Complex Numbers

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

May 2, 2024

Modified

December 18, 2024

Introduction

Inspiration

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.

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. Complex generative Art in p5.js. https://cratecode.com/info/p5js-generative-art-complex-functions

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

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

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

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

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

  7. https://clauswilke.com/art/

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

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

  10. https://nannou.cc

  11. https://openframeworks.cc/

  12. https://libcinder.org

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

  14. 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. Frank Farris. Creating Symmetry.

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

  3. Marcia Ascher, “Ethnomathematics”

  4. Claudia Zaslavsky, “Africa Counts”

  5. Ron Eglash, “Africa Fractals”

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

R Package Citations
Package Version Citation
ambient 1.0.2 Pedersen and Peck (2022)
mosaicCalc 0.6.4 Kaplan, Pruim, and Horton (2024)
plot3D 1.4.1 Soetaert (2024)
Kaplan, Daniel T., Randall Pruim, and Nicholas J. Horton. 2024. mosaicCalc: R-Language Based Calculus Operations for Teaching. https://CRAN.R-project.org/package=mosaicCalc.
Pedersen, Thomas Lin, and Jordan Peck. 2022. ambient: A Generator of Multidimensional Noise. https://CRAN.R-project.org/package=ambient.
Soetaert, Karline. 2024. plot3D: Plotting Multi-Dimensional Data. https://CRAN.R-project.org/package=plot3D.
Back to top

Citation

BibTeX citation:
@online{2024,
  author = {},
  title = {\textless Iconify-Icon Icon=“tabler:math-Xy” Width=“1.2em”
    Height=“1.2em”\textgreater\textless/Iconify-Icon\textgreater{}
    {Complex} {Numbers}},
  date = {2024-05-02},
  url = {https://av-quarto.netlify.app/content/courses/MathModelsDesign/Modules/2-Geometry/20-ComplexNumbers/},
  langid = {en}
}
For attribution, please cite this work as:
“<Iconify-Icon Icon=‘tabler:math-Xy’ Width=‘1.2em’ Height=‘1.2em’></Iconify-Icon> Complex Numbers.” 2024. May 2, 2024. https://av-quarto.netlify.app/content/courses/MathModelsDesign/Modules/2-Geometry/20-ComplexNumbers/.