2009-09-24T16:27:00.000-07:00

This is titled "Storm, Sepia Approach". The tile set could have made a hexameric tiling starting with the center hexagon. I forced the off-balance tri-radial asymmetry.

Gratuitous link: I suggest you look at the paintings of Janice Biala.

Pentamerism

2009-09-20T07:24:00.000-07:00

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

2009-09-14T18:27:00.000-07:00

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:

2009-09-02T18:38:00.000-07:00

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

2009-08-09T10:15:00.000-07:00

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

Beziér Curve Drawing

2009-07-08T20:13:00.000-07:00

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

2009-06-27T17:52:00.000-07:00

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

2009-06-05T08:27:00.000-07:00

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:

Radical Non-periodic Tiling

2009-05-28T03:59:00.000-07:00

This tiling is from a set of seven tiles, including the pentagon. It's the latest in a series of tilings.

Animated Radial Tiling

2009-05-04T21:06:00.000-07:00

Here's an initial test version of animated radial dart-rhombus tilings.

Tile Sets

2009-03-28T18:44:00.000-07:00

I have my Air tiling program under control, and I'm using it to generate asymmetrical tilings from tile sets I design myself. The tile sets are all periodic, could be used to generate a variety of symmetrical tilings, but I arrange them in asymmetrical and non-periodic tilings.

Tiling No. 2

2009-03-14T08:51:00.000-07:00

Possibly out of control. From the same tile set as No. 1.

Tiling No. 1

2009-03-13T13:42:00.000-07:00

This is the first successful test of a new tiling program. The program allows me to create custom tile sets, and tilings from the tile set. This is still just a preliminary test of the program, to show that the code generally works, so the colors are not important

This particular tiling is one I tried to draw by hand. The tiling could be extended to fill the plane. It's non-periodic, but the tile set itself is not aperiodic. This is a great improvement on my previous tiling program, which generated only radially symmetric, dart-rhombus tilings. Having the ability to do asymmetric tilings is liberating.

This is my first program with AS3, my first Air program, and my first OOP program. I'm excited.

Nanotube Forests and Diatoms

2009-03-04T06:57:00.000-08:00

Compare these images of carbon nanotube arrays, created by John Hart, of the University of Michigan, with the Ernst Haeckel drawings. Last year I covered a Scientific American magazine article on the 2008 BioScapes Photo Competition: story and photos, and the Nikon Small World Photomicrography competition. There are more amazing scanning electron microscope images, including nanotubes, here.

And, in the latest Scientific American on line there's a scanning electron microscope slide show by University of Georgia digital media professor, Michael Oliveri.

For another view of things small see SnowCrystals.com.

The 4th plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting diatoms (Diatomea).

Here's a gratuitous image, possibly tube-shaped like the nanotubes, from my sinusoidal grids project:
References
Bourzac, Katherine (2009). "Growing Nanotube Arrays", MIT Technology Review online, March/April, 2009.
http://www.technologyreview.com/computing/22095/?a=f

Tree of Life Diagrams

2009-02-10T20:04:00.001-08:00

In previous blogs about Ernst Haeckel I pointed to his amazing drawings of animals and sea creatures. He also published the evolutionary tree, below, showing how humans and animals evolved from single-celled creatures.

Haeckel's diagram was published in 1879, forty-two years after Charles Darwin wrote "I think" above his sketch, probably the first diagram of it's kind.

With DNA sequencing we now have this amazing diagram of the evolutionary tree. See the University of Texas source and a New York Times article.

Mac users can download a 3-D version of the diagram by M. J. Sanderson here: http://loco.biosci.arizona.edu/paloverde/paloverde.html

More on the tree of life here: http://tolweb.org/tree/phylogeny.html

References
Zimmer, Carl (2009). "Crunching the Data for the Tree of Life", The New York Times online, Feb. 9, 2009.
http://www.nytimes.com/2009/02/10/science/10tree.html?_r=1&pagewanted=1

Wikipedia contributors, "Kunstformen der Natur" Wikipedia, The Free Encyclopedia, http://commons.wikimedia.org/wiki/Kunstformen_der_Natur (accessed February 11, 2009).

Wikipedia contributors, "Charles Darwin", Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Charles_Darwin&oldid=269994913 (accessed February 11, 2009).

