This is a Charlieplexed 64 RGB LED 3D Cube. I found this website by Asher Glick, http://aglick.com/charliecube.html He has some good illustrations, instructions, and the software that makes it work.

The CharlieCube64 has a total of 192 LEDs and is controlled by 16 arduino pins wired through 16 spires of 4 LEDs and 4 connections. The 64 connections that 16 spires of 4 leads are wired in such a way that they only need 16 micro-controller (arduio) connections. 
The arduino (not pictured), connects directly to the LEDs without using any resistors, shift registers, LED drivers or any other electronic components. It is simply LEDs, micro-controller, and circuitry. It is sorta like applying geometry to circuitry.

This cube design uses complex wiring instead of parts. This type of wiring LEDs is known as charlieplexing, and it is a type of multiplexing, that arranges LEDs facing each other in groups. The result is that it requires less connections to control the LEDs. In the case of 64 RGB LEDs that is actually 192 LEDs that can be controlled with 16 tri-state micro-controller input/output connections (pins).

More about Charliplexing can be found here:

Building a Charlieplexed 64 RGB LED 3D cube can be broken down into several groups:

1) Assembling and preparing the parts
2) Assembling an LED spire
3) Assembling the cube
4) Programming
5) Alternatives and Modifications