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15.1: Fractals

Last updated

Jul 18, 2022
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- 15: Fractals
- 15.2: Iterated Fractals

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Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means through the rest of the chapter. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.

Self-similarity

A shape is self-similar when it looks essentially the same from a distance as it does closer up.

Self-similarity can often be found in nature. In the Romanesco broccoli pictured below[1], if we zoom in on part of the image, the piece remaining looks similar to the whole.

$clipboard_e4f4d3d1a6aa49cb35c9aa45f996359d1.png$

Likewise, in the fern frond below[2], one piece of the frond looks similar to the whole.

$clipboard_e1bf4eb5614e1f4c7b34a86870cc3cebd.png$

Similarly, if we zoom in on the coastline of Portugal[3], each zoom reveals previously hidden detail, and the coastline, while not identical to the view from further way, does exhibit similar characteristics.

$clipboard_e2bae29651141845db19e239d8a006205.png$

[1] en.Wikipedia.org/wiki/File:Ca...ractal_AVM.JPG

[2] http://www.flickr.com/photos/cjewel/3261398909/

[3] Openstreetmap.org, CC-BY-SA

Self-similarity

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