David M. Hillis, Derrick Zwickl, and Robin Gutell, University of Texas. "Download Graphic Images from the Hillis/Bull Lab, Tree of Life Poster and Other Graphic Images", http://www.zo.utexas.edu/faculty/antisense/DownloadfilesToL.html (accessed February 11, 2009).

Alicia Boole Stott

2009-02-08T13:05:00.000-08:00

Alicia Boole Stott, 1860-1940, was born in Cork, Ireland. Although she never studied mathematics she was able to visualize geometric forms in hyperbolic space. She is remembered for finding all three dimensional sections of the four dimensional polytopes and for discovering many of the semi-regular polytopes. She coined the term "polytope" to refer to a convex solid in four dimensions. She built beautiful models of polytopes.

"Perpendicular sections of the 600-Cell. Section number 7." Alicia Boole Stott
The University of Groningen, Netherlands
http://www.math.rug.nl/models/Alicia.html

References

Riddle, Larry (LRiddle@AgnesScott.edu). "Biographies of Women Mathematicians, Alicia Boole Stott". Agnes Scott College, Atlanta, Georgia.
http://www.agnesscott.edu/Lriddle/WOMEN/women.htm.

Photo Credit

Blanco, Irene Polo and van der Zalm, Lotte. "Mathematical models of surfaces, Alicia Boole Stott, 600(P), nr. 7". University of Groningen.
http://www.math.rug.nl/models/

Google Earth Oceans

2009-02-06T17:47:00.000-08:00

Two recent software releases have huge potential for influencing art based on nature and technology. The new release of Google Earth and Adobe's Creative Suite family with Flash CS4 will tempt artists open to the concepts and imagery of science. Google Earth gives us the ability to see the ocean scape, and Flash CS4 includes basic 3D object manipulation capabilities. Web designers and developers have increasingly influenced artists, opening their eyes to all sorts of possibilities. Designers and developers working with technologists are changing the way we access science like never before. The new release of Google Earth reveals huge, amazing geological patterns in the ocean scape. The animation, data visualization, and interactive graphics capabilities of software like but not exclusive to Flash are giving scientists powerful ways to communicate ideas to the rest of us. (Also see Processing, and Rhino.)

In "Art and Nature", Arcy Douglass (writing for PORT – www.portlandart.net) brought us up to date on how artists have used natural processes and math to influence their work. A few artists have been interested in and able to absorb what science and technology have to teach. Now, science becomes more accessible, through the efforts of huge undertakings like Google, and because on a small scale thousands of technical artists and designers are working with scientists and mathematicians to improve our visualization of natural processes. Our exposure doesn't stop with our last science or math class in school. Science journals and graduate level books aren't required to tap into a lot of the amazing work going on.

Sylvia Earle said to Google, “You’ve done a great job with the dirt. But what about the water?” (See http://www.nytimes.com/2009/02/03/science/earth/03oceans.html.) The February issue of Scientific American has an excellent article explaining the origin of continuous undersea "ridges that wind around the globe like seams on a baseball." See "The Origin of the Land Under the Sea", by Peter B. Kelemen.

From Google Earth. The Pacific Ocean floor, off the coast of Maui.

From Google Earth. The Pacific Ocean floor, off the coast of southeast Mexico.

This gratuitous image from my Squared Spiral series has nothing to do with the ocean scape, but it was built in Flash Actionscript, and is influenced by the math concept of tessellation or tiling of the plane.

Squared Spiral

2009-01-30T18:50:00.001-08:00

From a new series. Not quite as interesting as the Ulam spiral.

I've been intending to consolidate a list of Portland, Oregon artists that use some math in their work, and artists with a technical background. These are not geek artists. They are primarily interested in art and may be more or less inspired by math or technology.

I include Michael Knutson (paintings based on quasiregular rhombic tiling), Arcy Douglas (Sierpinski triangle fractals built with hexagonal tiles), Eva Lake (for her interest in the Richter Scale), and Julian Voss-Andreae (sculpture based on molecular structure, etc.). Also Stephan Soihl, and Martha Morgan (for "The Golden Ratio in Tryon Creek State Park"). More to come.

Earthquakes

2009-01-26T19:55:00.000-08:00

A sinusoidal ridge from Sinusoidal Grids. Earthquakes happen. A noticeable rise or fall in the normal depth of coastal waters is nature's tsunami warning. Move away from the shore immediately. (See the U.S. Geological Survey Earthquake Hazards Program).

