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Two Powerhouse Design Tools: Fibonacci and The Golden Mean

I was reading an online article about the Fibonacci sequence and the Golden Mean, and I was once again reminded what a powerhouse these two systems are for weavers. Before I even knew what they were, I discovered I was using them completely unintentionally. My “clever” stripes turned out to be Fibonacci. My asymmetrical blocks were close to the Golden Mean. These two constructs divide space in a way that is harmonious and interesting to the eye and regularly occur in nature.

Weavers are fond of symmetrical designs—elements that mirror each other—as a way to create harmonious designs. If you want to create an asymmetrical design, these two mathematical principles are my go-to design tools. Even if you don’t apply them perfectly, they can still give you interesting results.

Fibonacci Sequence

This numbering sequence was named after an Italian mathematician who introduced this concept to Western European mathematics. He used it to describe how rabbits multiply. It shows up in other natural phenomena, such as branching in trees. (Nature knew what it was doing long before we defined these phenomena as math.)

It starts with 0, then you add the two preceding numbers to get the next number. Start with 0+1=1, then 1+1=2, followed by 1+2=3, and so on.

Here is what that looks like:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

You can use this series to determine the size of your elements, then mix them in any order that fits in your space. Following the formula is a way to achieve harmony in an asymmetrical design in a systematic way. 

The units can be inches, ends, picks, repeats, or any kind of base you would like to use. For instance, if I use .25 as my “1” and built a series, it would look like this: 0, .25, .25, .5, .75, 1.25, 2. I arrived at this using the same methodology as the whole numbers: 0+.25 =.25, .25 +.25=.5, .25 +.5=.75, etc. I find this approach particularly useful when using the direct warping method that doesn’t lend itself to odd numbers, and there are many odd numbers in the Fibonacci sequence.

Here is a recent example. I’ve been working on writing the block weave section of the forthcoming new guide from Yarnworker, tentatively titled Rigid-Heddle Weaver’s Guide to Structures. (No pub date yet, but with the incredible support of the Yarnworker Patrons, it is closer to a reality.) In the Summer and Winter piece, I used Fibonacci numbers 2, 3, 8, and 13 to design the size of my warp elements. My base was a single structural repeat made of 4 ends. There are a total of 25 repeats in the warp that I divided into blocks of 2, 5, 3, and 13, and 21.

I wove my design using a different set of Fibonacci numbers: 3, 3, 8. Each group of weft repeats is a different arrangement of blocks, with structural repeats of either 3 or 8. (If I was using a shaft loom, it would take me 9 shafts and 6 treadles to weave this same piece.) This combination of warp and weft blocks made an asymmetrical design that appears like it may have taken a lot of brain work, but once you break down the method, is pretty straightforward. I could have easily made an arrangement using different Fibonacci numbers in the warp and weft and still come up with something that was interesting to look at.

 

 

Golden Mean

The Golden Mean, or Golden Ratio, is a way of dividing a unit of measure so the smaller unit is to the larger unit as the larger unit is to the whole. Think of it this way: There is a big piece, or the entire area, which is divided into two using the Golden Ratio, creating a medium piece and a smaller piece. You can keep dividing the smaller piece to create additional blocks or proportional relationships. Da Vinci famously used this technique in his work, as did many other artists.

To determine this ratio, you need the magic number, which is 1.618—technically, 1.6180339887. . . . , but 1.618 is close enough for our purposes. It the decimal that creates the Golden Mean. I use the ratio to divide my space this way: If “x” is the whole, then [x] times .618 is the asymmetrical sweet spot to divide the space.

Here is an example of how the Golden Mean can play out. In the Pleasing Proportion bag from Weaving Made Easy, there are four blocks on the front of the bag. It has a woven width of 7″ and length 8.5″. To determine the size of the blocks, I multiplied the longest dimension by the Golden Mean, in this case the length, to get the size of the medium block (8.5 x .619 = 5.25). Resulting in a medium block measuring 7″ by 5.25”, shown here in red, and a smaller block measuring 7” by 3.25″ (8.5 – 5.25). We will call this medium block A and the smaller block B. I often use this single division to create a horizon line in my designs, filling in different colors or structures above and below this line. The big block can be on the top or bottom of the piece.

To continue dividing the blocks, you will work with the smallest block and the longest dimension. The new small block is  7” x 3.25”, making 7 the longest dimension. Dividing 7 by the Golden Golden Mean creates a new set of medium to small block ratios, creating a new medium block of 4.3″ (7 x. 618 = 4.29) and a new smaller block of 2.7″ (7 – 4.3 = 2.7). We will call these blocks C and D, respectively. Since we can’t always weave to a precise measurement, I recommend rounding up or down to the nearest quarter inch. To continue to divide the space, keep working with the smallest block and the longest dimension to create new block ratios.

 

These two principles are tools that take the stress out of stripes and blocks. You can use computer drawing programs or simple graph paper to sketch out your ideas. When I’m struggling with designing a piece, I find if I fall back on these tools, it takes a lot of stress out of the process.

Heddles Up!

Liz

10 thoughts on “Two Powerhouse Design Tools: Fibonacci and The Golden Mean”

  1. It is wonderful but I don’t even think of math while actually weaving which creates problems and unweaving when I’m suppose to be following a pattern. LOL

  2. Nice article, Liz. I agree that the Fibonacci series is a useful design idea, though I sometimes laugh at what I’ve done with it. Maybe the total item is pictured in as divided in 2, then half of that is divided in 3 or some other number from the series. Could I not create a design by generating a random number series? That, too, would take the puzzlement out of each step in the distribution of design elements.
    There is a minor typographic error in the paragraph where you describe your “unit” sometimes used with Fibonacci: .5 + .75 = 1.25 not 1.75.
    Love reading your postings. Thanks for sharing so much with us.

    • Sometimes they don’t translate as we expect in soft materials. For instance, my bag really needed some seam allowance to make it a proper Golden Mean in the finished piece, but I’m not going to work that hard for it. These are certainly not the end all be all of design, but they show up often even when we don’t mean them to.

      I was having a hard time getting the Summer and Winter piece to look good to my eye on the loom. I had been working with even numbers and just wasn’t getting there. When I changed all the proportions to Fibonacci numbers I was much happier with what I was seeing. I rarely sit down and plan all these things out, but when I find I’m not getting what I want if I fall back on these two systems, I’m usually happier. These specific proportions are hard wired into our brains as harmonious. Thanks for the catch.

  3. I’m still following other people’s patterns mostly for now, but I’m hoping to get to the point where I can understand the structures that I’m weaving. I’m looking forward to your structures book but I’m a little intimidated! Thanks for the article. I always favor asymmetrical designs and Fibonacci and the Golden Mean are great assets to work from.

    • Structures are like music, you don’t have to comprehend everything about how music works to play a tune. Learning is layering. We take on bits at a time.

  4. Great article, Liz!!
    Could you check Golden Mean paragraph 3? (8.25 – 5.25) = 3.0. I believe should read (8.5 – 5.25) to equal the 3.25 block. Not sure, unless I missed something & being that I seem to have a propensity for typos all the time – thought to check??!! Great idea using visual dimensions!

  5. Great Blog Post, Liz. I’ve always loved the Fibonacci series as it’s found so much in Nature. Your use of the Golden Mean is really helpful too. Thanks.

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