The beauty of fractals : six different views by Denny Gulick, Jon Scott

By Denny Gulick, Jon Scott

With the arrival of the pc age, fractals have emerged to play an important position in paintings photos, clinical program and mathematical research. the wonderful thing about Fractals is partly an exploration of the character of fractals, together with examples which look in artwork, and partially an in depth examine recognized classical fractals and their shut kin. the ultimate essay examines the connection among fractals and differential equations. The essays that seem within the great thing about Fractals include views diverse sufficient to provide the reader an appreciation of the breadth of the topic. The essays are self-contained and expository, and are meant to be available to a wide viewers that incorporates complicated undergraduate scholars and lecturers at either collage and secondary-school point. The booklet can also be an invaluable supplement to the fabric on fractals that are present in textbooks

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This fractal consists of three self-similar pieces, each of which is half the size of the entire set, but the top piece is rotated by 180 degrees while the bottom two pieces are not rotated. Again, we can go backwards. 4 Thomasina’s Geometry How did Thomasina produce an algorithm that yields a natural form? We will illustrate this with the simplest case, a fern. A leaf is a little more difficult and a little less spectacular than a fern. We need to describe an algorithm that, when iterated as in the chaos game, yields an image of a fern.

Thus the line segment AD has length l C l = C l. One can show that the ratio of AD to AC is . Thus triangles 4ACH , 4BDF and 4IEG are similar to 4ADG with factor 1= . The iterated function system which consists of the three similarities that send the largest equilateral triangle 4ADG to three triangles scaled by 1= , namely 4ACH , 4BDF and 4IEG, corresponds to a variation on the usual Sierpi´nski gasket that is also self-similar [1]. 2. 25. 1= ; 120 ı /. 1= ; 120 ı / along with the self-contacting trees with branching angles 118 ı and 122ı.

The Sierpi´nski hexagon. 26 Chapter 2. Chaos, Fractals, and Tom Stoppard’s Arcadia by Robert L. Devaney the lifelike image that Thomasina promised us, but there is a hint of what is to come. Look at the boundary of the innermost white region in the Sierpi´nski hexagon (or, in fact, the boundary of any internal white region). Note how this curve resembles a snowflake. Indeed, this boundary curve is the von Koch snowflake curve, another very famous fractal. This particular algorithm can also be expressed in terms of linear contractions of a given square.

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