I've been trying to create a geometric representation of a landscape (mountain ridges) from a long distance, like a Google Earth image. Since I used a sinusoidal motif this reminded me of seismograph prints. So I'm combining a landscape with a seismograph-like motif.

Dart-Rhombus Recede

2009-01-23T16:15:00.000-08:00

This is a 3-fold, dart-rhombus, radial tiling. Each dart is decorated or marked by rhombi which recede from the adjacent rhombus. This divides each dart-rhombus into ever smaller copies. Each adjacent dart-rhombus is a trivially simple fractal.

I generated this image with six rows, but I have also done a 3 row version, both of which can be seen in my Tilings project. A description of the process for generating the basic dart-rhombus structure without the decoration is here.

In these digitial drawings I superimpose an image on each tile. Sometimes the image is programmatically varied from tile to tile. In these cases the tile with its marking is no longer a true tiling in the mathematical sense, though the underlying structure is. The marking or decoration of the tiles, other than by systematic coloring, often obscures the structure. Infinitely many interesting tilings are possible without this obfuscation. I choose this approach because I'm less interested in the math than I am in inventing a quasi natural image.

Pascal Cotte and Mona Lisa at OMSI

2009-01-23T07:05:00.000-08:00

Yesterday I was able to attend a presentation at OMSI (Oregon Museum of Science and Industry) by Pascal Cotte. OMSI is opening an exhibit on Leonardo da Vinci, featuring a whole room devoted to Cotte's 240-megapixel, multi-spectral scans of the Mona Lisa. Cotte not only invented the camera which uses 13 wavelengths from ultraviolet light to infrared; he also supervised the exhaustive analysis of the resulting images.

Through this analysis Cotte is able to reveal hidden details of how da Vinci painted the Mona Lisa, and how she appeared to da Vinci's contemporaries. The exhibit highlights Cotte's discovery of 25 secrets about the Mona Lisa, including evidence that she did in fact have eyebrows.

Figure/Ground in Similar Tilings

2009-01-17T17:32:00.001-08:00

These next two digital prints of tilings share almost the same code. Other than the colors, only a few minor changes were made to the program. In the first, the figure is paramount, and in the second it's the ground.

And both of the above two prints have a lot in common with the following, though the above are radial tilings, and the following is based on Fermat's spiral, not a tiling.

7-Fold Tiling

2009-01-12T20:51:00.000-08:00

This is a 7-fold radial tiling of the plane with a dart-rhombus tile set. It's a test example in my new series of n-fold, dart-rhombus radial tilings. Technically it's not a tiling since the image in each tile is slightly modified, but it is a tiling since it is constructed from congruent dart-rhombus tiles that fill the plane.

Piero della Francesca, Symmetry

2008-12-23T07:01:00.000-08:00

In several previous blogs (Haeckel, Calder, Jess, Merian, and Baer) I referred to artists with technical backgrounds, and to the connections between math and art. I have also quoted artists and mathematicians who sought to separate math from art: Robert Mangold, — "Abstraction is an idea. Geometry is not."; Mel Bochner, — "Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)."; and the 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."

I received a Kindle for my birthday, and the first book I read was Ian Stewart's Why Beauty Is Truth: A History of Symmetry. This is a fascinating history of the idea of symmetry, how central it is to mathematics, and of the mathematicians that worked out what we currently know about the concept. (Much of the math is over my head.) In one short section Stewart explains how artists Filippo Brunelleschi and others formulated, Leonardo da Vinci applied, and Piero della Francesca perfected the mathematics of perspective, and how this fits into the continuous thread of discovery.

"In those days [during the Renaissance] mathematics and art were rather close; not just in architecture but in painting. The Renaissance artist discovered how to apply geometry to perspective. They found geometric rules for drawing images on paper that really looked like three-dimensional objects and scenes. In so doing, they invented a new and extremely beautiful kind of geometry."

Flagellation of Christ by Piero della Francesca
Galleria Nazionale delle Marche, Urbino, Italy

In the 17th century, Girard Desargues used this math to develop a new non-Euclidean, projective geometry. Desargues' contribution was a key step in the path that leads to the mathematics of higher dimensions.

So despite what Mangold, Bochner, and Hardy had to say, there is an important connection between math and art. That relation may have been strongest during the Renaissance, but continues in the math if not the art of today.

