Working with veneer is a pain in the ass.  It's thin, light, and ridiculously fragile.  After spending lots of time trying to arrange up to 100 little wood pieces perfectly while minimizing gaps, I've figured out a reasonably good workflow for assembling the laser-cut spirals. 

Here's how it works:

Once I have all the pieces cut, I prepare a jig by laser-cutting a set of lines matching the final assembly into a piece of wood.  Then, I place a transparent sheet on top:



( Cut for people who just like the finished product :-) )