Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Adding layers to bumped-body polyforms with minimum perimeter preserves minimum perimeter. | Adding layers to bumped-body polyforms with minimum perimeter preserves minimum perimeter Winston C. Yang winston@ Submitted Dec 1 2004 Accepted Dec 8 2005 Published Jan 25 2006 Mathematics Subject Classifications 05B50 05B45 Abstract In two dimensions a polyform is a finite set of edge-connected cells on a square triangular or hexagonal grid. A layer is the set of grid cells that are vertex-adjacent to the polyform and not part of the polyform. A bumped-body polyform has two parts a body and a bump. Adding a layer to a bumped-body polyform with minimum perimeter constructs a bumped-body polyform with min perimeter the triangle case requires additional assumptions. A similar result holds for 3D polyominos with minimum area. 1 Introduction A polyform is a finite edge-connected set of cells in a grid of any number of dimensions. In two dimensions if the cell shape is a square triangle or hexagon the polyform is a polyomino polyiamond or polyhex respectively. In three dimensions if the cell shape is a cube the polyform is a 3D polyomino or a polycube. See Figure 1 for examples of 2D polyforms. In two dimensions the area of a polyform is the number of polyform cells and the perimeter is the number of edges belonging to only one cell. In three dimensions and higher the volume of a polyform is the number of polyform cells and the area is the number of faces belonging to only one cell. In all dimensions a layer is the set of grid cells that are vertex-adjacent to the polyform and not part of the polyform. Variations on the polyform theme exist researchers sometimes consider rotations or reflections as distinct polyforms or do not allow holes or do not require edge-connectedness or allow more than one cell shape. A revised classic reference on polyominos is Gol94 . Also see Cla02 . Throughout the paper BB means bumped-body. BCC93 considers polyhexes in the context of hydrocarbons in chemistry and it proves that if a polyhex can be circumscribed enough times the