Here's one of my paintings from last year. It's one of several I did based on the idea that I could break the rules by combining multiple horizon lines in a single projective plane.

There is more than one way to do it.

2008-12-14T20:22:00.000-08:00

"There is more than one way to do it" is an expression associated with Perl, a programming language developed by Larry Wall.

These two images are both modifications of grids mapped to a parabolic spiral. One line of Flash Actionscript code separates the two.

Here's the completed drawing, after several more modifications, but 95% of the code is the same as the above two:

Transfiguration of the Common Eye

2008-12-10T18:12:00.000-08:00

Transfiguration of the Commonplace, by Arthur Danto, was published in 1981. As Danto explained in 2007, it is:

". . . a contribution to the ontology of art in which two necessary conditions emerge as essential to a real definition of the art work: that an artwork must (a) have meaning and (b) must embody its meaning."

The meaning embodied by this image of eyes in the pattern of Fermat's spiral escapes me. So, though not art, I offer it as design potential. It's utility would need to be determined before I knew if the colors are wrong, or if there are too many/few eyes, etc. Without meaning, and without utility it's neither art nor design.

There are 232 eyes, the spiral grid corresponds to some natural disc phyllotaxis, and each eye is content for a single cell in the coordinate based grid.Therefore, at least it's an example of content presented in a non-Cartesian, coordinate based grid.

Developing the Grid

2008-12-02T19:03:00.000-08:00

Grids in art of the last century were mostly the Cartesian coordinate system kind, with orthogonal axes and rectangular cells. They were also usually module-based, having cells with common corner vertices, rather than coordinate-based. [1]. They didn't admit the possibility of overlapping and gapping of cells. Grids of many kinds – logarithmic, curvilinear, polar, geodesic, and unstructured grids like those used in surface modeling – are possible, but seldom used in visual arts.

A recent example of a grid-based image with a cubic function vertical axis.

Digital art is a recent exception. When artists have access to computer aided design systems they can use unstructured grids without having to calculate the data structure. Geometric primitives make up surfaces in the earliest stage of the graphics pipeline. Thanks to CAD, unstructured grids are commonly employed in architectural design, industrial design, and digital imaging. However, the fact that there is an underlying grid structure is usually lost in the final stages of rendering.

Victor Vasarely and Bridget Riley painted grids with quadrilateral cells. A few artists have been influenced by CAD imagery to create complex grid-like paintings without using a computer. Michael Knutson of Portland, Oregon, has painted grids that resemble polygon meshes or unstructured grids. They're based on the quasi regular rhombic tiling, but are mapped to freeform spirals that he lays out by hand. (See his paintings from 2007-2008 at the Blackfish Gallery in Portland, and this interview in Geoform.) The surfaces in the paintings of Vasarely, Riley, and Knutson undulate but never overlap or allow for gapping. They represent a development of the grid from flat surface to an optical bas-relief.

Simple programming, nothing as complex as a CAD system, can be used to generate grids other than unstructured and Cartesian grids. It's possible to create new designs using grids with nonorthogonal axes, and when coordinate-based, designs can include overlapping or gapping. Spirals can be used to map regularly spaced coordinates for grids resembling disc phyllotaxis. Coordinate or module-based grids can be generated from polynomial functions – sine waves, or cubic functions for example. This means that the grid no longer resists development. It's also neither flattened, nor antinatural. [2]

Grid scale also offers the possibility for development. Artists have just begun to use GPS systems. GPS systems use trilateration (not triangulation) to determine positions. If you think of the position of a receiver as a coordinate, the resolution or error in the system means there is a natural cell or area for each numerical position. Depending on the type of system, the accuracy may be from 10 meters to as little as a few centimeters. Appropriately larger cells of any size and shape could be calculated. The key concept though is that practical grids can now cover huge areas, such as an entire city. Alyssa Wright effectively used the entire city of Boston to map coordinates for "Cherry Blossoms". The coordinates make either an unstructured grid with significant coordinates as vertices, or a grid with all possible coordinates locating cells sized by the resolution of the system. For grid-like structures at the microscopic scale, look at diatoms and the drawings of Ernst Haeckel.

