Rigidity Results for Tilings by L-Tiles and Notched Rectangles

IntroductionOur Results/Contribution

Summary

Rigidity Results for Tilings by L-Tiles andNotched Rectangles

A. Calderon1 S. Fairchild2 S. Simon3

1University of Nebraska-Lincoln

2Houghton College

3Carnegie Mellon University

MASSfest 2013

Calderon, Fairchild, Simon Rigidity Results for Tilings by L-Tiles and Notched Rectangles


SummaryWhat is a tile?

Introduction to Tiling

Has been prominent in recreational math for decadesTiling of regions in a square latticeConsists of 1× 1 cells




Tilesets

Take a rectangle and make an L-shaped dissectionConsider the two pieces and the two obtained from areflection over the first bisector

C1 C2 C3 C4

(a) The dissections C1,C2,C3,C4.

R1 L1

R2

L2

(b) A C1 dissection

Figure: Our dissections and T (C1,6,3).




Rigidity

DefinitionA tiling by T (Ci , k ,n),1 ≤ i ≤ 4 and 3 ≤ n ≤ k , of a region inplane is called rigid if it reduces to a tiling by k × n and n × krectangles.




Problem Statement

ProblemInvestigate the rigidity properties of tilings of each quadrant byT (Ci , k ,n),1 ≤ i ≤ 4 and 3 ≤ n ≤ k



Summary

n × nmn × nNonrigidity

3× 3 Base Case

R1L1

R1

31

2

R2

L2

R1

L1



Summary


3× 3 Inductive Step

R1

L1

R2

L2

Xi−1

Xi

Xi+1



Summary


Tiling of Xi

R1

1

34

2

Xi+1

Figure: Tiling the corner of Xi with R1.



Summary


Tiling of Xi

R1

1

34

2

Xi+1




Summary


Tiling of Xi

R1

1

34

2

Xi+1




Summary


Tiling of Xi

14

56

(a) Propagation of thepattern up the staircase.

78

9

(b) The end of thepropagation.

Figure: Attempts to tile cell 1 with L2.



Summary


Tiling of Xi

R1or2

Xi−1

Xi+2

34

510

Figure: If we tile cell 1 by R1 or R2, we must tile cells 3 and 4 asshown.



Summary


Nonrigid Tilings by T (C2,3,3)

I

II

IIIIV

(a) A nonrigid tiling ofthe second quadrant.

II

IV

IIII

(b) A nonrigid tiling ofthe third quadrant.



Summary


T (C2,3,3) in the First Quadrant

1 1



Summary


T (C2,3,3) in the First Quadrant

1 1



Summary


Rigid Tilings of the Second and First Quadrants

12 3

Xi−1

(c) Tiling the corner of Xiwith R1.

12

(d) Tiling Xi on thestaircase line.



Summary


Definitions

DefinitionA gap is a n × k , k > 0, region in the first quadrant such thatthe x and y coordinates of its bottom left corner are bothdivisible by n. We call k the length of the gap.

DefinitionWe say an L1 tile is in an irregular position if its bottom leftcorner has both its x and y coordinates divisible by n and if alln × n squares below and to the left of the corner follow the rigidpattern.



Summary


Existence of an Irregular L1 Tile

Lemma

Any nonrigidly tiled gap of the i th quadrant by T (Ci ,mn,n)induces an L1 tile in an irregular position or a nonrigidly tiledgap closer to the y-axis.



Summary



Proof.Choose the gaps that are closest to the x-axis, and withrespect to those choose the gap closest to the y -axis. Considerthe bottom left corner of the gap

Case 1: L1 tiles the bottom left cornerCase 2: R2 tiles the bottom left corner

21

3R1 L1

R2

L2



Summary



Proof.Consider the bottom left corner of the gap

Case 3: R1 tiles the bottom left corner

12

3R1 L1

R2

L2



Summary


Impossibility of Irregular L1

LemmaTilings of the first quadrant by T (C1,mn,n) cannot contain anL1 in an irregular position.



Summary


Impossibility of Irregular L1

Proof.Assume for the sake of contradiction there exists a tiling of thefirst quadrant with an L1 tile in an irregular position.

1

2

3R1 L1

R2

L2



Summary


Rigidity results for tiling the first quadrant byT (C1,mn,n)

TheoremAny tiling of the first quadrant by T (C1,mn,n) is rigid.

Proof.Assume for the sake of contradiction that there is a nonrigidtiling of the first quadrant. This implies there is a nonrigid gap.By the first lemma, this implies the existence of an L1 tile in anirregular position or a non rigidly tiled gap closer to the y -axis.But the non rigidly tiled gap placed closer to the y axiseventually will force an L1 tile, so by the second lemma, weknow that such a tiling doesn’t exist.



Summary


Generalization of Rigidity Results

Remark

We have generalized this rigidity result for T (Ci ,mn,n) in the i th

quadrant for 1 ≤ i ≤ 4.



Summary


Nonrigidity for Rectangles of Coprime Dimension

TheoremIf p,n are coprime, then all quadrants have nonrigid tilings byT (Ci ,p,n),1 ≤ i ≤ 4.



Summary


Nonrigidity for Rectangles of Coprime Dimension

Proof.Assume that gcd(p,n) = 1. Then we know that p is a generatorin Z/nZ. Hence, for some x ∈ Z+, we have (x + 1)p ≡ 1mod n, which implies that xp + (p − 1) ≡ 0 mod n.

pn

n n n

p

p



Summary


Generalizing the Dissection

3 more independent parametersFor 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.

L1

IVI

III

II

V

rs

t



Summary

Summary

For n ≥ 4T (Ci ,n,n) is rigid in the i th quadrantT (C2,n,n) and T (C4,n,n) rigid in each quadrant

Special Case:T (C2,3,3) rigid in only the first quadrant



Summary

Summary

T (Ci ,mn,n) for 1 ≤ i ≤ 4 and m ≥ 2, n ≥ 3 is rigid in thei th quadrantT (C1,mn,n) is nonrigid in the second, third, and fourthquadrantsT (C3,mn,n) is nonrigid in the first, second, and fourthquadrantsT (C2,mn,n) and T (C4,mn,n) have currently eluded ourinvestigative measures



Summary

For p,n coprime,If p,n are coprime, then all quadrants have nonrigid tilingsby T (Ci ,p,n),1 ≤ i ≤ 4Algorithm generalizes to all possible L−shaped dissectionsof the rectangle



Summary

Applications


Appendix References

References I

M. Chao, D. Levenstein, V. Nitica, R. Sharp, A coloring invariant forribbon L-tetrominoes, Discrete Mathematics, 313 (2013) 611–621.

S. W. Golomb, Checker boards and polyominoes, AmericanMathematical Monthly, 61 (1954) 675–682.

S. W. Golomb, Replicating figures in the plane, Mathematical Gazette,48 (1964) 403–412.

S. W. Golomb, Polyominoes, Puzzeles, Patterns, Problems, andPackings (2nd ed.), Princeton University Press, NJ, 1994.

V. Nitica, Rep-tiles revisited, in the volume MASS Selecta: Teachingand Learning Advanced Undergraduate Mathematics, AmericanMathematical Society, 2003.

V. Nitica, A rigidity property of ribbon L-shaped n-ominoes andgeneralizations, submitted to Discrete Mathematics


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Rigidity Results for Tilings by L-Tiles and Notched Rectangles