This project is based heavily off AsherGlicks Charlie Cube. It uses a different construction method to achieve the same outcome. Instead of using additional supports the LED legs are used as the primary structure.
See the links for all the basic info on how the hardware and software works.
Components
64×
RGB LEDs
Common Cathode
1×
Perfboard/Prototyping board
1×
Arduino Pro Mini 5V
Or any other similar microcontroller
Here is a demo video of the cube in action. I couldn't get the lighting right to make it any clearer. However I'm sure you get the idea. There are the following animations.
All instructions are identical to here with the following exception. Instead of using additional supports I have just used the LED legs for the primary structure on the "spires". I found this was much easier to put together than having to hold other pieces of metal in place.