easy-peasy infinity mirror
I just threw together an infinity mirror with no carpentry at all, no glass cutting, in fact, naught but stuff you can buy (cheap) and some cyanoacrylate glue. The approximate cost is about $50 US if you're lucky and the shadow box is still on sale.
Note: Click on the pictures to get bigger versions of same.
The curvature you see in the display is because the plastic mirror at the back is slightly convex. Were it made of glass, this wouldn't happen, but then I'd have to cut glass, or pay someone else to.
For the lights inside the box, I used a 1 meter LED strip that I picked up when Radio Shack dumped a lot of stuff when they "transitioned". Any 1 meter LED strip should work for this project. banggood.com sells one like this for about $12 US.
For the box, I used an off the shelf "shadow box" that I picked up at JoAnne Fabric (a US chain). You can also find them on Ebay. They (were) on sale at JoAnne for $12 or so when I bought mine (early November, 2015). The brand is "Collector's Museum" and it's the 10" x 10" model. It comes in white, black and "bronze", I think, but the bronze was more expensive. Here's the current link.
This box is nice because it comes apart -- there's a "spacer" inside to separate the front glass from the rear piece of masonite (or cardboard, hard to say), and the spacer comes right out, to be replaced by your LED strip. Just pry up the little metal tabs on the back (on three sides is enough) and slide all the stuff out of the frame.
The shadow box is a 10" x 10" (inside dimension), which circumference is (duh) 40 inches, which means a 1 meter LED strip will fit about perfectly. I had to trim about a quarter of an inch off the "dead" end to get it to fit in the box.
The Radio Shack LED strip is covered in a silicone tube, and my favorite cement, Goop, won't stick to it, at all. Happily, cyanoacrylate will stick to it.
Additional parts you need: a piece of half-silvered "window tint" stuff. I got mine from an eBay vendor, but you could probably get it from a custom car shop or even an auto parts store, though there you might end up with a lot more than you need.
You'll also need a mirror for the back. I bought a plastic mirror "tile" off a vendor on Ebay, it's 1 foot by 1 foot, and since it's plastic, it's not too hard to cut it down to 10" x 10". I used a band saw, because the 1/8" thick mirror was taking too long to cut with a razor knife. A glass shop could probably cut you a proper glass mirror at the right size, if you think the plastic is too cheesy.
Disassemble the shadow box. Take the glass, clean it really well, and apply the half mirror plastic film to it according to the directions that should have come with the film.
Cut the mirror tile to 10" by 10" (the same size as the back cover of the shadow box).
Put the half mirrored glass in the shadow box (I don't think it matters which side is out, I put the glass side out for durability).
Test fit the LED strip around the inside of the box frame (LEDs pointing inwards, duh). If it doesn't fit, figure out how to trim it so it will. Once satisfied, glue the strip in. I used cyanoacrylate, but your strip might not have a silicone envelope, so you may have to experiment on what works best. Once the cyanoacrylate cures, you will probably have to wipe off the white residue that will be left on the half-mirror surface. I think the cyanoacrylate out-gasses as it cures.
Once the LED strip glue is cured, peel off the protective paper and lay the plastic mirror on top of the LED strip, mirror facing the glass. "Behind" that, put the original back of the shadow box. Fold the little metal tabs down to capture the back inside the shadow box frame. Be sure to snake the wire that drives the LED strip out. The fit between the mirror and the box frame was loose enough that I just ran the wire out through that gap.
Costs: (US dollar)