I found this website:
http://www.thegoldenmean.com/, which had a nice opening graphic. I was turned in for cheating on a base numbers math test in 7th-grade. The evidence: I got an A on it. I have no idea why they made sense to me when notheing else did...? Since I am an atheist, the patterns in nature are somehow comforting to me. I will study upon this. Please keep talking!
B"H
Earthmonkey, I find it odd and numinous that this is true:
Take a right-sided triangle with a base of 1 and a height of 2. The hypotenuse would be sqrt (5). We know this from the Pythagorean theorem, of course.
Now, add the base and the hypotenuse. In other words, measure the base and the diagonal. Then, divide the two of them by the height.
What number do we get?
We find this number embedded in to a mysteriously simple right triangle of a base and height of one and two, respectively.
What number? Go ahead and post it, Earthmonkey.
Then say that there are no "hidden clues" out in the open that we live in a mysterious Universe!
Awaiting your response.
All the best,
B"H
I will answer my own question. If we go across base=1, and go up the hypotenuse of sqrt 5. We get 1+sqrt 5. If we then divide by the height of 2, we get the Golden Mean! Also, if we take a similar right triangle with one fourth of the area, we have a base of one half, a height of one, and a hypotenuse of (sqrt5)/2. Trace our line along the whole perimeter of the triangle and we get 1+(1/2)+[(sqrt5)/2], which is the square of the Golden Mean!
Ponder that! Thank you again, Earthmonkey, for putting forth such excellent contributions.
All the best.
I love the golden ratio; I use it a lot in my artwork. However, I won't use it for the actual dimensions of a piece, since it seems to make the frame feel "invisible". My theory is that the brain is wired to recognize deviations from the golden ratio in living things, so that we aren't as overwhelmed by excess data. Beyond the beautiful mathematical and geometric properties of Phi, (after all, we were using it before we started approximating it with decimals), it is just plain everywhere.
If you haven't read about the 3-squares construction of the golden rectangle, you should definitely read this:
http://www.springerlink.com/content/l2g0...lltext.pdf
Other ratios I'm quite, quite fond of:
1:sqrt(2) - Silver/Roman rectangle - If you're an artist, do yourself a favor and throw out your letter-size paper. Get some A4 paper, which is based on this ratio.
1:sqrt(3) - Bronze rectangle - Divides repeatedly into thirds just like root-2 divides by halves.
3:4 - Divided in half, yields ratios of 2:3,3:4,2:3, etc. etc. Also forms the golden triangle: 3:4:5
^_^