Quilt Patterns

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Today we are going to talk about colorful abstract geometric patternssuch as quilts. For example, here is a beautiful hand-madequilt my sister Barbara Keenan created out of a variety of colorfulfabrics.

Certainly mathematicians are not the only ones to appreciate suchdesigns. Accordingto Wikipedia, peoplehave been making quilts at least since Renaissance times. Similargeometric designs also show up in architecture, oriental carpets andother art forms around the world. In part this comes from a deepseated human aesthetic appreciation for symmetry.

What is symmetry? Essentially it is a repetition in shape andcolor that our brains seem to enjoy, perhaps because it simplifieswhat would otherwise be a complicated jumble of unrelated visualelements. Most plants and animals have at least approximate left-rightmirror symmetry, so perhaps our brains evolved to notice symmetricforms because they help us interpret common scenes.

Barbara's fabric quilt is quite beautiful: one can enjoy looking at itfor an extended period of time, the eye following various paths,enjoying the contrasts between the black diamonds and the colorfultriangles. You can click the photo to see more detail.

What constitutes a visually "interesting" or "beautiful" pattern? Artists like my sister develop an intuitive understanding of what"looks" good, but is there a way to capture that intuition in moremathematical language?

After much searching, I read an interesting article on the webby NikosA. Salingaros called The "Life" of a Carpet: an Application ofthe Alexander Rules. While his focus is on Oriental Carpets, histheory and design guidelines apply much more generally. He provides adetailed description of the characteristics that make interestingdesigns.

We can find many of his ideas illustrated in Barbara's quilt. Some arepretty straightforward, like juxtaposing contrasts, e.g. a darkoutline around a light shape, such as her black diamonds and the outerblack border contrasting with the bands of colorful triangles.

Some of his other guidelines are more subtle, like the idea that thereshould be interesting detail at every distance scale, from the entirework down to very fine details. To my mind, one reason real quilts(and paintings) are more visually interesting than purely computergenerated geometric patterns is that fabric has texture and pattern ofits own. A small piece of fabric that forms a single triangle is oftennot one homogeneous color, but in fact a complex mix of colors andtextures on a much smaller, finer scale.

We see this too in Barbara's quilt. At the big-picture level, we havethe black diamonds against the colored diagonals. But within thecolored diagonals is a second layer of contrasts, this time mostly ofcolors (warm reds against cool blues) and values (dark against light),accentuated by the contrast in the orientation of the triangles.

The Carpet article also recommends including some randomness,but not too much. This also makes sense. A plot of random dots, linesor polygons looks like the visual equivalent of audio noise. This isnot surprising, since both have similar statistical properties. Incontrast, a modest amount of randomness - occasional deviations fromperfect symmetry - can actually add interest to a pattern.

Here too, the artist has done this intuitively. Part of the visualinterest in the quilt is that the colored triangles do not repeat inperfectly symmetric patterns, so that you keep coming back to them,looking for the pattern, catching glimpses of it but not quite fullyunderstanding its subtleties. There is symmetry at the high level; onecan imagine the pattern of black diamonds going on forever in alldirections; yet we would be hard pressed to say exactly what coloreach small triangle should be if we were to extend the design.

My children, who are much more musically talented than I, tell me thatthe same is true in classical music, where the listener comes toexpect certain patterns and can be pleasantly surprised when the musicoccasionally does something different.

A couple years ago, I decided to try out the design ideas inthe Carpet article by making a painting of a quilt-likepattern. I do not have the patience (or skill) to try actually sewinga quilt out of real fabric. As a prelude to painting, I wrote apattern generating program so I could experiment with some variations.

There are a large number of traditional quilt patterns, often formedby assembling right triangles of various colors into a square "block",and then repeating identical copies of that block in a grid layout,optionally rotating them to create different kinds of symmetries. Onevery interesting mathematical question is to try to enumerate, or atleast characterize, a wide variety of such patterns.

Mathematicianshave used group theory to classify repeating designs in theplane into 17 basic types calledWallpapergroups. However, when we allow patterns that do not repeatexactly (such as in our quilt picture) the possibilities are muchbroader, and I will not even begin to attempt a full description. Asjust the tip of the iceberg, let me point out that there are manyother ways to create geometric patterns besides using righttriangles. For instance, one can tile the plane using hexagons. Alsointeresting are patterns involving 5-sided shapes(e.g. PentagonalTesselations) and unusual configurations suchas Penrosetiling, in which two basic shapes can tile an arbitrarily largearea in a non-repeating (non-periodic) manner.

The "traditional" patterns have the advantage of simplicity:right triangles and squares are easy to draw and cut out without a lotof special measuring equipment or wasted fabric. Some of the "new"patterns (e.g. Penrose tiling) are a little more complicated, butshould I think be straightforward if one were to make a cardboardtemplate for each shape and then just trace them onto fabric.

I decided to stay within the traditional framework of righttriangles, but to try patterns that were not constrained by repeatingblocks, even though they should still involve some level ofsymmetry.

Computer generated patterns often look somewhat mechanical, e.g.

To me, this is not very interesting, aside from the novelty aspect: itwould be difficult to create this image as a fabric quilt, or as aconventional painting, because the visual impact depends so stronglyon the very precise, very mechanical progression in the colors.

My next step was to aim for images that looked more like traditionalquilts. I also experimented with adding randomness atvarious scales. Here are three example of designs I was able to create.

Ultimately, I decided that my painting would need morerandomness and less symmetry than these images. Actually making apainting is rather different from writing a computer program; you canmodify color as you go until it "looks right", and you can addtextures much like those of fabric. After a great deal ofexperimenting (and over-painting), the final result looked like this:

As you can see, I attempted to follow the guidelines of contrasts andrandomness at a variety of scales. You can judge for yourself whetherthe result is attractive.

I hope you enjoyed this article. While most of my posts are moretechnical, with source code to experiment with, I think it is fun tosometimes step back and enjoy the aesthetic side of math. To me, evenwithout "formulas", the images in this post are deeply mathematicaland quite intriguing. I certainly enjoyed creating my painting - it isfun to use a different side of the brain for a while! If you areinterested in these kinds of patterns, the links mentioned above arejust the beginning: there are a huge variety of web pages out therewith fascinating examples of periodic and non-periodic geometricpatterns described as Tesselations or Tilings.

As usual, please post questions, comments and other suggestions below,or email me directly at the address described at the end of theWelcome post. Remember you can sign up for email alerts about newposts by entering your address in the widget on the sidebar. Or follow@ingThruMath on Twitter to get a 'tweet' alerting you to each newpost. See you next time!