Working With Shape: Fractals

Abstract

A Geometric Jewellery Store

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A Geometric Jewellery Store

Photo by Mykola/Kolya Korzh on Unsplash

Inspiration

An Introduction to Fractals

Let us listen to the late great Benoit Mandelbrot, on the Art of Roughness.

Complex Planes: What am \(i\)?

Dude, what’s the square root of -1?? 🙀 🙀 🙀!! And, what can you do repeatedly to arrive at \([-1, 0]\)?

Let us head off to https://mathigon.org/course/fractals/mandelbrot and play with some iterated functions in the Complex Plane. This will lead us into an intuitive understanding of Julia and Mandelbrot Fractals.

Julia, and Julia, and still more Julia….

Jewellery Shopping with Mandelbrot

Ever gone shopping for jewellery? How is the place organized? That is a good metaphor for how Mandelbrot Set is generated!!

Using XaOS

Let us use the XaOS software to make different kinds of fractals. A sample screen to explore the Mandelbrot fractal is here:

Wait, But Why?

To be Written Up.

References

1. Batty, M. and Longley, P. A. (1994) “Fractal Cities: A Geometry of Form and Function”, London: Academic Press, 1994.

2. Wang H, Luo S, Luo T. Fractal characteristics of urban surface transit and road networks: Case study of Strasbourg, France. Advances in Mechanical Engineering. 2017;9(2). doi:10.1177/1687814017692289

3. https://www.reddit.com/r/CitiesSkylines/comments/9r0y4e/grid_idea_im_working_on_fractal_squares/

4. Menger Sponge: https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-menger-sponge/

5. <http://fractalfoundation.org/resources/fractal-software/.