Bibliography and resources

This page collects reading material and other resources related to the subject matter of the course. It is not intended to be an complete bibliography or an authoritative collection of links. It will be maintained only during the lifetime of the course, so if you're looking at this page beyond August 2025, don't be surprised if some of the links don't work.

Central to the page is a list of papers and articles that students might choose to read and present in class. These papers span a wide variety of topics, techniques, and levels of difficulty. The list is intended to provide more than enough material to keep the course running, but there's no strict requirement that we read all, or even any, of these particular papers. Feel free to propose other papers not listed here if you find something else that fits the scope of the course.

Meta

• Srinivasan Keshav's advice on How to Read a Paper. Read the two-page report at the "latest version" link.
• William Griswold's web page about How to Read an Engineering Research Paper. Also linked from a more general page of a broad advice page maintained by Michael Ernst.

Books

• Grünbaum, Branko, and G.C. Shephard. Tilings and patterns. Second Edition, Dover, 2016
• Adams, Colin. The tiling book: An introduction to the mathematical theory of tilings. Vol. 142. American Mathematical Society, 2022.
• Fathauer, Robert. Tessellations: mathematics, art, and recreation. CRC Press, 2020.
• Kaplan, Craig. Introductory tiling theory for computer graphics. Morgan & Claypool Publishers, 2009.
• Toth, Csaba D., Joseph O'Rourke, and Jacob E. Goodman, eds. Handbook of discrete and computational geometry. CRC press, 2017.
• The Tilings Encyclopedia

Software

• Tiled.art: a highly interactive site with galleries of Escher-like tessellations and web-based tile authoring software
• Conway's Magical Pen: a different authoring tool focused on demonstrating tiling criteria like the Conway Criterion
• Arnaud Chéritat's tiling applet: a useful sandbox tool for constructing polygonal tilings by hand. See also the instructions
• hypertiling: a Python library for drawing tilings of the hyperbolic plane in the Poincaré disk
• Tactile: my library for representing and manipulating isohedral tilings. There are ports to JavaScript, Rust, and Python
• heesch-sat: my suite of tools (in development) for computing tiling properties of polyforms, with an emphasis on computing Heesch numbers

Articles and papers

Generating Escher-like tilings

• Kaplan, Craig S., and David H. Salesin. "Escherization." In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 499-510. 2000.
• Koizumi, Hiroshi, and Kokichi Sugihara. "Maximum eigenvalue problem for escherization." Graphs and Combinatorics 27 (2011): 431-439.
• Nagata, Yuichi, and Shinji Imahori. "Escherization with large deformations based on as-rigid-as-possible shape modeling." ACM Transactions on Graphics (TOG) 41, no. 2 (2021): 1-16.
• Aigerman, Noam, and Groueix, Thibault. "Generative escher meshes." In ACM SIGGRAPH 2024 Conference Papers, pp. 1-11. 2024.

Tiles for image and geometry synthesis

• Cohen, Michael F., Jonathan Shade, Stefan Hiller, and Oliver Deussen. "Wang tiles for image and texture generation." ACM transactions on graphics (TOG) 22, no. 3 (2003): 287-294.
• Lagae, Ares, and Philip Dutré. "An alternative for Wang tiles: Colored edges versus colored corners." ACM Transactions on Graphics (TOG) 25, no. 4 (2006): 1442-1459.
• Fu, Chi-Wing, and Man-Kang Leung. "Texture Tiling on Arbitrary Topological Surfaces using Wang Tiles." Eurographics Symposium on Rendering 2005 (2005): 99-104.
• Meekes, Merel, and Amir Vaxman. "Unconventional patterns on surfaces." ACM Transactions on Graphics (TOG) 40, no. 4 (2021): 1-16.
• Akleman, Ergun, Vinod Srinivasan, and Esan Mandal. "Remeshing schemes for semi-regular tilings." In International Conference on Shape Modeling and Applications 2005 (SMI'05), pp. 44-50. IEEE, 2005.

