A random walk through mathematics -- mostly through the random part.
28 March 2008
The Mandelbrot monk
The Mandelbrot Monk, Udo of Aachen (1200-1270), began developing probability theory, computed an approximation of the Mandelbrot set, and like some other people got a suspiciously good approximation for π by stick-throwing.
This just screams hoax. He was working with complex numbers, and their geometric representation 300 years before Tartaglia and Cardano hesitantly advanced the former and 500 years before Argand advanced the latter? Not to mention that he's graphing algebraic statements 400 years before Descartes.
The Mandelbrot set hangs on so many other things than computer technology -- things almost everyone takes for granted. This story is at the very least horribly oversimplified, and I suspect is actually totally false. I'll check with BenoĆ®t, though.
Oh, and Blogger has broken OpenID. Big surprise: Blogger breaking something.
It shouldn't be that difficult to actually plot the Mandelbrot set using tools available to a medieval monk, even without invoking complex numbers.
Geometrically, computing z^2 is easier when viewed as r(cos θ+i sinθ). Doubling the angle is easy, and squaring the radius can also be easily done. Adding the +c is as simple as completing a parallelogram. All operations easily performed by someone adept with compass and straight-edge.
I could easily see showing how to do a single iteration to, say, Archimedes, and saying "If you start with some points, and iterate this procedure, it goes far away quickly. If you start with other points, it never gets very far away. I wonder which points stay close, and which points go far away? It's obvious that if you ever get to a point more than two units away, the procedure will just take you farther away, so we can give up on a point once the iteration falls outside the 2-unit circle."
If I have time today, I'll write up a full procedure on my blog and leave a link here.
(and, no, my observation isn't an April Fools Day thing, despite the timing.)
7 comments:
It is not yet the 1 April :)
stregatto,
I know that -- but it's close enough that people should be keeping their eyes open for such things.
This just screams hoax. He was working with complex numbers, and their geometric representation 300 years before Tartaglia and Cardano hesitantly advanced the former and 500 years before Argand advanced the latter? Not to mention that he's graphing algebraic statements 400 years before Descartes.
The Mandelbrot set hangs on so many other things than computer technology -- things almost everyone takes for granted. This story is at the very least horribly oversimplified, and I suspect is actually totally false. I'll check with BenoĆ®t, though.
Oh, and Blogger has broken OpenID. Big surprise: Blogger breaking something.
The "Harvard Journal of Historical Mathematics" doesn't seem to exist. All references to it in Google point to some reference to this page.
Yeah, BTW it is a really nicely crafted joke, the first time i've seen it i did not realize it. I was also too happy, being a fractal lover :)
Awww, c'mon! April 1 isn't close enough yet.
It shouldn't be that difficult to actually plot the Mandelbrot set using tools available to a medieval monk, even without invoking complex numbers.
Geometrically, computing z^2 is easier when viewed as r(cos θ+i sinθ). Doubling the angle is easy, and squaring the radius can also be easily done. Adding the +c is as simple as completing a parallelogram. All operations easily performed by someone adept with compass and straight-edge.
I could easily see showing how to do a single iteration to, say, Archimedes, and saying "If you start with some points, and iterate this procedure, it goes far away quickly. If you start with other points, it never gets very far away. I wonder which points stay close, and which points go far away? It's obvious that if you ever get to a point more than two units away, the procedure will just take you farther away, so we can give up on a point once the iteration falls outside the 2-unit circle."
If I have time today, I'll write up a full procedure on my blog and leave a link here.
(and, no, my observation isn't an April Fools Day thing, despite the timing.)
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