Subsections
• Example: 2010
• Example: Loch Ness Monster

  
2010 and other pretty curlicues

Every journey begins with a single step. It is made up of many steps. Just so, any drawing is only a collection of straight line segments. To detail each segment we need

• its starting point;
• the angle at which it is to be drawn, measured in fractions of a complete revolution;
• its length, in multiples of some suitably small unit; and
• the colour and thickness of the line.

That's how the laser-printer does its thing, using its PostScript$_{\hbox{\small{\tiny$\scriptscriptstyle\circledR$}}}$ language.

  
Example: 2010

Let's start at any old spot. The ``Strategic Review of Mathematical Sciences Research and Advanced Mathematical Services in Australia'' covered the time-span 1995-2010, so we might draw a segment of unit length and at angle $1/2010$. To that we adjoin a unit segment at angle $8/2010$, and to that another such segment at angle $27/2010$, next one at angle $64/2010$, and so on. It's obvious (at least to any mathematician) that after 2010 segments we're back to where we started. What we get is the cover picture .

It is fortuitous that 2010, with cubes in the numerator, happens to create an attractive logo. I have known this for quite some years and have been using it on my preprints. (Also it appears on the dust jacket of my book ``Notes on Fermat's Last Theorem'' , recently published by Wiley-Interscience). It's fortunate that the logo is a propos in respect of the Strategic Review.

  
Example: Loch Ness Monster

Actually, 2010 is not really all that interesting, mathematically. Instead the ``Loch Ness Monster'', constructed sequentially by adjoining segments at angles $(log2)^4$, $(log3)^4$, $(log4)^4$, $(log5)^4$, $(log6)^4$, $(log7)^4$ and so on, is much more interesting. Some 5,000 segments appear in the picture shown here. But it's not just laziness that made me stop after, say, 5000 segments. Eventually the black-hole comprising the monster's tail grows ever larger, swallowing-up the entire picture. It is an instructive exercise to work out, without resorting to the printer, just how many complete curls comprise the monster. For this one must compute the locations of those `stationary' points, and also find those points near which the curve moves most rapidly.