I'm sure someone on rentacoder can figure it out:

http://blog.willbenton.com/2008/11/rent-a-coder-hilarity/

If it's a limited number of states in the automaton, and a limited amount of "tape" in the canvas, it's not a Turing machine, it's a Finite-State Automaton, which is easy to analyze for loops.

Here's one that halts very late: http://wry.me/hacking/Turing-Drawings/#3,3,1,2,0,1,1,0,0,2,3...

And here's an amazing one: it fills the canvas, getting slower and slower. It will probably take days to complete fill it: http://wry.me/hacking/Turing-Drawings/#10,3,9,2,1,5,2,3,8,1,...

The world is finite, so every drawing eventually loops.

However, the state space is something like k * 2^18 * n^(2^18), where k is the number of states and n the number of symbols. For the default (4, 3) this is 3.125 * 10^125080, so you could be waiting for a while :)

that's a hard design decision.