Folding a better checkerboard

Abstract

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 2 [superscript 2]. Indeed, within a restricted class of foldings that match all previous origami models of this flavor, one can prove a lower bound of n 2 [superscript 2](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 n2 [superscript 2] + O(n) In particular, our construction strictly beats semiperimeter n 2 [superscript 2] for (even) n > 16, and for n = 8, we improve on the best seamless folding.