Fractal History
Well first off let's say that Euclidean shapes are objects such as the sphere, circle, cube, square, etc. They are ideal shapes that do occur (but
mostly of man-made objects). Now the universe is rough, broken, wrinkled, uneven shapes for the geometry of nature, but hold patterns. This is fractal geometry: fractal comes from the word
fractus meaning fragmented, broken and discontinuous.
Benoit Mandelbrot coined the term "fractal" in 1975 to describe shapes that were detailed at all scales; characterized by infinite detail, infinite length and the absence of smoothness. Fractal geometry is the geometry of the irregular shapes we find in nature. It takes Classical geometry one step further. Models can be made to describe all things of nature: clouds, shells, webs, galaxies, etc.
Now most of early mathematics when calculus was developed it could describe the smoothness of objects. Even when there were rough turns it could be split up to describe parts through calculus. This brought in the motions of orbits, electromagnetic fields, and rate of changes to the forefront described through calculus
Kepler, Newton, Leibniz, Maxwell, Laplace were the key figures in this material.
However this left out objects that were rough with corners but still could be continuous but couldn't be fully described by calculus. It then had to be described by numbers which is the birth of "analysis" by
Karl Weierstrass. He made a curve that consisted of nothing but corners with no smoothness anywhere (so it couldn't be taken apart). So analysis explained numbers through logic into "sets". Not going in to describe sets, but simply they are group of numbers, a collection let's say.
Now "topology" is the study of geometrical properties and spatial relations that remain unaffected by continuous stretching, bending, twisting....Dimensional studies of objects like
Giuseppe Peano's Curve that filled up space on a plane. It was a line and took up all the points in a plane. This was a type of "mapping" of a line to a plane, complicated shape it may be.
Felix Hausdorff described dimension through complicated shapes by fractional dimension, like one and half dimensions. Shapes that lie between dimensions are "fractals"
If any of you tried the applet for the Mandelbrot Set above you probably saw the set over and over again as you went smaller and smaller into it. That is called "self-similarity" A simple set to describe this is like below which is the
Cantor Set below. You can draw this out on paper. Take a line and then take the middle third out. Then you are left with two third parts that are lines again. Do that again and you split it up taken out the middle third. Here's how to make it and see it ->
http://www.mathacademy.com/pr/prime/articles/cantset/
If you did this infinitely you would have the Cantor Set. Well it has "zero measure", not literal zero measure but it's so far from dense you couldn't find points in being connected. Now this set has "self-similarity" too. This important word to describe Mandelbrot Set, Cantor Set, etc. Even the cauliflower idea from above post. You slice up a circle you have an arc. An arc is not a circle again. A side of a triangle is not triangular, etc. But if you took something from nature, a fractal, a break a piece off you still have the something. They are endless motifs "repeated at all scales". A cloud, tree, rocks, etc. all resemble smaller parts of themselves. Now these natural shapes are very hard to discuss through Euclidean Geometry and are called "pathological" shapes. A snowflake is a pathological shape, it succumbs to a bigger dimension then Cantor Set butit consists of the Koch Curve from
Helge von Koch. Everyone has made this curve in school. It's a infinite representation of the Cantor Set through angles. Here is easy link to how to make it ->
http://math.rice.edu/~lanius/frac/koch.html
Now mix this self similarity and "fractal" dimension and we are nearing the fascination of fractals. With a Euclidean shape like cube you can cut up cube divisible by 2. You'll have eight smaller cubes. The square will have four squares, the line have two lines. Cut it up like a
Rubik's Cube and have fractional pieces of smaller cubes, division by 3 and you have 27 cubes, nine squares, 3 lines moving down the dimensions. Basically any n-dimensional object will have m to the power of n 1/m sized copies of itself.
This is where we get technical: the Cantor Set has dimension 0.63, the Koch Curve is 1.26, clouds 1.35. Coastlines? 1.26. In fact Koch Curves when infinitely produced look like coastlines. Again
Felix Hausdorff's similarity dimension as the cutting up cube can be extended to objects NOT self-similar. Fractal shapes that lie somewhere between dimensions have dimensions known as "fractional" Hausdorff dimensions. And because we are talking about fractals it's sometimes called the fractal dimension. This was a theory of Hausdorff's until Mantelbrot revived it via the Julia Set.
Lewis Richardson, using idea of Jonathan Swift's quote:
So, Nat'ralists observe, a Flea Hath smaller Fleas that on him prey. And these have smaller Fleas to bite 'em. And so proceed ad infinitum.
took in natural occurrences like currents and turbulence as collections of smaller eddies and so on that he wrote like Swift as:
Big whorls have little whorls,
That feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.
Lewis Richardson asked many questions about nature, like how long is the coastline of Britain. With a map it's measured by the road near the coastline. But if you walked along the coast and measured it would be more accurate. If you held out a ruler to map it would be too straight. Make the ruler smaller and then you could accomplish measuring at angles. He did this for many coastlines around world comparing data from maps to the logarithm of the size of measuring stick. When Mandelbrot reviewed Richardson's findings he found them as Hausdorff dimensions of the coastline 1.26
Now cellular evolution spans from much of chaos theory of
Michael Barnsley in the 1980's. He worked from
Sierpinski's triangle (look this up here-->
http://math.rice.edu/~lanius/fractals/ )
Cellular evolutions follow patterns to become fractals because it becames layered like sea shells or fern leaves. His chaos game of rolling a dice to chance from points scribed would sooner or later produce Sierpinski's triangle if one initial point was taken. But in some instances it might produce a fern leaf if a different point was chosen as initial point. Once an initial point was chosen the fate of the game was decided what shape would be. This is the idea behind the "strange attractor".
It was known earlier but not overly focused on since chaos theory is fairly new. However because I'm getting tired of typing here are some names major to focusing on bifurcations (in population growths, yeah Differential Equations!), chaos theory, iterations and mapping, fractals! I am tired of typing seriously....here are important names to look up to learn more along the way. Most of these names are scientists noticing fractal and chaotic aspects to objects or systems. Everything from magnetic material to astronomical objects to cancer cells. I'll save to discuss more later--
Robert May, Thomas Malthus, Pierre Francois Verhulst, Henri Poincare, Edward Lorenz, Tien Yien Li, James Yorke, Kurt Godel, Mitchell Feigenbaum, Gaston Julia (Julia Set!)
, Pierre Fatou, Heinz-Otto Peitgen, Peter H. Richter, Tan Lei, Roger Penrose (will post Penrose Triangles in here sometimes)
, Arthur Cayley, Thomas Witten, Leonard Sander, Stephen Hawking, Alan Penn, Wilheim Olbers, Hannes Alfven, Ian Stewart, etc.....going back to bed.....lol
me --> :blah: