Here is the front view of the completed 8x8x8 LED cube.
This is the back view. You can see the wires going up to the eight layers.
Here is the bottom view so you can see the IC's.
This it the video of it on YouTube.
The design problem is that there should be a resistor in each LED leg, that would take 64 resistors. Instead they used 8 resistors in the common cathode circuit. As a result the brightness of each layer varies depending upon how many LED's are lit up. The more led's are lit the dimmer the layer becomes. I might try to fix this by adding a resistor on each of the 64 legs and removing the ones on the common cathode connection to each layer. If nothing else reducing the 500 ohm resistor to 100 or 220 ohms might do the trick.
Here is the quick fix just change 510 ohm resistors to 51 ohms. Then the display is much brighter. It is still not as bright as when you use 74595 shift registers.