Fractals

Abstract

A Geometric Jewellery Store

[“Write Something interesting here”]]{.f5 .f4-m .f3-l .lh-copy .measure .mt0}

—

Inspiration: A Geometric Jewellery Store

Photo by Mykola/Kolya Korzh on Unsplash

An Introduction to Fractals

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

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!!

Fractals with Code

Using p5.js

Following the development at Koch Curve:

Using R

library(blogdown)
library(tidyverse)
library(ggformula)
###
library(LindenmayeR)
library(LearnGeom)
library(TurtleGraphics)
###
library(mosaicCalc) # Analytic Calculus
library(ambient) # Noise generation in R
library(plot3D) # 3D plots for explanation
library(caracas)
library(downloadthis)
library(knitr)
library(kableExtra)
## Markdown boiler plate stuff!!
# ![An Elephant](elephant.png){#fig-elephant}
# This is illustrated well by @fig-elephant.
#
# ### Figure Panel Divs
#     ::: {#fig-elephants layout-ncol=2}
#     ![Surus](surus.png){#fig-surus}
#     ![Hanno](hanno.png){#fig-hanno}
#      Famous Elephants
#     :::
#     Adding download buttons
#     data that has been read in
#     {{< downloadthis ../../../../materials/Data/housing_train.csv dname="house_prices" label="Download the House Prices Dataset" icon="database-fill-down" type="info" >}}
#    existing file
#    {{< downloadthis Orange/grouped-summaries.ows dname="grouped_summaries" label="Download the Orange Workflow" icon="database-fill-down" type="info" >}}

Koch Snowflake

x_min <- -10
x_max <- 10
y_min <- -12
y_max <- 12
CoordinatePlane(x_min, x_max, y_min, y_max)

NULL

# Create three points, to make a triangle
P1 <- c(-5, 0) # points
P2 <- c(5, 0)
P3 <- c(0, 10)

# Parameters for a Koch Fractal
# Iterative Function
angle <- 60 # Angle of new linesegments
cut1 <- 1 / 3 # location along existing line segment
cut2 <- 2 / 3 # location along existing line segment
f <- 1 # Scale factor. Positive
it <- 6
FractalSegment(P1, P2, angle, cut1, cut2, f, it)
FractalSegment(P1, P3, -60, cut1, cut2, f, it)
FractalSegment(P3, P2, -60, cut1, cut2, f, it)

# Koch
x_min <- -6
x_max <- 6
y_min <- -12
y_max <- 6
CoordinatePlane(x_min, x_max, y_min, y_max)

NULL

P1 <- c(-5, 0)
P2 <- c(5, 0)
P3 <- c(0, -10)
it <- 4
Koch(P1, P2, it)
Koch(P3, P1, it)
Koch(P2, P3, it)

#########################################
# Turtle Graphics stuff

turtle_init(width = 150, height = 150, mode = "cycle")

turtle_setpos(x = 50, y = 50)

turtle_do({
  for (i in 1:180) {
    turtle_forward(distance = 2)
    turtle_right(angle = 2)
  }
})

Fractal Tree with Turtle Graphics

fractal_tree <- function(s = 100, n = 2) {
  if (n <= 1) {
    turtle_forward(s)
    turtle_up()
    turtle_backward(s)
    turtle_down()
  } else {
    turtle_forward(s)

    a1 <- runif(1, 10, 60)
    turtle_left(a1)
    fractal_tree(s * runif(1, 0.25, 1), n - 1)
    turtle_right(a1)

    a2 <- runif(1, 10, 60)
    turtle_right(a2)
    fractal_tree(s * runif(1, 0.25, 1), n - 1)
    turtle_left(a2)

    turtle_up()
    turtle_backward(s)
    turtle_down()
  }
}

set.seed(123)
turtle_init(500, 600, "clip")

turtle_do({
  turtle_up()
  turtle_backward(250)
  turtle_down()
  turtle_col("darkgreen")
  fractal_tree(100, 12)
})

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. Turtle Graphics in R: https://cran.r-project.org/web/packages/TurtleGraphics/vignettes/TurtleGraphics.pdf
4. https://www.reddit.com/r/CitiesSkylines/comments/9r0y4e/grid_idea_im_working_on_fractal_squares/
5. Menger Sponge: https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-menger-sponge/
6. http://fractalfoundation.org/resources/fractal-software/

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.