Many of his papercraft polyhedra are illuminated, as you can see in the flickr set I linked to. I decided to replicate and illuminate one of his designs, the "Tri-Star Fruit."
My Papercraft Lantern utilizes surface mount LEDs for illumination, and flexible self-adhesive copper foil as conductors. A single 9V battery provides power, for an estimated run time of about 40 hours.
Step 1: Materials and Tools
Three sheets of 8.5 x 11" card stock (Or get fancy with patterned or textured paper)
Glue (glue stick or white glue)
Copper foil tape (used for stained glass)
Some clear tape (packing tape or Scotch tape)
3 LEDs (I used Digikey p/n 475-2542-1-ND)
A 9V battery clip
One 9V battery
Computer (I assume you've got one of those already!)
A printer (laser is preferable)
A soldering iron
A good, sharp knife with fresh blades (X-acto will work fine)
Wire stripper and other soldering hand tools