# Group Theoretic Origin of the Domino Height Functions

In order to learn about domino shuffling, I have asked around various times for implementations of the domino shuffling algorithm.

- Mathoverflow 40445 (Looking for Implementations of Domino Shuffling)
- Mathoverflow 81009 (Height Function from Max-Flow/Min-Cut)
- Mathoverflow #78302 (Rhombus Tilings with More than Three Directions

The original program was implemented by Sameera Iyengar who was a student at MIT in the mid 90′s and has since gone into theater. Programming has changed a bit in the past 20 years, and I have posted my own python implementation on the internet.

The height of the domino goes up and down around each tiling. What happens when we join some tiles together?

1-0-1-0-1-0-1-0 | | | | | | 2-3-2 I 2-3-2 I | | | | | | 1-0-1-0-1-0-1-0

Here `I`

means -1 (so as to only take up one space in the `ASCII`

art).

### Tilings and 3D surfaces

If you tile a rectangle with dominos, the height function defines a surface over the edges of the tiling.

Here are pictures of 3D surfaces you can get with the domino tiling projected to various angles.

3D partitions `------`

Lozenge tilings

### How do we get height functions in the first place?

The domino group is:

The lozenge group is:

These were calculated by William Thurston in his math monthly article. Cayley graphs are graphical ways of drawing the elements of a group presentation. Thurston takes it a step further and draws Cayley 2-complexes.

Tilings by T-tetraminos were analyzed by Korn and Pak.

_ _ _ _ |_ _| | | |_|_ | | _| |_| |_|_ _ _|

Are there any other cases with interesting height functions? It seems very difficult to solve the group equations outside the realm of domino’s or lozenges.

### Example: Tri-ominos

At one point I tried to work out the relation for the L-triomino in the plane. Any lattice polygons can be presented in a group with two generators.

_ _ _ | |_ | _| |_ _| y^(-2)x^2yx^(-1)yx^(-1) |_| xyxyx^(-2)y^(-2) _ _ _ _| | |_ | |_ _| x^2y^2x^(-1)y^(-1)x^(-1)y^(-1) |_| y^(-1)xy^(-1)xy^2x^(-2)

Manipulating the group words looks very difficult. Instead, we can visually merge two triominos to get

_ _ |_| _|_| |_| yxyx^{-1}y^{-1}xy^{-1}x^{-1} = 1 |_| xy^2xy^{-1}x^{-2}y^{-1} = 1

Then taking three of these pieces we can show a loop has order 3, `xyx^{-1}^{-1})^3=1`

. The resulting group is .

This non-commutative loop model looks very interesting… maybe a way of studying non-abelian cohomology of .

### Further Reading

- Tiling with Polyominoes and Combinatorial Group Theory
- On Conway’s Tiling Groups
- Squaring Rectangles with Squares (for Dummies
- Domion Tilings and Planar Algebras Generated by a 3-Box
- Laying Train Tracks
- Rotation Numbers and the Jankins-Neumann Ziggurat
- Geodesics in Cat(0) Cubical Complexes
- Crystals, quivers and dessins d’enfants
- Cut and Project Tilings

## About this entry

You’re currently reading “Group Theoretic Origin of the Domino Height Functions,” an entry on monsieurcactus

- Published:
- April 12, 2012 / 3:32 pm

- Category:
- Domino / Dimer, Geometric Group Theory

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