Crochet By Numbers: Applications of Math to Fiber Arts
The Fibonacci Sequence
The Fibonacci Sequence was popularized by Leonardo Fibonacci. He did not discover it, but wrote about it in his book, Liber Abaci.
A picture of Fibonacci
It was actually Acharya Hemachandra , an Indian scholar, that wrote the relation that defines the Fibonacci sequence:
F(n) = F(n−1) + F(n−2)
Where n is a whole integer and F(N) is the nth number in the sequence.
In other words, two consecutive numbers are added, then the sum is added the the next consecutive number:
The sequence then reads: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765…
What is the significance of the sequence then? It is commonly found in nature. The number of petals a flower has is often based on the sequence. (Even a flower who has had all its petals pulled off in a game of he loves me, he loves me not has a Fibonacci number of petals - zero.)
Also,when Fibonacci numbers are consecutively stacked on top of each other, they create a spiral
This spiral resembles the nautilus shell found in nature:
The spiral is also applied in art; notice how this photograph follows the spiral:
The sequence can also be used in clothing, specifically in crochet. I made a scarf and chained (representing row 0) 144 stitches, which is one of the Fibonacci numbers. I then made one row of red, one row of orange, two yellow rows, three green rows, and five blue rows.These are the first 6 number of the Fibonacci sequence.
Scarves can also be made with random orders of fibonacci numbers. A television show on DYI Network called Knitty Gritty featured a fibonacci scarf made from non- consecutive fibonacci numbers, using different yarns and working horizontally. The designer of this scarf says this "creates a seemingly random, but always beautiful scarf".
The first picture above is my scarf, and the second scarf is the scarf from Knitty Gritty. The pattern for my scarf can be found here, and the Knitty Gritty scarf here.
The fibonacci sequence is repeated everywhere: in nature, in culture, in architecture, and even on popular television shows. On a recent Criminal Investigations episode, a serial killer murdered consecutive fibonacci numbers of victims in several cites. On a map, the cites formed a Fibonacci spiral. The FBI team was then able to use the spiral to predict the next attack and save lives.
The Mobius Strip
The mobius strip was discovered by both August Ferdinand Möbius and Johann Benedict Listing in 1858, but they each discovered it independently of each other. A mobius strip is easy to make by taking a strip of paper, making a half twist , then taping the two shortest sides together.
This simple shape has several interesting properties. If a sentence is written on one side, following the strip will take you back to the beginning of the sentence. When cut in half, it forms one long strip with two twists in it.
Chances are, a person will run into a Mobius strip at least once a day. The most common one is the recycling symbol.
The Mobius strip can also be used in jewelry. Generally something is inscribed on the mobius, such as a bible passage, to represent how indefinite it is.
The Mobius strip can also be used in clothing. I made a mobius shrug with a pattern from Interweave Crochet. The garment is made just like the paper mobius. It is crocheted up a a strip, then given a half twist, and then seamed together.
As an extension of my project, I've decided to start a crochet club at ASU. The club is open to any student who wants to learn to crochet or knit. At least one knit or crocheted charity project per semester is mandatory for membership, but can be as small as a square for Warm up America, or an entire Afghan for Afghans for Afghans. I am donating the Fibonacci scarf from this project to a charity in Flagstaff called The Dulaan Project that gives scarves, hats, blankets and sweaters to Mongolians in the coldest part of the world.