There seems to be an infinite number of things possible with a laser cutter. I first wanted to make an automata of sorts, then moved on to thinking about origami, and what interesting structures could be possible... though it seemed like the laser cutter would only really be useful for scoring, so this didn't seem like the best route – though perhaps there's more to explore here (at some other time), with rigid folding. Origami made me think of something mathematical, which made me think of doing something involving code. The helicone desktop toy came to mind, which has a harmonic-y feel to it.. they use some stoppers or something to neatly settle at each state though, which also  felt like it would involve a lot of gluing and a big time investment.  

helicone desktop toy

I liked the idea of doing something with waves / multiple units slightly offset to create additional structure. I found a lot of parametrically designed lamp shades like this –

https://www.etsy.com/listing/181366392/twisted-lasercut-wooden-lampshade-no2
https://www.etsy.com/listing/169151657/twisted-lasercut-wooden-lampshade-no1?ref=related-3

I could figure how to do this with a circular base, but for a square or triangle base it seemed it would be hard to get the angles of interlocks just right without some software designed for this sort of thing (edit: later figure out how to do this – project the angles from a circumscribed circle to the square/triangle). Instead of vertical pieces, I wondered if I could manage with horizontal pieces, stacking up – vertical pieces would have to be pretty close together to get a nice effect, and the light coming through could stilll look a little harsh

stacking creates nicer curves (though this is 'horizontally stacked', if thats a thing)
since the gaps between slices are more even, you don't see the harsh light of the bulb directly

I wasn't sure how to create support panels with the correct angles, but tried visualizing with code a bit, which made me realize that I can safely assume that there will be a circle at the center of a full rotation

And so I could create any form/shape, leaving a circle hole in the middle, and rotating the notches that interlock with the support pillar.

While trying to find some references of different material, I found some really interesting stuff done with acrylic :

'Gweilo' lights by The Partisans (http://www.welovelamp.ca/gweilo/)
Leandro Erlich’s “La Vitrina Cloud Collection” 

I will probably end up going with ply/mdf for this, but a trip to canal plastics is due to see if there's some acrylic that jumps at me – maybe etch the layers as well..

Updates:

I ended up buying some white frosted acrylic from canal plastics – I didn't like the plastic-y look of the clear acrylic, and the frosted sheets still allowed light to hit the edges in a nice way. For the supports, I bought a strip of walnut wood from blick.

materials 

Initially I had something biggish in mind, but I realized how much material it would require and decided to scale down, with the constraint of trying to fit all layers of the stack on a single sheet of 24"x12" acrylic. I had wanted to try a gentle change in size of each layer, but figuring out a good packing method to get it all in one sheet at different sizes seemed like a task for another time, once I had the basics figured. I went with 1.5mm acrylic sheet instead of the 3mm I was initially thinking of, as it felt more proportionate for the anticipated final size. I decided to go with 1/4" thick wood, to provide as much support material as possible for the acrylic to slide on to, and hopefully to prevent breakage of the 'teeth' – since they would be very short in height, I wanted to max out on the width. The thickness of the wooden strip would become the 'spine' of the support pillar.

It took a while of measuring and lots of scribbles and notes to figure out the exact dimensions of all the parts – budgeting for some wiggle room. While initially I was going for a rounded triangle, I found it to be too geometric and drew out more organic curves instead.

1/4" wood was actually 1/4"!

The rotation+repetition would have been a perfect task to automate, but figuring out clean SVG exports seemed like a whole other thing. After wrestling with illustrator for a bit, I finally had my files ready. I made copies of the files and reduced the number of elements – to use as test files.

The design I had for the support pillars translated to extremely thin wedges that would fit between the material – and so I started with testing the feasibility of this to determine if I needed larger gaps between each layer of acrylic. The recommended settings for the wood didn't really work – the cuts didn't reach the other side. On Danny's advice, I tried a second pass at slower speed (lowering the speed from 10 to 4). This created very charred wood, but it worked.

cuts didn't go all the way through

I then went on to cut with my test file for acrylic. Good thing I made a test file – I had messed up on the measurements! For each of my file, I had planned for a .2" margin on the edges, to account for errors in measurement of the acrylic or slight offset with the laser. I accidentally set up one of my guides wrong, and so lost half a piece.

incorrect measurements on illustrator! 

I tested the pieces with the wooden support and the two acrylic pieces that cut fine — everything seemed to fit well, so measurements seemed good enough to proceed with.

testing with 2 layers

I got about to fixing the illustrator file, and struggled with getting all my pieces to fit in the reduced measurements. So I reduced the margin from .2" to .1". This just about worked – the edges of the cuts were right on the edge of acrylic – something to keep in mind for next time.

the edge of the bottom row is right on the edge of the acrylic sheet

As Ben promised, the laser is weaker on the bottom-right corner. The last piece took a bit of effort to push out, and I ended up cracking it while pushing the center circle out.

sad cracked piece of acrylic

While cutting the other 2 supports from the wood, I thought perhaps one pass at slower speed would be enough. The laser reached the back this time, but the cuts were too faint for me to push the piece out. I ran another pass at a higher speed (15), to closely replicate the 2 passes I used for my first piece. I thought increasing the speed of the fast-pass might reduce the chance of charring. This led to an interesting outcome – the thin teeth of the wood burnt and turned to ash a bit. I'm not sure why this happened – even though I had changed the order of my passes and increased the speed for the fast-pass, intuitively it seemed that the results should be the same. Something to learn more about.. how to make adjustments based on what you see.

wood is turning to ash :/
assembling

Assembling the piece took a bit of effort, but I managed finally, and like the way it looks with light. I realized I messed up on the measurements for the supports – the idea was for them to be flush against the acrylic on the inside, and not extrude on the outside – simple mistake of not accounting for the diameter of a circle being double the radius! The wood is a little heavy looking for the size.

Since the acrylic was so thin, the edges ended up been smooth curves – probably melting a bit with the heat of the laser – not exactly the crisp hard edges I had in mind, but nice in its own way. I'm going to see if I can get a small bulb holder to fit well with this. Also glad it ended up being small so I could see the potential issues (teeth too thin, trying to cram too much stuff onto one sheet, figuring out setting for cutting through hardwood, not quadruple-checking my measurements). I would have preferred a more pronounced angle of rotation for each layer, amplifying the twist effect, and modulation in the size of each layer as well. I also learnt about the software Slicer by Autodesk after I finished this – definitely need to look into this for future iterations.

the unintentionally extruding wood could work as a technique for some interesting effects

Some other interesting references I came across:

the sharp edge and visible weave is interesting (https://www.techzug.com/furniture-tech/amazing-aurora-lamp.html).. didn't notice this till later, would have been cool to try a sharper edge. not sure how this is supported..
material reflects/diffuses the light nicely (http://www.theshift.in/portfolio/partart)
etching on multiple pieces that then interlock, amplifying the optical-illusion feel (https://retaildesignblog.net/2016/09/15/uri-led-light-bulb-by-nap/)
etching generative textures
feels a bit like something from a hacker movie, but clear acrylic edge-lighting is interesting