Working with Shape: Iterated Functions

Iterated Functions
Fractals
L-Systems
Mandelbrot
Julia
Algorithmic Patterns
Symmetry
Published

May 2, 2024

Modified

November 11, 2024

Introduction

Topics that may be covered here:

  • Iterated Function Systems and Fractals
  • Symmetries in Space
  • L Systems, Mandalas and Kolams
  • Logistic Functions
  • Projections

Inspiration

This equation will change how you see the world. It’s about the Logistic Map, bifurcation diagrams, the Mandelbrot set, animal populations, dripping faucets, neuron firing rates and more.

So let us see how we can construct this algorithmically, the different parts of it. And of course why we should bother!

What is an Iterated Function?

Well, we know what a function is don’t we? A relationship between numbers. One varies as the others, and the relationship is specified by the function. E.g. \(y = \sin(x)\).

So what is iteration then, and how do you do that to a function? It is applying the same function over and over again, by piping the results of the previous step i.e. iteration back into the function. Let us hear from Ron Eglash, the author of African Fractals: Modern Computing and Indigenous Design

Wait, But Why?

References

  1. Working with Shadows. https://www.wikiwand.com/en/Map_projection

  2. Working with Fourier Series and Epicyles http://www.jezzamon.com/fourier/index.html and https://alex.miller.im/posts/fourier-series-spinning-circles-visualization/

  3. https://twitter.com/i/status/962449509782495232 https://codegolf.stackexchange.com/questions/36374/redraw-an-image-with-just-one-closed-curve

Generative Art

  1. Art from Code: Danielle Navarro. https://art-from-code.netlify.app

  2. https://generative.substack.com/p/generative-art-and-r

  3. https://github.com/cutterkom/generativeart

  4. https://paulvanderlaken.com/2020/05/02/generative-art-computer-design-painting/

  5. https://www.bigbookofr.com/art.html#thinking-outside-the-grid---a-bare-bones-intro-to-rtistry-concepts-in-r-using-ggplot

  6. https://clauswilke.com/art/

  7. https://generatecoll.medium.com/how-i-used-excel-to-create-abstract-album-artwork-fee740d4414f

  8. Random Digital Beauty. https://anaselk.com/p/generative-r/

  9. https://nannou.cc

  10. https://openframeworks.cc/

  11. https://libcinder.org

  12. https://cgjennings.ca/articles/l-systems/

  13. https://paulbourke.net/fractals/

Kolams, in R and other tools

  1. Reddy, Anu, and Alex McLean. 2024. “Drawing Kolam Patterns in Stitches and Code.” Algorithmic Pattern, March. https://doi.org/10.21428/108765d1.53f112a6.

  2. Ezine for Kolam: ( Bad Link) https://files.cargocollective.com/c989887/Kolam_zine--1-.png

  3. https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_designs

  4. https://www.atlasobscura.com/articles/indian-rice-art-kolam

  5. https://www.sciencedirect.com/science/article/abs/pii/0146664X74900112

  6. https://www.sciencedirect.com/science/article/pii/S0166218X09000845#b26

  7. http://interdigitation.embodimentlabs.org/

  8. https://www.jeremykun.com/2014/09/29/hybrid-images/ Using Fourier Series to morph images !!!

  9. https://nb.paulbutler.org/l-systems/

  10. http://algorithmicbotany.org/papers/#abop

  11. https://acko.net/blog/how-to-fold-a-julia-fractal/

  12. https://algorithmicpattern.org

  13. Steven Witten. https://acko.net/

  1. Brian Eno. 77 Million Paintings. https://www.wikiwand.com/en/articles/77_Million_Paintings

Book References

  1. Julien Sprott. https://sprott.physics.wisc.edu/fractals/booktext/SABOOK.PDF

  2. Marcia Ascher, “Ethnomathematics”

  3. Claudia Zaslavsky, “Africa Counts”

  4. Ron Eglash, “Africa Fractals”

  5. Paulus Gerdes, “Adventures in the World of Matrices”

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