You're probably right. As I look at mine further, the result is a larger square that has a Fibonacci sequence, but doesn't form the spiral as well as your solution. But one could apply the modular idea to your setup.
I think it would be neat to go from make a series of larger modules that just went from 1 to 8 and see what kinds of designs you get by putting them together in different ways.
But I'll show off a little ... there is a strong interrelationship between the numbers in the Fibonacci sequence. For a start, you get the next Fibonacci number by adding the two previous ones together, so there is already a relationship between them. For example, if you square the first hundred Fibonacci numbers and add them together it is the same result as the hundredth Fibonacci number times the 101st Fibonacci number. It is curious that these connections between the squares of Fibonacci numbers connect with later numbers in the sequence and in fact, there are quite a lot of these weird connections between the numbers. For example, if you have the 100th Fibonacci number and you want to find out the 300th Fibonacci number there is a very simple way to do it. You cube the 100th Fibonacci number, multiply by 5 and then add 3 times the 100th Fibonacci number. This will give the answer to what the 300th Fibonacci number is.
It would be fun -- not that I have any plans to do this, as I've got half-a-dozen other projects I should really be looking at -- to do a series of different afghans or suchlike, each one exhibiting a different Fibonacci identity. Any identity where you multiply two or fewer terms together should have a pretty straightforward plane-geometric interpretation.
There's a fine series of books by Roger Nelson called 'Proofs Without Words'. I'm wondering how many of his elegant pictures could be rendered in knit or crochet: the square-spiral design is an excellent demonstration of the identity: f_0^2+f_1^2+f_2^2+...+f_n^2 = f_n * f_(n+1)
while the one I suggested (with rectangles) was f_0*f_1 + f_1*f_2 + f_2*f_3 + ... + f_n*(f_(n+1)) = f_(n+1)^2
These have certainly been explored in the context of visual design, but less so with fibercrafts.
logarithms - I loved them ayt school. I don't think they're used any more, no need with calculators but it deprives students of part of the fascination of maths.
Yes I agree I've still got my log books lol DH asked a couple of weeks back, why I still had them...there isn't any real answer to this, only, '''cos I have''
I ain't getting rid of them either....they're mine.
No need to stitch the modules together. Knit a 1x1 square (A), then change colors and knit another 1x1 square (B). Bind off. Pick up stitches along the long side (across A and B) and knit a 2x2 square (C). Bind off. Pick up stitches along C and A and knit a 3x3 square. Repeat.
Would that end up being something like entrelac? I've never used that method before, but I thought there must be a way to use it for fibonacci. Thanks for the idea.
The Fibonacci series is also interesting because the higher the series goes, the ratio between one number is to the next-lowest one is what I think is called 'phi', or the so-called Golden Ratio. As I understand it, the Golden Ratio was used as the ratio between the width of the Greek Parthenon to its height. And is also part of nature although I don't know specifically how or where.
Phi has its own interesting qualities - phi squared (phi time phi) is phi plus 1, and 1 divided by phi is also phi minus 1. It's approximately 1.618034, or exactly
Actually, I've been thinking ever since I read this; your idea would be very easy to knit. There is a technique that I've been fiddling with a little bit...I've seen it in two or three recent knitting books/magazines. It's a style of modular knitting that involves garter stitch squares. Garter stitch is kinda neat in that it really does form a square, with the row measurement the same as the stitch measurement.
In this case, the square is made by casting on twice the number of stitches you need for the length of one side; that is, if you get five stitches per inch, and you want a four inch square, you cast on 40 stitches. Then you knit every row, with a double decrease (slip two stitches, knit one, pass slipped stitches over, for example) in the middle on every other row. You end up with an almost perfect square with what looks like two right triangles...and no binding off.
If you begin each row with a slip stitch, you end up with a perfect selvedge edge for picking up stitches. So..you pick up stitches on the edge of the previous square, cast on an equal number of stiches, and go again.
If you want the spiral look (well, it wouldn't be too curvy...perhaps someone could figure that out?) simply use two colors; cast the first half of the square with one, and the second half with the other. The spiral will grow naturally as the knitting progresses.
(s'cuse me, let me check to see how that would work...)
According to my "paint" program, you'd have to be careful about the direction you picked up your stitches, but it works!
No binding off, and no joining. 'course, it only works with garter stitch, but hey, at least it's easy!
....just a thought experiment...hmn. I'm going to give that a try.
Similar, except that you wouldn't be joining at the end of the rows as with entrelac, since each square would be building OUT from the rest of the piece. The only "seam" would be where the stitches are picked up.
Thank you. That was the method I was attempting to describe in my post. If this isn't a case of "a picture is worth a thousand words," I don't know what is!
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