Fractal Art

At one time I was an art student; however, they showed us a film on fractal geometry in a design class, and I went out forthwith and changed my major to mathematics. (Actually it wasn't Most of the fractals below are described in Benoit Mandelbrot's seminal work, Additional information on fractals and creating fractal art is forthcoming. For the time being, you can view my interactive explorations via my profile page at GeoGebraTube, and my GeoGebra images on my blog. Here, rather than explaining what a fractal is, I shall content myself with the following quatrain from Alexander Swift, and then proceed to the pictures:
The
Each iteration in the recursive construction of the
Plane-filling curves like the dragon sweep were discovered in the late nineteenth century and caused great consternation in the mathematical community, showing as they did that a planar or solid region can be "filled in" by an infinitely thin, winding curve. The first continuous curve was described by Guiseppe Peano (1858 – 1932), who'd been inspired to look for one by Georg Cantor's findings in set theory. A better known example was discovered by David Hilbert (1862 – 1943). These curves fill a square; the one shown above fills a Koch snowflake. The first iteration consists of a seven-segment polygonal curve connecting points on the trisected sides of an equilateral triangle. Each segment is then replaced by a copy of the original figure, and so on,
Filling in the plane on one side of the snowflake-filling Peano curve above forms what Mandelbrot calls an
Though considered by some to be a fractal, the
The great Greek geometer Apollonius of Perga developed an algorithm which, given three mutually tangent circles (the large green circles), constructs two other circles tangent to the three (the blue-green circle at the center, and the large circle bounding the other four). The construction of the "inner" circle can then be repeated for each of the six circular "triangles" that result, and so on,
Beginning with a
Begin with a Poincaré chain of four circles with centers at the vertices of a rhombus, and a rectangle of lines, each of which passes through the center of a circle; these are shown faintly in the background. Construct the six colored circles so that each is orthogonal to a triple of the given objects. Reflect each of these across each of the original circles and lines, and so on, |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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Copyright © 2013 Michael Luis Ortiz. All rights reserved.