"Great Britain Box" by Prokofiev - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

Inspired by the Introduction to Complexity course and the unit on *Factals*, I though it would be fun to make a rough calculation of the fractal dimension of the Greek coastline using the box counting method.

The box counting method goes as follows:

- Split the 2D map that depicts the coastline into squares (boxes) of a certain side size (
*r*) and count the number of boxes (*n*) that include the coastline.
- Decrease the size of the boxes and go to step 1.
- When finished for a series of box sizes do a linear regression of log(n) given log(1/r).
- The slope of the line fitting the points on the plot is the fractal dimension of the object.

For a map of Greece, I used the one from

Ginko maps licensed under the Creative Commons Attribution 3.0. Via an image editor, I removed the frame with the infobox and geolocation axes, plus the borders not residing by the sea to facilitate further image processing. The retouched image was cropped to 1600×1600 pixels. Both images are shown below.

## Original map

## Retouched map

The R script below implements the box counting method on the coastline jpeg picture (make all values > 0.5 white) using boxes with sides that are divisors of 1600.

## The coastline map

## The R script

```
library(jpeg)
rm(list=ls())
img = readJPEG("coastline.jpeg")
# filter out mainland
img[img > 0.5] = 1
# divisors of 1600
# 1,2,4,5,8,10,16,20,25,32,40,50,64,80,100,160,200,320,400,800,1600
boxSizes = c(50, 40, 32, 25, 20, 16, 10, 8, 5, 4)
h = img[,,1]
data = data.frame()
for(k in 1:length(boxSizes)) {
b = boxSizes[k]
x = dim(img[,,1])[1]
ratio = x/b
# https://stat.ethz.ch/pipermail/r-help/2012-February/303163.html
k = kronecker(matrix(1:(ratio^2), ratio, byrow = TRUE), matrix(1,b,b))
g = lapply(split(h,k), matrix, nr = b)
counter = 0;
for(i in 1:length(g)) {
counter = counter + any(g[[i]] < 0.999)
}
data = rbind(data,c(log(counter),log(1/b)))
}
names(data) = c("Y", "X")
model = lm(Y~., data=data)
cat(coef(model), "\n")
```

## The plot

With this rough approximation, the calculation yielded that **the fractal dimension of the Greek coastline is 1.25**. Great Britain’s was measured to be 1.25 and Norway’s 1.52 [source].

## Comments