Solder the ends of each ring of WS2811's together, making sure to have common ground/single source power or the other way around to ensure proper power distro.
Serially connect the digital in/out lines to make one large chain.
Wire up the buck regulator to power and the strips, and the digital in to the Arduino (or teensy, or micro of your choice).
Have fun coding some things. Adafruit Neopixel code should work with these LEDs, http://learn.adafruit.com/adafruit-neopixel-uberguide/arduino-library
If somebody wanted to recreate this ball? is there any place that one might be able to find the dxf's for the lasercut pieces, The code to enable and react to tilt, or any other fun details?
https://drive.google.com/file/d/0B_2xkbzRozdxbHZLQVBoMUtVQXc/edit?usp=sharing