If you have exceptional patience and good soldering skills, perhaps this is the perfect project for you!
Nick Schulze recently finished an 8x8x8 LED cube (totalling 512 LEDs). Sure it's designed around an Arduino, although it could be adapted to any microcontroller.
Here's a video of the cube:
Nick also explores some animation algorithms that result in the above patterns. On the right is the plot of the following function:
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The code is written in C++, and is well explained throughout the write-up.

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