Actually, you need some mechanical skills to produce one. There are ready-made ones, but they are extremely expensive. So, if you want to build one, look for an extra long weekend.
When you finished the mechanical part think about how to control the LED. In case of a 3x3x3 cube you will have 27 LEDs (that is more than the amount of pins of the standard Arduino, but you can easily reduce the number of pins needed to 9 + 3 by using a multiplex control.
And when it comes to display different planes try to remember your math lessons.