So I was bored and decided to stuff around in paint (yes i know, I was pretty bored), I started with a circle, added a midpoint, then made all kinds of links and connections to come up with what is attached.
Then I noticed that the rectangles sorta looked familiar. Are they the golden ratio???
Last edited by Toast (2006-11-27 02:25:16)
Ooh, very interesting. Let's see if we can work out the lengths of that highlighted rectangle.
If we say that the circle has a radius of 1, then its equation would be x² + y² = 1.
The upper-left corner of the rectangle lies on the circle and on a line drawn from the centre at 45° to the horizontal. From standard trigonometric triangles, this means that the co-ordinates of that point are (-1/√2, 1/√2), which in turn means that the length of the rectangle is 1/√2.
The lower-left corner has the same x co-ordinate as the upper-left corner, and it also lies on the line y = 1+x. We know that at that point, x = -1/√2, so that means that y = 1 - 1/√2.
So the difference in y-values of the upper-left and lower-left corners of the rectangle is given by (1/√2) - (1 - 1/√2) = √2 - 1.
We have now worked out that the highlighted rectangle has dimensions of 1/√2 and (√2 - 1).
To find the ratio of these, we need to divide one by the other.
(1/√2)/(√2 - 1) = 1/(2-√2) = (2+√2)/2 = 1+1/√2 ≈ 1.70
So, unless I've made a mistake somewhere, that rectangle isn't golden. Just quite close to it.
Very nice diagram though. It's amazing what you can produce with a simple drawing tool and a bit of boredom.
Why did the vector cross the road?
It wanted to be normal.
but it begs the question... if you keep seperating it into smaller rectangles in the same fashion, will it remain ~1.7, or does it maybe diverge to phi?
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