Introduction to Fractals

A short while ago I was contracted to write a short piece entitled “Introduction to Fractals”. The article can be found here. Admittedly it is hard to do justice to the topic in less than 1000 words. Both of the illustrations were created with R.

Mandelbrot Set

The Mandelbrot Set image was created using the Julia package.

library(Julia)

First an image of the entire set showing a region that we will zoom in on later.

npixel <- 2001
#
rshift = -0.75
ishift =  0.0
#
centre <- rshift + ishift * 1i
width <- 3.0
#
mandelbrot <- MandelImage(npixel, centre, width)

gray.colors <- gray((0:256)/ 256)

mandelbrot = t(mandelbrot[nrow(mandelbrot):1,])

x = 0:npixel / npixel * width - width / 2 + rshift
y = 0:npixel / npixel * width - width / 2 + ishift
#
image(x, y, mandelbrot, col = rev(gray.colors), useRaster = TRUE, xlab = "Real", ylab = "Imaginary",
      axes = FALSE)
box()
axis(1, at = seq(-2, 2, 0.5))
axis(2, at = seq(-2, 2, 0.5))
rect(-0.325, 0.75, 0.075, 1.15, border = "black", lwd = 2)
points(c(0.5, 0), c(0, 1), pch = 19)
text(c(0.5, 0), c(0, 1), labels = c("A", "B"), adj = c(1.55, -0.3))

mandelbrot-full

Then the zoomed in region.

npixel <- 2001;
#
rshift = -0.125
ishift =  0.95
#
centre <- rshift + ishift * 1i
width <- 0.4
#
zoom.mandelbrot <- MandelImage(npixel, centre, width)

zoom.mandelbrot = t(zoom.mandelbrot[nrow(zoom.mandelbrot):1,])

x = 0:npixel / npixel * width - width / 2 + rshift
y = 0:npixel / npixel * width - width / 2 + ishift
#
image(x, y, zoom.mandelbrot, col = rev(gray.colors), useRaster = TRUE, xlab = "Real",
      ylab = "Imaginary", axes = FALSE)
box()
axis(1, at = seq(-2, 2, 0.05))
axis(2, at = seq(-2, 2, 0.05))
points(c(0.5, 0), c(0, 1), pch = 19)
text(c(0.5, 0), c(0, 1), labels = c("A", "B"), adj = -0.75)

Cantor Set

The Cantor Set illustration was naturally created with a simple recursive algorithm.

cantor.set <- function(x) {
    y = list()
    for (n in x) {
        nL = n[1]
        nR = n[2]
        nl = nL + (nR - nL) / 3
        nr = nR - (nR - nL) / 3
        y[[length(y)+1]] <- c(nL, nl)
        y[[length(y)+1]] <- c(nr, nR)
    }
    return(y)
}

C = list()
#
C[[1]] = list(c(0, 1))
C[[2]] = cantor.set(C[[1]])
C[[3]] = cantor.set(C[[2]])
C[[4]] = cantor.set(C[[3]])
C[[5]] = cantor.set(C[[4]])
C[[6]] = cantor.set(C[[5]])
C[[7]] = cantor.set(C[[6]])
C[[8]] = cantor.set(C[[7]])
C[[9]] = cantor.set(C[[8]])
C[[10]] = cantor.set(C[[9]])

par(mar = c(4.1, 0.0, 2.1, 0.0))
plot(NULL, xlim = c(0,1), ylim = c(0, 7), axes = FALSE, xlab = "", ylab = "")
axis(1)
#
for (n in 1:7) {
    for (p in C[[n]]) {
        # print(p)
        lines(p, c(8-n ,8-n), lwd = 8, lend = "butt")
        text(x = 0, y = n, 7 - n, adj = c(3, 0.5))
    }
}

Categorically Variable