Going Bananas With Hilbert

February 1, 2016
By

(This article was first published on Ripples, and kindly contributed to R-bloggers)

It seemed that everything is in ruins, and that all the basic mathematical concepts have lost their meaning (Naum Vilenkin, Russian mathematician, regarding to the discovery of Peano’s curve)

Giuseppe Peano found in 1890 a way to draw a curve in the plane that filled the entire space: just a simple line covering completely a two dimensional plane. Its discovery meant a big earthquake in the traditional structure of mathematics. Peano’s curve was the first but not the last: one of these space-filling curves was discovered by Hilbert and takes his name. It is really beautiful:
hilbert_n5

Hilbert’s curve can be created iteratively. These are the first six iterations of its construction:
hilbert

As you will see below, R code to create Hilbert’s curve is extremely easy. It is also very easy to play with the curve, altering the order in which points are sorted. Changing the initial matrix(1) by some other number, resulting curves are quite appealing:

hilbert3
hilbert5
hilbert4
hilbert1
hilbert8
hilbert7
hilbert6
hilbert2
hilbert9

Let’s go futher. Changing ggplot geometry from geom_path to geom_polygon generate some crazy pseudo-tessellations:

hilbert_polygons5
hilbert_polygons8
hilbert_polygons4
hilbert_polygons9
hilbert_polygons1
hilbert_polygons7
hilbert_polygons6
hilbert_polygons3
hilbert_polygons2

And what if you change the matrix exponent?

hilbert_mix4
hilbert_mix7
hilbert_mix6
hilbert_mix5
hilbert_mix1
hilbert_mix8
hilbert_mix2
hilbert_mix9
hilbert_mix3

And what if you apply polar coordinates?

hilbert_polar8
hilbert_polar3
hilbert_polar4
hilbert_polar7
hilbert_polar6
hilbert_polar2
hilbert_polar9
hilbert_polar1
hilbert_polar5

We started with a simple line and with some small changes we have created fantastical images. And all these things only using black and white. Do you want to add some colors? Try with the following code (if you draw something interesting, please let me know):

library(reshape2)
library(dplyr)
library(ggplot2)
opt=theme(legend.position="none",
          panel.background = element_rect(fill="white"),
          panel.grid=element_blank(),
          axis.ticks=element_blank(),
          axis.title=element_blank(),
          axis.text=element_blank())
hilbert = function(m,n,r) {
  for (i in 1:n)
  {
    tmp=cbind(t(m), m+nrow(m)^2)
    m=rbind(tmp, (2*nrow(m))^r-tmp[nrow(m):1,]+1)
  }
  melt(m) %>% plyr::rename(c("Var1" = "x", "Var2" = "y", "value"="order")) %>% arrange(order)}
# Original
ggplot(hilbert(m=matrix(1), n=1, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(1), n=2, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(1), n=3, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(1), n=4, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(1), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(1), n=6, r=2), aes(x, y)) + geom_path()+ opt
# Changing order
ggplot(hilbert(m=matrix(.5), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(0), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(tan(1)), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(3), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(-1), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(log(.1)), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(-15), n=5, r=2), aes(x, y)) + geom_path()+ opt
ggplot(hilbert(m=matrix(-0.001), n=5, r=2), aes(x, y)) + geom_path()+ opt
# Polygons
ggplot(hilbert(m=matrix(log(1)), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(.5), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(tan(1)), n=5, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-15), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-25), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(0), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(1000000), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-1), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-.00001), n=4, r=2), aes(x, y)) + geom_polygon()+ opt
# Changing exponent
gplot(hilbert(m=matrix(log(1)), n=4, r=-1), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(.5), n=4, r=-2), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(tan(1)), n=4, r=6), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-15), n=3, r=sin(2)), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-25), n=4, r=-.0001), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(0), n=4, r=200), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(1000000), n=3, r=.5), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-1), n=4, r=sqrt(2)), aes(x, y)) + geom_polygon()+ opt
ggplot(hilbert(m=matrix(-.00001), n=4, r=52), aes(x, y)) + geom_polygon()+ opt
# Polar coordinates
ggplot(hilbert(m=matrix(1), n=4, r=2), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(-1), n=5, r=2), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(.1), n=2, r=.5), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(1000000), n=2, r=.1), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(.25), n=3, r=3), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(tan(1)), n=5, r=1), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(1), n=4, r=1), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(log(1)), n=3, r=sin(2)), aes(x, y)) + geom_polygon()+ coord_polar()+opt
ggplot(hilbert(m=matrix(-.0001), n=4, r=25), aes(x, y)) + geom_polygon()+ coord_polar()+opt

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