Birthday Cards

But I've become too serious, hence the preceding unexplained decoration. When I first wrote on this subject in 1988 with my colleague Ross Moore, we began by analysing John Loxton's monster and continued by explaining renormalisation. We concluded by displaying some attractive curlicues--one of which was 2010. That idea has been refined by David Angell (from UNSW) who supplied me with the pictures shown in this presentation. More than that, he had the felicitous idea of using the angle rule $n/d+n^2/m+n^3/y$, thereby creating a curve for each date $d:m:y$. This allows one to create designs for individualised birthday cards. My birthdate is unattractive; I'm jealous of the 10,620 segments of David's.
Other rules are possible; the next examples include things like $n/d+n^2/m+n^4/y$ or $n/d+n^3/m+n^5/y$ and changing the order of the powers of $n$.

(Click images for a larger version.)

${1\over21}n+{1\over9}n^2+{1\over57}n^3$

${1\over12}n+{1\over8}n^2+{1\over95}n^3$

${1\over16}n+{1\over5}n^3+{1\over42}n^4$

${1\over16}n+{1\over5}n^4+{1\over42}n^5$
Christmas 1994: ${1\over25}n+{1\over12}n^2+{1\over94}n^4$

...more good tidings: ${1\over25}n+{1\over12}n^3+{1\over94}n^5$

Here are some more examples using different formulæ. The variability attainable in the nature of the curves is quite striking.

${1\over7}n+\cos(n)$

${1\over515}n^3$

${1\over16}n+{1\over5}n^2+{1\over42}n^4$