Unfolding an 8-high Square, and Other New Wrinkles

A geometric dissection is a cutting of one or more geometric gures into pieces that we can rearrange to form one or more other geometric gures [3]. When applied to two-dimensional gures, dissections are not only demon-strations of the equivalence of area but also a source of enchantment since ancient times. In the last hundred years, there has been an increasing em-phasis on nding dissections that use the fewest possible number of pieces. Various dissections have remarkable properties. One such property is the hingeability of pieces, which has been explored in some depth in [4]: Swing the pieces one way on their hinges to form one gure, say an equilateral triangle; swing them another way to form another gure, say a square. Dif-ferent types of hinges eect dierent types of motion, ranging from swinging to twisting, and most recently, to folding [5]. In this article, we focus on a piano hinge, which connects two pieces along a shared edge and allows a folding motion. In [5], we dened a piano-hinged dissection as a cutting of one or more gures that are two levels hig