Fractals

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Fractals are shapes which exhibit self-similarity. We’ll explore whatthis means shortly. The term fractal was coined in 1975 by themathematician Benoit Mandelbrot. The term roughly refers to beingbroken, or fractured.

Part of the motivation for studying fractals is to understand complexshapes, such as jagged coastlines.

One application of fractals is to image compression. Since graphicfiles are often quite large, figuring out a way to save the informationin a small format is important. Old formats saved color informationpixel by pixel, which is very inefficient. By using the self-similaritynature of fractals, the hope was images could be saved with muchsmaller file size.

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Self-Similarity

Self-similarity refers to an object that is exactly or approximatelysimilar to a piece of itself.

A boring example is a straight line. No matter how much you zoomin, it looks exactly the same.

The Wikipedia page en.wikipedia.org/wiki/Self-similarity has a goodgraphic for showing self-similarity of a fractal called the KochSnowflake.

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A circle is an example of an object that is not self similar as we nowsee.

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These pictures shows a circle and successive images obtained byzooming in further and further. The circle looks more and more like astraight line when zooming.

We experience this everyday by living on a sphere which feels mostlyflat to us.

We will mostly focus on the artistic aspect of some fractals.

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Clicker Question

Does the following picture exhibit some self similarity?

A Yes

B No

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The Mandelbrot Set

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The Mandelbrot Set is one of the most famous fractals. It is namedafter Benoit Mandelbrot. It has a complex mathematical description.Roughly, by starting with a given point and performing an interactiveprocess, points are colored by whether the sequence of points goes offto infinity and how fast this happens.

Here are some more images of the Mandelbrot set.

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Clicker Question

Does the following picture exhibit some self similarity?

A Yes

B No

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The Dragon (aka Jurassic Park) Fractal

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This fractal, shown on the previous screen, got its name by the factthat early iterations appeared in the book Jurassic Park, by MichaelCrichton.

In spite of the complex look of the fractal, it is built from very simplepieces. In fact, when one zooms in far enough, one sees that it isbuilt from line segments of the same length, connected at a 90 degreeangle.

We’ll illustrate the first dozen or so iterations. Going from oneiteration to the next is accomplished, roughly, by duplicating thedrawing and connecting the two halves.

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Fractals in the Flatland and Sphereland Movies

Fractals were used as background art in these movies. Here are acouple stills from Flatland.

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The following fractal shows up as an asteroid field in the movieSphereland.

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The Sierpinski Triangle

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The website

www.shodor.org/interactivate/activities/SierpinskiTriangle

shows how this fractal is built.

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The Koch Snowflake

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The following picture shows the self-similarity aspect of this fractal.

www.shodor.org/interactivate/activities/KochSnowflake shows how tobuild the snowflake.

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Clicker Question

If we draw the snowflake so that it is 1 inch across, how long is itsperimeter?

A 10 inches

B 100 inches

C 1,000 inches

D 10,000 inches

E Infinite

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Answer

Each time we iterate the construction the perimeter increases by 4/3The actual Koch snowflake then has infinite perimeter.

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Fractals and Randomness

Fractals can be created through certainrandom processes. We’ll illustrate thiswith some mathematical software. Oneof the images we’ll create is a fractalfern.

By plotting just a few points we’ll seerandom shapes. When plotting morewe’ll see the full shape.

We’ll see four fractals, including theSierpinski triangle and the fern.

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Next Time

We’ll end the semester by identifying some mathematical andscientific ideas that show up in an episode of the Simpsons. Duringthe episode Treehouse of Horror VI, one of the segments showsHomer getting trapped in the third dimension. We’ll watch thesegment and then discuss briefly the ideas the writers brought in.

The (optional) project and Assignment 9 are due on Friday.

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