Building Affine Transformation Fractals With R

Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line (Benoit Maldelbrot)

Fractals are beautiful, hypnotics, mysterious. Cantor set has as many points as the real number line but has zero measure. After 100 steps, the Koch curve created from a 1 inch segment is long enough  to wrap around the Earth at the equator nearly four thousand times. The Peano Curve is a line that has the same dimension as a plane. Fractals are weird mathematical objects. Fractals are very cool.

One way to build fractals is using the Multiple Reduction Copy Machine (MRCM) algorithm which uses affine linear transformations over a seed image to build fractals. MRCM are iterative algorithms that perform some sort of copy+paste task. The idea is quite simple: take a seed image, transform it (clonation, scalation, rotation), obtain the new image and iterate.

To create the Sierpinsky Gasket Fractal you part from a square. Then you divide it into three smaller squares, locate them as a pyramid and iterate doing the same with avery new square created. Making these things is very easy with grid package. Defining the division (i.e. the affine transformation) properly using viewPort function and navigating between them is all you need. Here you have the Sierpinsky Gasket Fractal with 1, 3, 5 and 7 levels of depth. I filled in squares with random colours (I like giving some touch of randomness to pictures). Here you have pictures:

Rplot01 Rplot03

Rplot05 Rplot07

And here you have the code. Feel free to build your own fractals.

library(grid)
rm(list = ls())
grid.newpage()
pmax <- 5 # Depth of the fractal
vp1=viewport(x=0.5,y=0.5,w=1, h=1)
vp2=viewport(w=0.5, h=0.5, just=c("centre", "bottom"))
vp3=viewport(w=0.5, h=0.5, just=c("left", "top"))
vp4=viewport(w=0.5, h=0.5, just=c("right", "top"))
pushViewport(vp1)
m <- as.matrix(expand.grid(rep(list(2:4), pmax)))
for (j in 1:nrow(m))
{
for(k in 1:ncol(m)) {pushViewport(get(paste("vp",m[j,k],sep="")))}
grid.rect(gp=gpar(col="dark grey", lty="solid",
fill=rgb(sample(0:255, 1),sample(0:255, 1),sample(0:255, 1), alpha= 95, max=255)))
upViewport(pmax)
}
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