# Fractals with R

by Allan Roberts

Many biological objects have structure that, at least at a glance, may appear rather fractal-like. A brief discussion of the applicability of fractals and fractal dimension to biology can be found in Vogel’s ‘Comparative Biomechanics‘ (2003, pp. 84–86). While noting that there seem to be few truly fractal objects in biology, Vogel speculates, “I think, […], that sponges may be fractal, with large ones organized mainly as foldings and refoldings of proliferated small ones, thereby making transitions between the organizational grades we learned as ascons, sycons, and leucons” (p. 86).

Figure 1. Four iterations in the construction of a Sierpinski triangle-like object. I produced these graphics using the statistical programming language R (R Development Core Team, 2011).

I’ve included an image, representing the construction of a fractal largely similar to a famous fractal, the Sierpiński triangle. A mathematical description of the Sierpiński triangle can be found in Vejnar (2012). (In contrast to the actual Sierpiński triangle, my example is based on right-angled triangles, rather than on equilateral ones.) These images were produced using the statistics application and programming language R (R Development Core Team, 2011). I’ve also included the R script that I used to produce the graphics. For more, see Fractals with R, Part 2.

```R Script
IterateTriangle <- function(A){
B <- cbind(A,0*A);
C <- cbind(A,A);
D <- rbind(B,C);
return(D);
}

par(mfrow=c(2,2))
for (i in 1:4){
T <- matrix(1,1,1)
for (i in 1:i) T <- IterateTriangle(T);
image(T,col=c("white","black"),axes=FALSE);
text(0,1,i,col="black")
}```

References

R Development Core Team, 2011. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Vejnar, Benjamin. 2012. A topological characterization of the Sierpiński triangle.       Topology and its Applications, 159 (5), 1404-1408.

Vogel, Steven. 2003. Comparative Biomechanics: Life’s Physical World. Princeton University Press: Princeton, New Jersey.