Rigidity Results for Tilings by L-Tiles and Notched Rectangles

Transcription

1 Rigidity Results for Tilings by L-Tiles and Notched Rectangles A. Calderon 1 S. Fairchild 2 S. Simon 3 1 University of Nebraska-Lincoln 2 Houghton College 3 Carnegie Mellon University MASSfest 2013

2 Introduction to Tiling What is a tile? Has been prominent in recreational math for decades Tiling of regions in a square lattice Consists of 1 1 cells

3 Tilesets Introduction What is a tile? Take a rectangle and make an L-shaped dissection Consider the two pieces and the two obtained from a reflection over the first bisector L 2 C 1 C 2 C 3 C 4 R 1 L 1 R 2 (a) The dissections C 1, C 2, C 3, C 4. (b) A C 1 dissection Figure: Our dissections and T (C 1, 6, 3).

4 Rigidity Introduction What is a tile? Definition A tiling by T (C i, k, n), 1 i 4 and 3 n k, of a region in plane is called rigid if it reduces to a tiling by k n and n k rectangles.

5 Problem Statement Introduction What is a tile? Problem Investigate the rigidity properties of tilings of each quadrant by T (C i, k, n), 1 i 4 and 3 n k

6 3 3 Base Case Introduction m R 1 L 1 R L 2 R 2 R 1 L 1

7 3 3 Inductive Step m R 1 L 1 X i+1 L 2 X i X i 1 R 2

8 m Tiling of X i 4 X i R 1 2 Figure: Tiling the corner of X i with R 1.

9 m Tiling of X i 4 X i R 1 2 Figure: Tiling the corner of X i with R 1.

10 m Tiling of X i 4 X i R 1 2 Figure: Tiling the corner of X i with R 1.

11 m Tiling of X i (a) Propagation of the pattern up the staircase. (b) The end of the propagation. Figure: Attempts to tile cell 1 with L 2.

12 m Tiling of X i X i R 1or2 X i 1 Figure: If we tile cell 1 by R 1 or R 2, we must tile cells 3 and 4 as shown.

13 Nonrigid Tilings by T (C 2, 3, 3) m II III IV IV II I (a) A nonrigid tiling of the second quadrant. III (b) A nonrigid tiling of the third quadrant. I

14 m T (C 2, 3, 3) in the First Quadrant 1 1

15 m T (C 2, 3, 3) in the First Quadrant 1 1

16 m Rigid Tilings of the Second and First Quadrants X i (c) Tiling the corner of X i with R 1. (d) Tiling X i on the staircase line.

17 Definitions Introduction m Definition A gap is a n k, k > 0, region in the first quadrant such that the x and y coordinates of its bottom left corner are both divisible by n. We call k the length of the gap. Definition We say an L 1 tile is in an irregular position if its bottom left corner has both its x and y coordinates divisible by n and if all squares below and to the left of the corner follow the rigid pattern.

18 m Existence of an Irregular L 1 Tile Lemma Any nonrigidly tiled gap of the i th quadrant by T (C i, mn, n) induces an L 1 tile in an irregular position or a nonrigidly tiled gap closer to the y-axis.

19 m Existence of an Irregular L 1 Tile Proof. Choose the gaps that are closest to the x-axis, and with respect to those choose the gap closest to the y-axis. Consider the bottom left corner of the gap Case 1: L 1 tiles the bottom left corner Case 2: R 2 tiles the bottom left corner L R 1 L 1 R 2

20 m Existence of an Irregular L 1 Tile Proof. Consider the bottom left corner of the gap Case 3: R 1 tiles the bottom left corner L 2 R 1 L 1 R2

21 m Impossibility of Irregular L 1 Lemma Tilings of the first quadrant by T (C 1, mn, n) cannot contain an L 1 in an irregular position.

22 m Impossibility of Irregular L 1 Proof. Assume for the sake of contradiction there exists a tiling of the first quadrant with an L 1 tile in an irregular position L 2 R 1 L 1 R 2

23 m Rigidity results for tiling the first quadrant by T (C 1, mn, n) Theorem Any tiling of the first quadrant by T (C 1, mn, n) is rigid. Proof. Assume for the sake of contradiction that there is a nonrigid tiling of the first quadrant. This implies there is a nonrigid gap. By the first lemma, this implies the existence of an L 1 tile in an irregular position or a non rigidly tiled gap closer to the y-axis. But the non rigidly tiled gap placed closer to the y axis eventually will force an L 1 tile, so by the second lemma, we know that such a tiling doesn t exist.

24 m Generalization of Rigidity Results Remark We have generalized this rigidity result for T (C i, mn, n) in the i th quadrant for 1 i 4.

25 m for Rectangles of Coprime Dimension Theorem If p, n are coprime, then all quadrants have nonrigid tilings by T (C i, p, n), 1 i 4.

26 m for Rectangles of Coprime Dimension Proof. Assume that gcd(p, n) = 1. Then we know that p is a generator in Z/nZ. Hence, for some x Z +, we have (x + 1)p 1 mod n, which implies that xp + (p 1) 0 mod n. n n n p p n p

27 Generalizing the Dissection m 3 more independent parameters For yp xn = r t, region I is infinite strips of length xn, region II is a strip of width p. For ap bn = s, region III is an infinite strip of width (a 1)p, region V is a rectangle of size bn yp. I s r IV L 1 II V t III

28 Introduction For n 4 T (C i, n, n) is rigid in the i th quadrant T (C 2, n, n) and T (C 4, n, n) rigid in each quadrant Special Case: T (C 2, 3, 3) rigid in only the first quadrant

29 Introduction T (C i, mn, n) for 1 i 4 and m 2, n 3 is rigid in the i th quadrant T (C 1, mn, n) is nonrigid in the second, third, and fourth quadrants T (C 3, mn, n) is nonrigid in the first, second, and fourth quadrants T (C 2, mn, n) and T (C 4, mn, n) have currently eluded our investigative measures

30 For p, n coprime, If p, n are coprime, then all quadrants have nonrigid tilings by T (C i, p, n), 1 i 4 Algorithm generalizes to all possible L shaped dissections of the rectangle

31 Applications Introduction

32 Appendix References References I M. Chao, D. Levenstein, V. Nitica, R. Sharp, A coloring invariant for ribbon L-tetrominoes, Discrete Mathematics, 313 (2013) S. W. Golomb, Checker boards and polyominoes, American Mathematical Monthly, 61 (1954) S. W. Golomb, Replicating figures in the plane, Mathematical Gazette, 48 (1964) S. W. Golomb, Polyominoes, Puzzeles, Patterns, Problems, and Packings (2 nd ed.), Princeton University Press, NJ, V. Nitica, Rep-tiles revisited, in the volume MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics, American Mathematical Society, V. Nitica, A rigidity property of ribbon L-shaped n-ominoes and generalizations, submitted to Discrete Mathematics