December 4th, 1999
Greg N. Frederickson
Purdue University
Geometric Dissections That Swing and Twist
Abstract
A geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure. Some dissections can be connected with hinges so that the pieces form one figure when swung one way, and form the other figure when swung another way. This talk explores two fundamental ways to hinge dissections of 2-dimensional figures such as regular polygons and stars. The first way uses "swing hinges", which allow rotation in the plane. Techniques based on tessellations and infinite strips are presented and analyzed. The second way relies on "twist hinges", which allow one piece to be turned over relative to another, using rotation by 180 degrees through the third dimension. Techniques are introduced to convert swing-hingeable dissections to be twist-hingeable, to change the length (and thus the height) of a parallelogram, and to apply "pseudo-tessellations". The natural goal of minimizing the number of pieces, subject to the dissection being hingeable, is used throughout. The generality of such hinging schemes is discussed.