So this thing is pretty cool, and seems not horribly hard to clone.
It’s the printer Staples is going to have in stores. It just cuts shapes in paper, inkjets them for full color then glues them together. They have special ink to ensure the color bleeds into the sides of the paper but I bet just spraying extra ink has the same effect.
As for actually cutting it the most obvious route is a laser. I mean the cheapy little things that can only cut paper are maybe $40? One would need to cut in a pattern that could still go through the rollers of the inkjet after, since if you colored first I think you would end up with burnt paper colored edges. Maybe something like this for one layer of a sphere? http://imgur.com/VcAyL
Then all that’s left is gluing the pages together. The little bridges in that picture could be staggered so they don’t stick the print to the scrap paper. I think some sort of sprayable glue is the easiest route here, but I don’t know much about glue.
I’ve found paper hard to cut with a low-power (~250mw) that will cut foam and plastic just fine. I don’t know enough about lasers to know why.
It might be easier to repourpose an existing x-y paper cutting machine (the Cricut comes to mind, but in know there is a more hackable one as well) which uses a blade to cut paper pretty effortlessly.
In fact, I think there is a Cricut model that already incorporates an inkjet head, so you’re 2/3rds of the way there already
It looks like it’s just using a needle to punch holes and basically rasterize the cut lines. I feel like any needle small enough to make smooth edges would be really slow and wear out quickly.
edit: I just thought of a gloriously hackish way to test this on my mendel. Just put a needle on one of these and a small motor. http://www.dtonline.org/areas/6/1/22/p.gif
Lift the Z to stop cutting, lower to start. Run the needle without control at a constant speed.