sequences in textiles

Alcuin's Sequence

(and Some Thoughts on Repeats and Symmetry Disguised in Polynomial Sequences)
by Deanna Deeds

Approach.

My approach to the application of sequences to weaving was to take one interesting-looking sequence and see how many different ways I could think of to apply it to weaving. Some applications are listed in Brezine [1] and treated in some depth by Brezine [2]. Using a single example sequence, the methods I chose to address are (a) shaft number, (b) block number, (c) point twill reversal length, (d) stripe width and (e) color of individual thread.

I chose Alcuin's sequence, which begins

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, ...

I found the description of this sequence at the Mathworld website (Weisstein [3]). This is a "cool" sequence because, like a lot of mathematical expressions and sequences, it describes two seemingly unrelated phenomena. It's the Maclaurin expansion coefficients for the expression

1/(1-x2)(1-x3)(1-x4).

It's also the number of different integral-sided triangles with perimeter n for n=3,4,5,... Moreover (and important for our discussion), it has a couple features conducive to using it for design: (1) it doesn't blow up too quickly, and (2) it isn't monotonic; that is, every element isn't larger than the last - instead it has a nice little sawtooth deal going on (Fig. 1).

Shaft number assignment.

As pointed out in [2], for an unbounded sequence, the values can get large: in the case of shaft number assignment, too large to be applied directly at all. The values in the example sequence do so (especially for those of us who only have 8 shafts). For a sequence that's generally increasing, you might not even use all your shafts before you've passed them all by! There are at least two ways of solving these problems, dependent on the the sequence itself as well as the application of the sequence to the design: you can (1) choose a point to stop and repeat or mirror your sequence, or (2) use the congruence common residual of each element, modulo m (or in computer-ese, "take modulo m"), as described in [2].

The second solution, modulo, works well for this sequence; it yields values covering all of your shafts. The only dilemma right to deal with when applying directly to shaft number is that I have no shaft number I call "zero". Adding one to each term is equivalent to (and easier to remember than) renaming my shafts 0 through 7, so that's what I did. The result is interesting pattern (fig 2), but there are some awfully long floats. I think this is one of those sequences that almost works. Playing with the tie-up made the floats more reasonable, but in the process washed out the design.

Block number assignment.

Of course, a better fabric structure resulted from substituting block numbers for shaft number (fig 3). The number of possible loom-controlled blocks is also dependent on (and limited by) the number of shafts you have, so the modulo method is appropriate here as well.

Point twill reversal.

The other solution, a repeat or mirror imposed artificially by the designer, works well for point twill reversal. Since you aren't physically limited (except by the total number of ends you can fit in your fabric width), the choice is a subjective "judgement call" based on how much difference in relative scale you can tolerate in your design. I chose to repeat rather than mirror the design (Fig. 4).

Stripe width.

The artificially-induced repeat also is appropriate for stripe width. Throwing some color in for fun now, using 5 colors in a fixed order, the Alcuin's sequence values were used to determine the number of ends to draw through or weave of each color. I skipped a color for the zero in the sequence. This yielded the plaid in Fig. 5, where again a subjective decision about relative tolerable scale was used to determine the stop point.

Color assignment.

For color assignment, depending on the coverage of the sequence and your palette size, either modulo or repeat/reverse could be used: your palette limit determines the modulus or the repeat/reverse point. I used the same 5-color palette as above. Taking mod 5 of each sequence element, then assigning colors thread by thread, one thread per sequence element, yielded a surprisingly pretty plaid (Fig. 6).

An observation.

While playing with this sequence in the drawdowns, I was amazed to discover that not only does the modulo-ed incarnation repeat itself, but it mirrors itself before doing so. What a nice property for the designer/user: a natural mirror or repeat, already built right into your sequence. As was remarked in [2], the sequence n2 and others also have the repeating property. On further research I learned that in fact all polynomials P(n) do. This wasn't immediately obvious to me, but I was able to convince myself with an informal proof why it is so. Alcuin's is formed using powers of two. Further playing leads me to believe that if you take a polynomial and give it a coefficient less than 1, and then take the nearest integer (as in Alcuin's), the result still retains the periodic nature, but the repeat is lengthened and the pattern within the repeat is much more visually complex. Alcuin's has the added interest that n odd is treated differently than n even (hence the sawtooth).

For further study.

Further sequence applications I'd like to try are color blending in each warp end (two or three threads threaded as one end), applying the sequence in the tie-up, or assigning blocks or shafts via an algorithm similar to name drafts.

Deanna's intro:

My background and vocation is engineering, with a B.A. in Physics (math minor), and a B.S. in Engineering. I currently write embedded software for guidance and navigation systems. Weaving is my hobby, and though it is a passion... it doesn't pay the bills, so my progress has been slow. I think my first clue that people actually did handspinning and handweaving was in 1986. From then on I was off on this great adventure. I'm just now feeling as if I'm getting a grasp on the mechanics and becoming a better craftsman/technician at weaving, and I'm loving the process. But I have always been fascinated by the connection between weaving and math. Recently several events (including the startup this study group) have occurred concurrently for me, opening up a door...with the help of others I hope to begin to learn in earnest the application of mathematics to the design, analysis and implementation of weaving. I am looking forward to seeing what others have thought of that I haven't, and hope that I have some ideas to share as well.

References:

[1] Carrie Brezine, Mathematics and Textiles Preliminary Study Group Preliminary Newsletter
[2] Carrie Brezine, "Beyond Fibonacci", Complex Weavers, Sept. 1997
[3] Eric Weisstein, website at http://mathworld.wolfram.com, sponsored by Wolfram Research (Mathematica).

Notes:

1. The elements of Alcuin's sequence can be formed using [n2/48] for n even A(n) = [(n+3)2/48] for n odd, for n = {3,4,5,...}, and where [x] denotes taking the nearest integer (rounding).

2. Note: this sequence seems to have a property that eight sequential values can be found all divisible by the same integer. I found this true for any modulus from 2 to 10, and suppose that it's generally true. This yields a section of 1's all in a row. I omitted these 1's for the shaft number assigment exercise because they aggravated the long float problem, but left them in for the block number and thread color assignment, where instead of being annoying they added to the design interest.

3. It is a property of congruence modulo m that P(n+m) is congruent to P(n) for all polynomials P(x). This can be shown informally with an example and then extrapolating to the general case. As an example take the polynomial P(n) = n2. Now start with an n such that n is congruent to 0 (mod m), that is, n is divisible by m. Then so are n2 and (n + m)2 = n2 + mn + m2. They're all multiples of m. What happens with the congruence of the terms subsequent to those for n and (n + m)? Well, (n+1)2 = n2 + 2n + 1 is congruent to 1 (mod m). And ((n + m) + 1)2 = (n + m)2 + 2(n+m) + 1 is also congruent to 1 (mod m). And so on when you add 2, 3,...m-1. You can do the same exercise with P(n) = n3 and see the same result. A P(n) from sums of powers of n is going to do the same thing. Another interesting property is that if P(n) only contains even powers of n, then not only will it repeat modulo m, but it is also symmetric with itself within that repeat (this is because (-x)2 =x2).