There are lots of great projects you can build with laser-cut and hand-cut PMMA plexiglas (acrylic) projects. Sometimes these projects don't look great because of the ends - cut, broken, sanded; these ends aren't completely transparent and professional.
Step 1: Prepare your edge - surface preparation
Most PMMA (Acrylic) edges that are laser cut are automatically smooth, mostly due to the laser beam thinness and the fact that the cut is made with heat from the UV laser light doing the cutting. But, if the cut is uneven or if some finishing or adjustment is needed, or if acrylic is cut by hand, then the edge will exhibit cut marks and other surface roughness. It won't be shiny, but instead matte and uneven.
Project aesthetics is affected by the smoothness of the surfaces when working with acrylics, and if you use the acrylic to carry light (as in another Instructable) then its critical to make that light-carrying edge as smooth and flat as possible. Fire-polishing does just that.
Because the process slightly melts and reforms the rough material, it's absolutely critical to minimize the roughness of the edge before fire-polishing. It's not so different from other types of surface polishing -- to get the final mirror shine, you need the most rough parts sanded down flat.
So, first get some sandpaper or a sanding block. Make sure the surface of the sandpaper or the block are completely flat, and begin sanding slowly against the surface. Since plastic is affected by heat, make sure to sand with a bit of water, which also carries away the waste material and keeps the paper or block clear and continuing to sand. Also, where possible keep the protective film on the main surface of the item so it won't become accidentally scratched as you grip it.
I started with 400 grit material until the surface was evenly flat and matte - I regularly reviewed the edge under magnification and used a light to reflect and reveal the unsanded surfaces. Once that was complete, I moved to 800 grit sandpaper, again maintaining a flat sanding surface. Check the edge regularly, as this process doesn't require a terribly large amount of sanding or great force to achieve the effect. I stopped at 800, but you could easily go on to 1500 grit for an even better and smoother effect.
Once the edge is very flat and regular on its surface, you clean the surfaces with denatured (or 91%) alcohol, and then move to the next step.