I think that I have looked over this instructable before, i liked the way that he build the cube.
It looks like he's using the TLC 5940, and I dont see any decoupling caps in the BOM. I cant say for certain that every time absolutly... and there is precious little documentation, but the rumor going around is that you should use a 47uf cap on each chip.
The cube I am building is unfinished, im still working on the sourcing side of the design using mosfets, but my cube is very similar to that cube, only its running on a nano, rather than stand alone.
I have a website that I am developing (too slowly, but some days I make progress...) and I will eventually have four different designs that run 4x4x4 RGB LED cubes.https://sites.google.com/site/rgbledcubes/
There will be:
Charlieplexed Cube (done and documented)
Constant Current Cube (using the TLC5940 chip)
Shift Register Cube (currently under re-design)
Cascaded Transistor Cube (getting closer to working)
This is my list of the more common ways of solving the same problem, but it turns out that there are many ways to control LEDs
Cube building is rather difficult, it might be best to start with a small 3x3x3 single color cube, so that you can figure out what is easy/hard for you to do, so you can practice in advance of the final assembly.
There are many ways to build the cube, and many ways to build the control circuitry. I do not know the best way, but I have tried several approaches, and I hope to have a better understanding once I have all 4 cubes running.
I've included support documentation, like good soldering techniques, how to prep parts to make soldering easier/better, LED details and info, ...
The site is all about making it easier to build cubes, The only thing I dont have is specific parts lists/data or any for sale, but maybe in the future.
Keep in mind that a 4x4x4 RGB LED cube is 192 LEDs, which is kind a lot, and not as simple undertaking, its probably not the best first project as it may discourage you with a lot of difficulty. Try not to get too frustrated.