Another obvious area for development of grid-based art is in cell content. Until recently, grid cells were often modular, without gapping or overlapping, and filled with color (Ellsworth Kelly), texture (Nevelson), images (Warhol), or space (some Sol Lewitt structures). The natural world, from the atomic scale and up suggests that coordinate-based systems can position a variety of phenomena in grids. Disc phyllotaxis and social networks are two examples.

The more we consider algorithms, scale, or content (biological, social, political...) as means to develop the grid, the more we discover and invent underlying structures, the more possibilities the grid should offer artists.

Here are some graphic examples of new grids:

Cubic Functions
Sinusoidal (an almost sinusoidal grid in use, here)
Parabolic Spirals
Curvilinear (also here)
Curvilinear Triangles

References:
1. Jack H. Williamson. “The Grid: History, Use, and Meaning.” Design Issues, Vol. 3, No. 2. (Autumn, 1986), pp. 15-30.
2. Rosalind Krauss . "Grids." October, (Summer 1979), pp: 50-64. Reprinted in The Originality of the Avant-Garde and Other Modernist Myths. Cambridge, Massachusetts: MIT Press, 1985.

Photography, Science and Jazz

2008-11-29T17:26:00.000-08:00

In several previous blogs (Haeckel, Calder, Jess, Merian, and Baer) I referred to the occasional overlapping of art and technology.

The recent Scientific American magazine has an article on the 2008 BioScapes Photo Competition: story and photos. Entrants were allowed to use computers to enhance the images, turning scientific study into computer photo art. If you're still not sufficiently impressed, see the Nikon Small World Photomicrography competition.

For a completely different take on revealing the art in science, see the December 1 issue of The New Yorker magazine, which has an article on the collaboration between a musician and a biologist: Swing Science.

My latest series of digital prints is about grids other than the Cartesian. So far, I've covered curvilinear, parabolic spiral, sinusoidal, and cubic functions. Here's the last print, a grid following a cubic function in one axis:

Cubic Function Grid

2008-11-21T21:18:00.000-08:00

Here's my first example of a cubic function grid. The vertical axis of the grid is a polynomial of degree 3.

A Sinusoidal Grid

2008-11-19T18:25:00.000-08:00

Sinusoidal Grids: It's possible to create coordinate-based grids based on functions. In the example below I'm using a sine wave. I can simulate the module-based grid, with common corner vertices, or use coordinates to position cells without common vertices, overlapping and gapping the cells.

This image was inspired by columnar basalt.

Plant Senescence, Sample, Remix

2008-11-13T07:44:00.000-08:00

Here's a new print, "Plant Senescence, Pixelated Sample, Vector Remix":

Pierre de Fermat

2008-10-20T17:01:00.000-07:00

Malcolm Gladwell writes in the October 20, 2008, New Yorker, that in our accounting of creativity we have forgotten to make sense of late bloomers. We expect poets, artists, and mathematicians to do their best work before middle age. We tend to accept the conventional wisdom that age is the enemy of creativity.

Mathematician Pierre de Fermat (1601–1665) had completed a manuscript for his pioneering work in analytic geometry by the time he was 35, but he also helped lay the groundwork for probability theory when he was 53. Fermat's example contradicts what G. H. Hardy said in his essay, A Mathematician's Apology (full text here):

"No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." G. H. Hardy (1877 – 1947)

Hardy may have been right that original mathematics is the most difficult discipline to continue into middle age and beyond, but Gladwell says something else is going on. Gladwell's article covers the work of University of Chicago economist, David Galenson, who showed that there are two different life cycles of artistic creativity — the conceptual and the experimental. Gladwell writes:

"The Cézannes of the world bloom late not as a result of some defect in character, or distraction, or lack of ambition, but because the kind of creativity that proceeds through trial and error necessarily takes a long time to come to fruition."

Fermat was a lawyer first, and though devoted to mathematics, he never felt the need to publish. His work survives through correspondence and notes he made, rather than finished writings. It seems to me that he chose to be the experimentalist, and wasn't distracted by his own early success. Like Cézanne, Fermat was more interested in the process of discovery, and less distracted by his own success.

I usually end a post with a gratuitous link to some new artwork of mine. In this case, I'm connecting Fermat's spiral, which he discussed in 1636, to the first drawing for a new series — Fermat's Spiral. (More to come. . .)

