Note: Apparently, some of the links for the supplies and code aren't working, if you need them send me an email at email@example.com.
You can find a video of the working LED cube on youtube here:
And here's another:
Step 1: Supplies
-30x 150 Ohm resistors
-5x 3-8 Line Decoders
-5 NPN Transistors
The manufacturer numbers and prices can be found in the attached document though the cost was about $40.
Additional required materials:
-Sturdy wire (for structure of cube)
-sockets (for decoders)
Step 2: Cube Construction
I began by assembling the 5 layers or planes of the cube. This involved connecting all the cathodes of the LEDs together. Id did this by poking 25 LEDs (about 1 inch apart) into a piece of styrofoam. I bent all the anodes out of the way and soldered the LEDs in 5 columns and then two rows on the end (see picture). Be careful when soldering because the entire wire becomes hot and I accidently burned myself multiple times by touching the end of one of the wires.
The next step in cube construction is even trickier. You need to connect the anodes of the LEDs in 25 vertical columns without letting any of them (or their connecting wire) touch the cathode planes. The way I did this was leaving the top plane in the styrofoam and placing ~6 inch wires next to the anodes. I used needlenose pliers to curl the anodes into loops so that it was easier to solder. Alternate the directionality of the wires of the cathode planes for stability. See photos.