Thursday, August 18, 2016

Fibonacci and Lucas and crochet

Over the years I've made soooo many scarves and in a few of them I've used the Fibonacci sequence. You may have heard the term Fibonacci numbers. Well, about a dozen years ago I researched this and wrote an article on it. I thought it might be interesting to revisit it so here it is again. 

Fibonacci and Crochet
©2004, 2016 Sandra Petit,
What exactly is Fibonacci?

Well, Fibonacci is not a WHAT, but a person. His name was actually Leonardo of Pisa, or Leonardo Pisano, also called Leonard Bigollo. Bigollo means traveler. According to what I’ve read, he called himself Fibonacci, short for Filius Bonacci. I think he did this to make it hard for us to look him up. LOL He was born in Italy in 1170, and died in 1250. That makes him a ripe old 80 at his death, depending on the actual date of birth.
So why do we care anyway?
Well, Fibonacci was a famous mathematician and he wrote some well known books such as Liber abaci. Until this time folks in Italy used Roman numerals. Ever try to add and subtract Roman numerals? Major pain. In this book, which was basically a primer in our current ten based (decimal) number system, he posed problems which were to be worked out by the reader. One of these problems gave birth to the now famous Fibonacci numbers.
In the Fibonacci sequence you add two numbers to get the third in the sequence, as I’ve done below. Note that Fibonacci excluded 0 in Liber abaci. I don’t know when it was added, but somewhere, somehow it was. This did not change the sequence since 0 + anything is that number.
0 + 1 = 1
Sequence: 0, 1, 1
1 + 1 = 2
Sequence: 0, 1, 1, 2
1 + 2 = 3
Sequence: 0, 1, 1, 2, 3
2 + 3 = 5
Sequence: 0, 1, 1, 2, 3, 5
3 + 5 = 8
Sequence: 0, 1, 1, 2, 3, 5, 8...
And so on. Those are the Fibonacci numbers. Of course, this can go on indefinitely, but for the purposes of crochet, there is a limit to how far we want to go. Even for an afghan we would hardly go over a certain number of rows. Can you imagine working 17711 rows or rounds?
Here are the first two dozen numbers so you can see what I mean. The sequence quickly escalates.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657
The Fibonacci sequence of numbers is quite useful for planning a pleasing-to-the-eye sequence of colors. It can help you in designing your crocheted items. Yep. It’s true.So you’re tapping your foot and wondering again, WHY do I care?
For example, squares. Given you have a round 1" x 1" square, and your gauge doesn’t change, you place two of these side by side. That’s the first two numbers (because you obviously can’t have a square 0" wide). The next number is two. You can place a 2" square on top of the two 1" squares. Now you have a square with sides 2" + 1". That would be 3, right? Place that 3" square on one of the sides. Now you have a new side with a 3" and a 2" square. The next number in the sequence is...5. So each new square has a side which is as long as the sum of the last two squares added. Say that real fast three times. Not easy, is it? Makes you sound like the Scarecrow in the Wizard of Oz. Remember that your tension has to be even so your gauge is correct, or it won’t work.
Another way to use the sequence is in determining the length of your afghan. If it is 34" wide, you can make it 55" (the next number in the sequence) long. Or is it’s 55" wide, you can make it 89" long. 21" x 34". You see what I’m saying. Does that mean you HAVE to make it 55" long if it is 34" wide. No. It is just one way of determining a length that might give a nice look. It is by no means the ONLY way.
There is another set of numbers we can also use. These were defined by the French mathematician, Edouard Lucas (1842-1891). As I understand it, he gave the name Fibonacci Numbers to the above sequence. Then he developed his own sequence. He began with 2, 1 rather than 0, 1. Then continued in the same manner as Fibonacci, adding the two digits to get the next. You get a different list of numbers, of course.
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123
Just because I think it’s cool, note this. From the Fibonacci list, take one number, skip a number and add the next number. Your answer will be the the number between those two numbers on the Lucas row. For example, on the Fibonacci row start at the second 1, skip 2, add 3. 1 + 3 = 4, which is the number in the middle of 1 and 3 on the Lucas row.

Let’s see how else we might use this number sequence in our crochet projects. By the way, an afghan 47 x 76 would, to me, be perfect for a twin size bed. (The two numbers in the Lucas sequence)
One way we make our projects unique is in the colors we choose. Stripes is an oft used method of alternating colors. Why not try using the Fibonacci or Lucas numbers to vary the colors? There are so many different ways to do this.

Two rows of blue, one of white, three of red, four of white, seven of purple, then you might start over or go backwards. That’s the Lucas sequence. Of course you can choose different colors as well. How about the colors of the rainbow?
Rainbow Over You, God’s Blessings In View  - red, orange, yellow, green, blue, indigo, violet.
In case you’re wondering, indigo is the shade of color between blue and violet. According to (definition #3) it is: "The hue of that portion of the visible spectrum lying between blue and violet, evoked in the human observer by radiant energy with wavelengths of approximately 420 to 450 nanometers; a dark blue to grayish purple blue."
How else might you use the Fibonacci or Lucas numbers in your crochet? You could vary the stitches on each row. You might do 1 row of sc, 1 row of dc, 3 rows of sc, 5 rows of dc.
Or you could use different stitches. 1 sc, 1 puff, 2 sc, 3 puffs, 5 sc, 8 puffs ...
You can substitute any other stitch for the sc and puff. Try it with the cross-stitch or the front post stitches or any other. Of course if you are using a pattern with alternating stitch types on one row you have to make sure your foundation chain is correct.
If you’re varying the length of your stitches, as in making sc in one row and dc in the next, you will be able to see the change easier if you use different colors. However, you will still see a difference even if you use the same color. You might consider using the same color but an entirely different sort of stitch.
You don’t have to go the entire sequence either. You could choose two numbers from the sequence. Say 3 and 5 and alternate those. 3 cross-stitches, 5 dc, 3 cross-stitches, 5 dc and so on...
There are just any number of ways to use this sequence with a beautiful result.
There is one other sequence of numbers I read about but it is not very useful in crochet. It is called the Golden String, or the Infinite Fibonacci Word or the Fibonacci Rabbit Sequence.
The sequence is:   1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...

How do you get the numbers? Good question. I’m not sure I understand that myself, but here’s how I think it is done.
You start with 1 and 0. That’s easy enough to remember since we work with a decimal system.
Now rather than adding the numbers as you do in Fibonacci, you follow the rule.If the last digit is a 0, make the next digit a 1. If it’s not, then make the next digits 10
Let’s see if that works.
First numbers are 10. The last number in that sequence is a 0, so you add 1.
Sequence 101.

Now the last number is a 1, so you add 10.
Sequence: 10110
Now the last number is a 0, so you add 1
Sequence: 101101
Yep, that seems to work.
Since you can’t do ZERO rows of crochet, it’s not very helpful but I would feel remiss if I didn’t mention it since it is associated with the Fibonacci and Lucas numbers.
So there ya go. Fibonacci, Lucas, and Crochet.

Happy crocheting!

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