Folding a Better Checkerboard

Folding an n× n checkerboard pattern from a square of paper that is white on one side and black on the other has been thought for several years to require a paper square of semiperimeter n. Indeed, within a restricted class of foldings that match all previous origami models of this flavor, one can prove a lower bound of n (though a matching upper bound was not known). We show how to break through this barrier and fold an n×n checkerboard from a paper square of semiperimeter 1 2 n + O(n). In particular, our construction strictly beats semiperimeter n for (even) n > 16, and for n = 8, we improve on the best seamless folding.