Many of the subjects I've worked on can be illustrated with nice pictures. Below is a collection of mathematical images related to papers I've written. See the references at the bottom for links to the relevant papers and additional information.

**NEW** (June 2016): I've created a web page about the moving sofa problem, which has additional fun mathematical pictures and animations.

**NEW** (September 2021): This page has additional pictures and animations related to the Tower of Hanoi puzzle

**1.**A random square Young tableau of order 25, shown on the left as a discrete stepped surface. The smooth surface on the right is the limit shape of such random Young tableaux. The coloring scheme highlights the level curves of both surfaces. [References]

**2.**The half-way permutation of a random sorting network with 2000 particles. [References and movies]

**3.**A random domino tiling of the Aztec diamond of order 50. [References]

**4.**A random Erdos-Szekeres permutation of order 50, and the limit shape of such permutations, an algebraic curve of degree 4. [References]

**5.**Connectivity patterns in loop percolation on a cylinder. [References]

**6.**A fully packed loop configuration of order 8. [References]

**8.**A tesselation of the hyperbolic plane by quadrilaterals. This is related to a natural dynamical system associated with Pythagorean triples. [References]

**9.**Limit shapes of bumping routes in the Robinson-Schensted correspondence. [References]

**References**

**1. **Random square Young tableaux

- B. Pittel, D. Romik. Limit shapes for random square Young tableaux.
*Adv. Appl. Math.***38**(2007), 164-209. - See also Chapter 3 in my book.
- Random square Young tableaux can be simulated using my Mac simulation app MacTableaux.

**2. **Random sorting networks

- O. Angel, A. Holroyd, D. Romik, B. Virag. Random sorting networks.
*Adv. Math.***215**(2007), 839-868. - D. Dauvergne. The Archimedean limit of random sorting networks. Preprint, 2018.
- Random sorting networks can be simulated using my Mac simulation app MacSort.
- Alexander Holroyd's sorting networks page.
- A movie of a 2000-particle sorting network (.mov file, 3MB)
- Another movie of a random sorting network (time-stationary version; .mov file, 4.3MB)

**3. **Domino tilings

- D. Romik. Arctic circles, domino tilings and square Young tableaux.
*Ann. Probab.***40**(2012), 611-647. - The arctic circle theorem (Wikipedia).
- See also my Mac simulation apps MacTableaux and ASM Simulator.

**4. **Erdos-Szekeres permutations

- D. Romik. Permutations with short monotone subsequences.
*Adv. Appl. Math.***37**(2006), 501-510. - See also Chapter 3 in my book.

**5-6.** Connectivity patterns in loop percolation, fully packed loops

- D. Romik. Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities.
*Commun. Math. Phys.***330**(2014), 499-538. - Fully packed loops are also studied under the name
**alternating sign matrices**, which I studied in the papers:- I. Fischer, D. Romik. More refined enumerations of alternating sign matrices.
*Adv. Math.***222**(2009), 2004-2035. - M. Karklinsky, D. Romik. A formula for a doubly refined enumeration of alternating sign matrices.
*Adv. Appl. Math.***45**(2010), 28-35. - A. Ayyer, D. Romik. New enumeration formulas for alternating sign matrices and square ice partition functions.
*Adv. Math.***235**(2013), 161-186.

- I. Fischer, D. Romik. More refined enumerations of alternating sign matrices.

**7.** Bootstrap percolation

- A. Holroyd, T. Liggett, D. Romik. Integrals, partitions and cellular automata.
*Trans. Amer. Math. Soc.***356**(2004), 3349-3368. - Alexander Holroyd's bootstrap percolation page.

**8. ** Tesselation of the hyperbolic plane by quadrilaterals

- D. Romik. The dynamics of Pythagorean triples.
*Trans. Amer. Math. Soc.***360**(2008), 6045-6064.

**9. ** Bumping routes in the Robinson-Schensted correspondence

- D. Romik, P. Sniady. Limit shapes of bumping routes in the Robinson-Schensted correspondence.
*Random Struct. Algor.***48**(2016), 171-182. - This paper and the above picture answer a question posed by Cris Moore in 2006 on his gallery page (scroll to the section titled "Flows in Young diagrams").
- The bumping routes can also be experimented with using my Mac simulation app MacTableaux.