## A Conversation for Mathematics in the Arts

### Golden ratio and fractals

jeenius Started conversation Jul 28, 2004

Sorry to nitpick.

The golden ratio you gave is an approximation, I think. The mathematical definition is phi=1/(sqrt(1/(sqrt(1/sqrt...)+1)+1+)+1)+1, making it the "most irrational" number possible.

I've read that the "beautiful rectangle" study couldn't be replicated. Does anyone know more about this?

The mathematical definition of fractals doesn't actually include the "repeating shape" quality found in natural or computer-generated fractals. A fractal is defined as a shape that has a constant ratio of filled to unfilled space, no matter what scale you are looking at. Say you have a square -- if you look at the whole square, it is 2-dimensional (all the space is filled in two dimensions). If you look at a quarter of the square, it is 2-dimensional. If you look at 1/16 of the square, it is 2-dimensional. This is a special case where a fractal has a whole-number "fractional dimension" of 2. If you look at a fractal with fractal dimension of 1.5, then half of the space in 2 dimensions is filled, whether you are looking at it on a scale of 1, 1/4, 1/16, or whatever. Naturally, mathematical dimensions must be infinitely large and extend down to infinitesimally small scales that exceed the smallness of particles, so all real fractals are approximations of a mathematical fractal.

An earlier book of Crichton's, The Andromeda Strain, had a picture of a fractal at different scale at the beginning of each chapter. I think he was getting at chaos theory (as in Jurassic Park) even at that early point, and referring to the fact that chaotic attractors are fractals.

### Golden ratio and fractals

flyingtwinkle Posted Jul 29, 2004

a very brainy and interesting topic for an article!In the second paragraph 7th line should read ''tenuous link "" without any coma in 5th para instead of ""an c"" it should be ""a c ""

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