Institution: | aMIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., 200 Technology Square, Cambridge, MA 02139, USA bDepartment of Information and Computer Science, University of California, Irvine, CA 92697, USA cDepartment of Computer Sciences, Purdue University, West Lafayette, IN 47907-1398, USA dMathematics Department, Stetson University, DeLand, FL 32720, USA |
Abstract: | A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be rotated into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to k/2(n−1). In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q. |