Ernst Haeckel

2008-10-19T07:43:00.000-07:00

In previous blogs (More. . ., and Artist-Scientist Maria Sibylla Merian. . .) I referred to the artist/naturalist. Here's a link to 100 images by Ernst Haeckel, whose work survives alongside the prints of Maria Sibylla Merian. Here's a link to an exhibit of natural history books by ten authors, including Merian. Here's a link to the work of Richard Hertwig, a scholar of Ernst Haeckel.

Here's just one image of radiolarians from Haeckel's Kunstformen der Natur (Artforms of nature), followed by my own interpretation, "Radiolarian Skeleton."

Alexander Calder

2008-10-17T07:18:00.000-07:00

Continuing my search for artists with science or math training, while reading about the Whitney Museum of America Art exhibit "Alexander Calder: The Paris Years, 1926-1933", I discovered that Calder trained as an engineer. Previously, I identified Jess Collins, Jo Baer, Portland artists Julian Voss-Andreae and Stephan Soihl, and the artist and entomologist, Maria Sibylla Merian (1647-1717). Calder went to the Stevens Institute of Technology in New Jersey, and worked for a short time as a hydraulics engineer and a draughtsman for the New York Edison Company.

Surely the other end of the art spectrum from an exhibit at the Whitney would be a gallery show that accepts all entrants and charges a fee. For the time being, I'll have to be content with the later — Snap to Grid 2008, at the Los Angeles Center for Digital Art includes one of my "Canopy" prints.

Extensions

2008-10-15T11:37:00.000-07:00

The digital art at joebartholomew.com is grouped by project. Many of these projects share a common purpose — to extend a graphics system in some new way and thereby provide a tool for creating new imagery. The Rectangles and Spirals project extends features of the golden rectangle to other rectangles by generalizing the golden ratio formula and the Fibonacci number sequence. The Small Programs and Plane Symmetry Groups projects relate to plane symmetry or wallpaper groups, but differ in ways that disqualify them from, while extending the mathematical form.

The Coprimes and Greatest Common Divisor (GCD) projects plot the GCD of integers as colored cells within a grid. Patterns are generated by extending the integers provided to the GCD calculation beyond the grid column and row numbers. These numbers figure in a pair of formulas to generate different GCDs.

I based the Perspectives project on a simple two-point perspective drawing function. Instead of using the tool to create visually accurate perspective drawings I mix several viewpoints within the same drawings.

The Cubes and Cabtaxi Fleet project is based on the recreational math idea of taxicab and cabtaxi numbers. It suggests possibilities for representing the numbers in sculpture.

Here's my latest Small Programs drawing, "Apse". The apse, a frequently vaulted recess at the sanctuary end, may extend the exterior of a church.

Four Similar

2008-10-13T07:06:00.000-07:00

Here are four new images from my Small Programs series. They were developed with mostly common code. As usual, I generate the unit cell with a small program. The program randomly varies the resulting pattern. I like to consider this technique an extension of plane symmetry or wallpaper groups in that instead of translating, rotating, or reflecting the primitive cell, the small program regenerates a varying image each cell. This gives me control over the total image. For example, I did a series of three entitled "Lessness", I, II and III, with clumped images in the center, in a grid, and along diagonal rows. In the first image below the program generates sparse cells toward the center.

Lessness III

2008-10-10T07:35:00.000-07:00

Here's the final drawing of three entitled "Lessness", I, II and III. "Lessness" is a short story by Samuel Beckett in which he randomly ordered the sentences. The Lessness drawings are from my series, Small Programs, which relate to plane symmetry or wallpaper groups except I generate the unit cell with a small program. The program randomly varies the resulting pattern.

More Art and Science

2008-10-04T07:28:00.000-07:00

Last June we went to see the Getty Center exhibition, "Maria Sibylla Merian & Daughters: Women of Art and Science". Merian was an artist and entomologist, with a special talent for natural history illustration and printmaking. I have an interest in artists with scientific or technical backgrounds such as Jess and Jo Baer, or artists that understand how to use technology as well as Larry Bell does. Last night I saw Susan Murrell's exhibit at galleryHomeland, in Portland. Murrell recreates the effect of scientific illustration and museum display without the actual science. She fabricates displays and specimen boxes of fantastic faux creatures. She paints multi-layered abstracts on transparent acetate that hint at purposeful geological renderings and maps. The whole presentation is tied together by call out label leader lines, installation style with tape applied directly to the wall. It's similar to installations and sculpture that reference without being informed by architecture. Eric Zimmerman is one of many young artists taking architecture and doing installations and drawings similar to what Murrell does with biology or geology. (See www.ezimmerman.org and Art Lies, Issue 58.)

