Stat 45: Our Fractal World? Topics for Today. Lecture 1: Getting Started. David Donoho Statistics Department Stanford University. Sierpinski Gaskets
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1 Stat 45N: Our Fractal World? Lecture 1 1 Stat 45N: Our Fractal World? Lecture 1 2 Stat 45: Our Fractal World? Topics for Today Lecture 1: Getting Started What Are They? David Donoho Statistics Department Stanford University Where Do They Occur? Why are they Controversial? Real Ideas behind them... Look Ahead Stat 45N: Our Fractal World? Lecture 1 3 Stat 45N: Our Fractal World? Lecture 1 4 Sierpinski Gaskets Examples of Fractals Sierpinski Gasket Menger Sponge Koch Curve (a) Two-D (b) Three-D
2 Stat 45N: Our Fractal World? Lecture 1 5 Stat 45N: Our Fractal World? Lecture 1 6 W. Sierpinski Menger Sponge Stat 45N: Our Fractal World? Lecture 1 7 Koch Curve Stat 45N: Our Fractal World? Lecture 1 8 Some Properties Multiscale Recursive Fractional Dimensional
3 Stat 45N: Our Fractal World? Lecture 1 9 Stat 45N: Our Fractal World? Lecture 1 10 G. Peano Where do they occcur? Math Analysis Heroes: Cantor, Peano, Koch, Menger, Cesaro Curves with no tangent Space-Filling Curves Chaos Theory Computer Graphics Stat 45N: Our Fractal World? Lecture 1 11 Stat 45N: Our Fractal World? Lecture 1 12
4 Stat 45N: Our Fractal World? Lecture 1 13 Stat 45N: Our Fractal World? Lecture 1 14 Where are they thought to occcur? How Long is the Coast of Britain Coastline of Britain Lungs, Arteries Galaxy Distributions DNA Physics, Chemistry Internet Art(?) (a) Polygon Approx (b) Blowing-Up Stat 45N: Our Fractal World? Lecture 1 15 L.F. Richardson Stat 45N: Our Fractal World? Lecture 1 16 Related Concepts Chaos Wavelets Multifractals
5 Stat 45N: Our Fractal World? Lecture 1 17 Stat 45N: Our Fractal World? Lecture 1 18 Benoit Mandelbrot Why are They Controversial? Late 19th Century Monsters in Analysis Perrin Quote (Handout) 1970 s Mandelbrot Nature vs. Euclid Fractals Everywhere but education obscures them Revolutions across all sciences (Kuhn, Paradigm Shift) Stat 45N: Our Fractal World? Lecture 1 19 New Yorker magazine, July 31, 2000 GORE WITHOUT A SCRIPT Stat 45N: Our Fractal World? Lecture If you look at a map of the coastline of New Jersey, and then magnify that a thousand times, the basic design of the ins and outs of the coastline will be the same at every level of magnification. And they call that the self-sameness principle. I don t understand it. It s way beyond my depth. But I do believe there s something about our world that He began another long pause. I m searching for the right word here that manifests that self-sameness principle in a lot of different ways. And when we find a brand-new understanding of the world that comes out of a powerful new discovery in science, it often allows us to look at social and political matters and find ways toconnect the dots that haven t made sense before. THE 2000 CAMPAIGN: A TEST OF CHARACTER From the NY Times June 21, 2000 At one point in a conversation about some the big think questions that intrigue him, he began to moan, I can t say this, it s going to sound so weird. He went on that way for a minute, as if he were about to admit to something really twisted, before he finally revealed the big secret: Oh, O.K.! I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world.
6 Stat 45N: Our Fractal World? Lecture 1 21 Stat 45N: Our Fractal World? Lecture 1 22 Real Ideas? What will we use? Fractional Dimension Multiscale Analysis Book: Fractals: Images of Chaos Lauwerier Reserve: Mandelbrot, Fractal Geometry of Nature Recursive Synthesis Edgar, Measure, Topology, and Fractal Geometry Multiscale Processes in Nature Schröder, Fractals, Chaos, and Power Laws Mathematical Modelling Barnsley, Fractals Everywhere Data onslaught Computer Resources Scientific Articles Stat 45N: Our Fractal World? Lecture 1 23 Stat 45N: Our Fractal World? Lecture 1 24 Course Structure First 4 weeks: elementary properties of fractals Next 4 weeks: applications of fractals in science Next 4 weeks: cognate areas (wavelets, refinement schemes, multifractals, etc.) Not a math course Course Philosophy Mixture of Lecture and Discussion each meeting Readings each week Stanford since 1990 I Knew Nash Awards, Honors About Me
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