Using stiff paper and transparent tape, Craig Kaplan assembles a beautiful roundish shape that looks like a Buckminster Fuller creation or a fancy new kind of soccer ball. It consists of four regular dodecagons (12-sided polygons with all angles and sides the same) and 12 decagons (10-sided), with 28 little gaps in the shape of equilateral triangles. There’s just one problem. This figure should be impossible. That set of polygons won’t meet at the vertices. The shape can’t close up.
Kaplan’s model works only because of the wiggle room you get when you assemble it with paper. The sides can warp a little bit, almost imperceptibly. “The fudge factor that arises just from working in the real world with paper means that things that ought to be impossible actually aren’t,” says Kaplan, a computer scientist at the University of Waterloo in Canada.