Fibonacci
Leonardo of Pisa, aka Fibonacci, was a late medieval Italian mathematician whose writings helped bring Arabic numbers, the digit ‘zero,’ and decimals to Europe along with the famed number sequence, which carries his name.
The Fibonacci sequence starts with 1 and the following number is the sum of the previous two, hence: 1, 1, 2, 3, 5, 8, 21, 34, 55….
Leonardo of Pisa, aka Fibonacci, was a late medieval Italian mathematician whose writings helped bring Arabic numbers, the digit ‘zero,’ and decimals to Europe along with the famed number sequence, which carries his name.
The Fibonacci sequence starts with 1 and the following number is the sum of the previous two, hence: 1, 1, 2, 3, 5, 8, 21, 34, 55….
“Arabic” numerals actually originated in ancient India, where the Fibonacci sequence was also already well known, although not by that name, of course. Supposedly the sequence had something to do with Sanskrit poetry, but since it appears repeatedly in nature, from the spiral of a nautilus shell to the patterns of petals on a flower, my guess is that it probably was ‘discovered’ rather than invented for linguistic purposes.
The sequence has all sorts of interesting mathematical properties and quirks, which I will not go into here but which can be found on the following Internet sites:
- One of my favorites because it has plenty, but not too much, info is: http://library.thinkquest.org/05aug/00565/
- If you want more details try: http://library.thinkquest.org/27890/mainIndex.html
- And, of course, Wikipedia has lots about both the mathematics of the sequence and Leonardo of Pisa.
Design applications and the golden ratio
My interest in the Fibonacci sequence for knitting is its design applications. The numbers can be used for both striping patterns and proportions. The ratio of two consecutive Fibonacci numbers, especially the larger numbers, generate the golden ratio (aka golden mean) and using two consecutive Fibonacci numbers for the dimensions of a rectangle creates the golden rectangle. The golden ratio (which is 1.610833… an irrational number like pi) and the golden rectangle are believed to have universally aesthetically pleasing proportions and have been used in classic art from painting to architecture.
I like doing my own designs in knitting projects, whether by modifying an existing pattern or creating my own. Fibonacci numbers provide both a good starting point and lots of fun in experimentation.
Some examples
The tea cozy shown here is Fibonacci-striped both with the larger reverse stockinette "welts" and in the stockinette "troughs."
The Fibonacci numbers (in bold) are from the base up-- 1 dark green stripe, 1 blue, 2 more dark green, 3 light green, 5 gold. The trough striping is the same but offset by one (just for fun).
The double-knit, felted hot pad below right was designed using Fibonacci numbers and "golden" proportions. The chart below left shows the design in a grid. The "golden" proportions come automatically from using Fibonacci numbers. Note the following Fibonacci numbers (in bold below) in the design:
- The top turquoise squares constitute 1 design row (as opposed to knitting rows) and are 5 knitting rows deep.
- The next 2 design rows comprise 3 knitting rows each.
- The middle 5 design rows comprise 2 knitting rows each.
- The pattern is then reversed for symmetry.
- The vertical pattern comprises 5 design columns.
- The only place where a Fibonacci number is not used is in the number of vertical stitch columns for each design column, but as the design columns are all the same size, that does not matter.
- I could have left 3 turquoise squares in the top and bottom design rows, but thought that taking the middle ones out to form an "x"-shaped pattern created a more interesting design. However, since that resulted in a void of 8 knit rows next to space of 5 knit rows (not counting the base row) the proportions are pleasing.
Other knitting projects using Fibonacci numbers (including the golden spiral), as well as other "geeky" knitting or crochet ideas can be found:
- On Ravelry at groups: "Geekcraft" and "Woolly Thoughts"
- And on the Woolly Thoughts Web site: www.woollythoughts.com/
Moral: Make numbers fun, go Fibonacci.
