Saturday, March 30, 2013

Fractal Geometry -- a Follow Up

It is amazing to me that of the  904 posts on this blog, the most visited is the one on Fractal Geometry.  Out of the nearly quarter million visits to this blog, nearly 19,000 have been to the September 8, 2011 posting.

I decided that since the first posting on fractal geometry only gave a little bit of background, that it is worth posting more detail as well as some more terrific photos.

Definition
Fractals are geometric figures -- just like rectangles, squares, and circles are geometric figures. They form in nature -- in fact  most objects in nature do not from in squares or triangles, but instead take the form of complicated geometric fractal shapes.

Some refer to the result of the pattern as the fractal; others refer to the mathematical formula that determines the pattern as the fractal.  Mathematicians can create fractal sets that demonstrate self-similar patterns such as the Cantor dust image shown below.

Sometimes the mathematical process that a natural form develops is in part random, but does have a branching pattern that has a fractal element, such as seen in the two photos below.

Another way to think about what fractal really means is to compare the mathematics of fractal geometry with the process of measuring the length of a coastline. If you were to measure the shoreline of Lake Superior using a mile-long ruler, you would get an approximate measurement of the length of the shoreline.  Then, if you instead used a half-mile long stick to measure that same shoreline -- you would get a longer measurement because the shorter measurement stick would more efficiently measure some of the shoreline's sections.  If you then used a 1,000-foot stick to measure that same shoreline -- the length would again increase because you would be able to measure most every cove and inlet.  If you then used a yard stick to measure the circumference of Lake Superior's shoreline -- the length would again increase.  The accuracy of the measurement increases with the preciseness of the measurement length.  That is fractal.

Self-Similarity
To have a fractal pattern, there has to be self-similarity.   In life, we use the word "similar" to mean that things have at least something in common.  In geometry, the term "similarity" is very specific and precise.  Geometric figures are similar if they have the same exact shape, although the size of the shape can change.  So the two triangles shown below that have different shapes are not really similar from a mathematical point of view.  But the two squares shown below are similar.

Mathematics can play even more of a role in self-similarity.  The two shapes must have the same similar shape, but one shape can be a different proportion as determined by a ratio, or scale factor.  The corresponding sides must be mathematically linked while the corresponding angles must remain of equal measure.  For example, the second rectangle on the right below is twice as high and twice as wide as the rectangle on the left.  Both rectangles are similar.

Notice in the picture below that there are several different sizes of equilateral triangles that make up a large triangle.  If you magnify any of the small triangles, you get back to the original big triangle! In other words, this object is self similar. Self similarity is an important property of fractal objects.

Now for some intricate fractal geometry images.  Enjoy.

Fractal limestone crystals....

Fractal Romanescu....

Fractal honey crystals....

Fractal copper crystals....

Fractal broccoli...

Fractals carved in stone....

Fractal peacock...

Fractal cactus...

Fractal octopus...

CITES:
http://math.rice.edu/~lanius/fractals/iter.html
http://commons.wikimedia.org/wiki/File:Julia_set_(highres_01).jpg
http://commons.wikimedia.org/wiki/File:Mandel_zoom_12_satellite_spirally_wheel_with_julia_islands.jpg
http://commons.wikimedia.org/wiki/File:Grotte_du_Grand_Roc_-_Fractal_limestone_crystals_-_20090923.jpg
http://commons.wikimedia.org/wiki/File:Crystallized_honey.jpg
http://commons.wikimedia.org/wiki/File:DLA_Cluster.JPG
http://commons.wikimedia.org/wiki/File:Fractal_Broccoli.jpg
http://commons.wikimedia.org/wiki/File:Romanescu.JPG
http://commons.wikimedia.org/wiki/File:Fractals_in_stone.JPG
http://commons.wikimedia.org/wiki/File:Peacock_Closeup.jpg
http://commons.wikimedia.org/wiki/File:Cactus_-_close_up_(6843283770).jpg
http://commons.wikimedia.org/wiki/File:Cooked_Octopus.jpg