Renormalisation

Yet more interesting cases are those for which the curve occupies more than a finite region. Pick any nasty irrational number; $\pi$ is a good example. Now let the rule for the angles be $n^2\pi$ -- so the angles are sequentially $\pi$, $4\pi$, $9\pi$, $16\pi$, ...revolutions.

There are two difficulties in drawing the complete picture:

• there are too many segments--infinitely many of them;
• the complete picture takes up too much space--an infinite amount.

Nevertheless, what to do is clear. Suppose we have just a few (say 10,000) segments in our field of view. Now change the magnification of our microscope by pulling it away. Then we may be able to see perhaps 100,000 segments; but their detail is blurred. What we actually can see is easily portrayed using just some 10,000 `renormalised' segments. Here are some of the first few renormalisations of $\pi$.

(Click images for a larger version.)

The preceding pictures are the 0$^{\mathrm{th}}$, 2$^{\mathrm{nd}}$ and 4$^{\mathrm{th}}$ renormalisations of $\pi$. Each consist of some 4000 actual segments. Now the third picture corresponds to many millions of segments of the original $\pi$ curlicue. Indeed the first black `blob' on the right, together with its initial looping tail, is its starting tens-of-thousands of segments. The first two pictures are effectively magnifications of this portion.

(Click images for a larger version.)