This forces me to think about my own pursuit of art grounded in math. In my work I use programming, simple math, elementary algebra, trigonometry, and Euclidean geometry. Yet, the closest I've come to actually practicing math was with my project on a generalization of the golden ratio formula. So, what's the point in mimicking, paralleling, or otherwise borrowing from math, science, and architecture to create art? I've noticed that many artists want to separate themselves from the technical. They blatantly use technical imagery while distancing themselves from the technology. Both Robert Mangold and Mel Bochner de-emphasized the math in their art. Susan Murrell takes this much further. Joel Courtney said of Murrell in the 6/20/2008 Las Cruces Bulletin, "Even though science is a large influence, Murrell goes out of her way to avoid studying the sciences further so as not to cloud over the messages of her artwork with the particulars of the technical world." Murrell was quoted, "My ignorance (of natural science) is part of the equation in a way. The abstraction does dumb it down a bit, but it allows my work to explain it to others."

In contrast, I prefer the technical illustration of Merian to the parody. This inspires me to renew my efforts toward taking a stand opposite to mimicry and parody. Merian used artistic talent to advance science. Can we use math or computer science to advance art without trivializing the technical aspect? Mel Bochner said,

"Mathematical thinking is generally considered the antithesis of artistic thinking, but it is not. The two aspects of mathematical thinking that interest me are its clarity and rigor. These are also the characteristics of the best art." Mel Bochner — in his “ICA Lecture”, 1971. [Reprinted in Bochner, Solar System & Rest Rooms, Writings and Interviews, 1965-2007, Cambridge, MA: The MIT Press, ISBN 978-0-262-02631-4. Page 90-92.]

So Bochner seems to say that art should be precise and valid like mathematical thinking, but I think there's much more opportunity here than Bochner's statement implies. I think twentieth century art opened up possibilities for artists to use materials, skills, and techniques from all professions. I hate to see artists stuck within the limitations of pre-twentieth century art simply mimicking other professions when they could employ the techniques and knowledge of those professions to make a new art.

I'm sure there are other artists with a technical background and a portfolio of work that advances rather than parodies science in art. In Portland we have Julian Voss-Andreae who studied physics at the universities of Berlin and Edinburgh and did graduate research in quantum physics. Also, there's Stephan Soihl (B.A. in Physics) of Mt. Hood Community College and Blackfish Gallery. Coincidentally, Soihl also does botanically accurate watercolors.

Here's a second version of a drawing from my Small Programs project.

Magnetic Field

2008-10-03T20:12:00.000-07:00

Here's another drawing from my Small Programs project. This image depends heavily on the program's ability to concentrate images toward the center of the grid.

Chirality

2008-10-01T07:22:00.000-07:00

An object or a system is called chiral if it differs from its mirror image, that is it cannot be superimposed on its mirror image.

The Small Programs are digital drawings that I program instead of using traditional artist's media. They grew out of my Plane Symmetry Groups project. These drawings relate to plane symmetry or wallpaper groups, but they differ in ways that disqualify them from, while extending the mathematical form. The primitive cell in each of these patterns is generated by a small program. Like plane symmetry groups, I can transform, rotate, or reflect the cell. However, the drawings differ from the mathematical form in that I generate the unit pattern programmatically. I have replaced the static cell with a small program which may randomly vary the resulting unit pattern. The patterns are isosceles trapezoids arranged radially in tracks around a center point. The patterns grow in width as the tracks radiate out.

I explore small programs in somewhat the same way I would use traditional media. I have a general idea of the result I am attempting to achieve before I start coding. I work through various challenges during the process. I selectively discard or keep elements as I approach the final rendering.

The programs are Flash ActionScript. I often use random selection within the program, so the result varies each time an image is rendered.

Here are two new closely related prints in the Small Programs series.

"Prebiotic Earth"

"Mass Ascension"

Devil's Claw & The Far Shore

2008-09-28T07:26:00.001-07:00

Here are two new digital prints, continuing a theme of images inspired by the natural world. First is "Devil's Claw":

"The Far Shore":