Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior.

Here's an example:

enter image description here

(Note that I'm considering the grey "negative space" here, not the graph made by the black "walls.")

In particular, this breaks an rectangle up into a number of disjoint paths. What can be said about the length of such paths, say asymptotically as $m, n \to \infty$ while $m / n$ remains constant? (To make things a bit cleaner, it may make more sense to identify the edges to make a torus instead of leaving the edges ragged.

We can get some initial results just by looking at Euler characteristic, as described by Anders Kaseorg at the link below

See also the discussion here:

Example in UNICODE if you can't see the picture:


(Thanks to Alon Amit for the sample picture.)

share|cite|improve this question
can't see the picture... – gt6989b Dec 3 '12 at 23:22
Maybe you got it right while I was editing? Does it show up if you reload? – Daniel McLaury Dec 3 '12 at 23:23
Found the problem, it was on my side, sorry for the trouble... – gt6989b Dec 3 '12 at 23:25
OT, but I cannot help myself... This reminds me of one of the many, many interesting things in the book "Programming the PET/CBM" by Raeto West. It was also my first introduction to systems of orthogonal polynomials. Is that Commodore BASIC by the way? ;-) – WimC Dec 11 '12 at 21:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.