Tilings (Math 285)
Pak, MS 6125, pak@math.
Class Schedule: MWF 12:00-12:50, MS 7608.
We will give an introduction to the subject,
covering a large number of classical and a few recent results.
The emphasis will be on the main ideas and techniques rather than
proving the most recent results in the field. The idea is to to give
a guided tour over (large part of) the field to prepare for
more advanced results in the future.
I will be posting most of the literature I will be covering, numbering them roughly in order
of the lectures (some lectures are collapsed into one item). A few of these papers will not be taught
- they are included the source of additional reading closely related to lecture material.
- Domino tilings
- Combinatorial group theory
- Ribbon tilings of Young diagram shapes
- Tile invariants and ribbon tilings of general regions
- Rectangles with one side integral
- S. Wagon, Fourteen Proofs
of a Result About Tiling a Rectangle (1988); original article with 14 proofs (not all of them clear or valid)
- D.J.C. MacKay, Simple Proofs of a Rectangle
Tiling Theorem, useful variations and additional details on these proofs.
- P. Winkler, Integers and Rectangles,
in Mathematical Puzzles (2004), another proof.
- R. Kenyon, A note on tiling with integer-sided rectangles (1994);
a new group theory style proof with algorithmic applications.
- N.G. de Bruijn, Filling Boxes with Bricks (1969), the original motivation behind this study.
- V.G. Pokrovskii, Linear relations in dissections into n-dimensional parallelepipeds (1985);
an interesting but little cited generalization of de Bruijn's theorem
(free access Russian original)
- Tilings with two bars
- T-tetromino tilings
- Further applications of height functions
- Tilings of rectangles
- Tilings of rectangles with rectangles
- Order of tiles
- Yang's paper (see above); hardness of augmentability and decidability of augmentability with rectangles.
- Korn's thesis (see above), Section 11; negative solution of augmentability for dominoes.
- Counting domino tilings and perfect matchings
- L. Lovász, M.D. Plummer, Matching Theory, Chapter 8; among other things, general Pfaffian-Determinant approach.
- R. Kenyon, An introduction to the dimer model (2002);
variations on Kasteleyn method, isoradial graphs.
- R.W. Kenyon, J.G. Propp, D.B. Wilson, Trees
and Matchings (2000); Temperley's bijection and applications.
- Aztec diamond
- A. Björner, R.P. Stanley, A combinatorial miscellany (2010);
- N. Elkies, G. Kuperberg, M. Larsen, J. Propp, Alternating-sign matrices and domino tilings;
the original article with four proofs.
- Jim Propp, San Diego talk slides, outline of a short proof based
on the "urban renewal".
- S.-P. Eu and T.-S. Fu, A
simple proof of the Aztec diamond theorem (2005), an elegant short proof based on the non-intersecting paths principle.
- K.P. Kokhas, Domino tilings of aztec diamonds
and squares (2008); yet another beautiful proof of the Aztec diamond formula via a
general recurrence relation (§ 2.1 and 2.2)
- L. Pachter, Combinatorial
Approaches and Conjectures for 2-Divisibility Problems Concerning Domino Tilings of Polyominoes (1997); simple
proof of divisibility of the number of domino tilings of a even square.
Warnings: Most of these links are external, some are by subscription, some can be broken; occasionally, their content is unverified.
Also, the explanations are NOT review, but rather quick summary of material I used from the sources; often there is wealth of other work
presented there as well.
- Federico Ardila and Richard P. Stanley, Tilings (2005), an introductory article.
- Branko Grunbaum and G. C. Shephard, Tilings and Patterns (1986), an instant classic. A generic introduction to the subject.
- Solomon W. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, true classic in the recreational literature (Second Ed., 1995).
to return to Igor Pak Home Page.
Last updated 3/8/2013