Update 11/22/2013: Thanks to everyone who voted for this project in the Microcontroller Contest! It was one of three first-prize winners.
Update 9/17/2013: Thanks to everyone who voted for this project in the Arduino contest (I was one of ten "second prize" winners)! If you want to try this project with an addressable LED strip instead of an analog strip, check out the Rainbow Jar project (also an Arduino contest winner).
This is my take on a combination of two classic projects: RGB LED control with an Arduino, and an Infinity Mirror. It's an RGB LED infinity mirror that lets you toggle between an adjustable-speed color-fade mode and a direct-control mode where you individually set the red, green, and blue LED brightness levels. The primary inspiration for this particular project comes from this infinity mirror Instructable and Adafruit's RGB LED Strip tutorial, but there are many more quality resources out there on both projects.
I've done my best to gear this project towards newbies by providing an exact list of materials I used and the exact procedure that I followed. One recurring theme I've noticed in comment sections for other infinity mirrors is a lack of links to specific parts (e.g. exactly what type of LEDs or LED strips were used, what power supply, where to buy the mirrors, the enclosure...). Clearly, if you know what you're doing and want to spend more (or less) money to design a slightly different mirror, you can adjust your materials as needed, use a different Arduino board, etc. You can skip the Arduino entirely and make a pretty simple, cheap infinity mirror if you want (just search Instructables for "infinity mirror" and you'll find a few), or go crazy and spend hundreds if not thousands of dollars (search YouTube for "infinity mirror table" and you'll get the idea).
So, on to the materials list. Remember that this is an exact list of parts that I used, but I gradually cobbled together the supplies for this project over a long period of time. I didn't sit down, compare vendors (e.g. Adafruit vs. Sparkfun) and find the absolute cheapest way to build this. So, feel free to shop around to bring down the cost (and post links in the comments if you find a better/cheaper version of a certain part!). Quantities are just 1 (one) unless otherwise noted, prices are rounded to the nearest dollar as of September 2013.
Update Feb. 2016: A kit with all the electronic materials for this project is now available from Jameco Electronics. Please note that this kit does not include materials for building the mirror frame so you will still need to purchase those separately (see below). I earn a commission on sales of this kit as described at the bottom of this page.
Materials: Building the Mirror
Important: there are three main parts that need to fit together to build this: the regular mirror, the frame, and the one-way mirror. First, it's easiest if you can find a cardboard/paper mache lid and a regular mirror that will fit snugly inside it - the parts I bought didn't fit together perfectly, so I had to use a workaround (see Step 6). Second, cutting acrylic can be a pain depending on the tools you have available, so plan accordingly (see Steps 9 and 10). There's also an important consideration regarding the LED strip, which can't be cut to any length - it has to be cut in multiples of 3-LED segments, which are just shy of 2" long - so you want the inside perimeter of your mirror frame to be a multiple of that length. So, I'll link to the parts I used to build my mirror, but you can still follow these directions to build a mirror of a different size or shape.
Got all that? Time to start building!