Tilings and sampling

• Ostromoukhov, Victor, Charles Donohue, and Pierre-Marc Jodoin. "Fast hierarchical importance sampling with blue noise properties." ACM Transactions on Graphics (TOG) 23, no. 3 (2004): 488-495.
• Ostromoukhov, Victor. "Sampling with polyominoes." ACM SIGGRAPH 2007 papers. 2007. 78.1–78.6.
• Kopf, Johannes, Daniel Cohen-Or, Oliver Deussen, and Dani Lischinski. "Recursive Wang tiles for real-time blue noise." In ACM SIGGRAPH 2006 Papers, pp. 509-518. 2006.

Aperiodicity

• Robinson, Raphael M. "Undecidability and nonperiodicity for tilings of the plane." Inventiones mathematicae 12 (1971): 177-209.
• Socolar, Joshua ES, and Joan M. Taylor. An aperiodic hexagonal tile." Journal of Combinatorial Theory, Series A 118, no. 8 (2011): 2207-2231.
• Walton, James J., and Michael F. Whittaker. "An aperiodic tile with edge-to-edge orientational matching rules." Journal of the Institute of Mathematics of Jussieu 22, no. 4 (2023): 1727-1755.
• Frank, Natalie Priebe. "A primer of substitution tilings of the Euclidean plane." Expositiones Mathematicae 26, no. 4 (2008): 295-326.
• Culik II, Karel. "An aperiodic set of 13 Wang tiles." Discrete Mathematics 160, no. 1-3 (1996): 245-251.
• Jeandel, Emmanuel, and Michael Rao. "An aperiodic set of 11 Wang tiles." Advances in Combinatorics (2021).
• Smith, David, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. "An aperiodic monotile." Combinatorial Theory 4, no. 1 (2024).
• Smith, David, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. "A chiral aperiodic monotile." Combinatorial Theory 4, no. 2 (2024).

Computational geometry and complexity

• Langerman, Stefan and Winslow, Andrew. "A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino." In Sándor P. Fekete & Anna Lubiw, editors: 32nd International Symposium on Computational Geometry (SoCG 2016), LIPIcs 51, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 50:1–50:15.
• Winslow, Andrew. "An optimal algorithm for tiling the plane with a translated polyomino." In International Symposium on Algorithms and Computation, pp. 3-13. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015.
• Keating, Kevin, and Andrew Vince. "Isohedral polyomino tiling of the plane." Discrete & Computational Geometry 21 (1999): 615-630.
• Brlek, Srecko, Xavier Provençal, and Jean-Marc Fédou. "On the tiling by translation problem." Discrete Applied Mathematics 157, no. 3 (2009): 464-475.

Undecidability

• Ollinger, Nicolas. "Tiling the plane with a fixed number of polyominoes." In Language and Automata Theory and Applications: Third International Conference, LATA 2009, Tarragona, Spain, April 2-8, 2009. Proceedings 3 , pp. 638-647. Springer Berlin Heidelberg, 2009.
• Demaine, Erik D., and Stefan Langerman. "Tiling with three polygons is undecidable." arXiv preprint arXiv:2409.11582 (2024).

Tiling properties of polyforms

• Rhoads, Glenn C. Planar tilings by polyominoes, polyhexes, and polyiamonds." Journal of Computational and Applied Mathematics 174, no. 2 (2005): 329-353.
• Myers, Joseph. "Polyform tiling". 1996–2025.
• Mann, Casey, and B. Charles Thomas. "Heesch numbers of edge-marked polyforms." Experimental Mathematics 25, no. 3 (2016): 281-294.
• Kaplan, Craig. "Heesch numbers of unmarked polyforms." Contributions to Discrete Mathematics 17, no. 2 (2022): 150-171.
• Bašić, Bojan. "A figure with Heesch number 6: pushing a two-decade-old boundary." The Mathematical Intelligencer (2021): 1-4.

Tilings by rhombs and their friends

• Kenyon, Richard. "Tiling a polygon with parallelograms." Algorithmica 9 (1993): 382-397.
• Kari, Jarkko, and Markus Rissanen. "Sub Rosa, a system of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry." Discrete & Computational Geometry 55 (2016): 972-996.
• Gähler, Franz, Eugene E. Kwan, and Gregory R. Maloney. "A computer search for planar substitution tilings with n-fold rotational symmetry." Discrete & Computational Geometry 53, no. 2 (2015): 445-465.
• Pautze, Stefan. "Cyclotomic aperiodic substitution tilings." Symmetry 9, no. 2 (2017): 19.

Pentagonal tilings

• Mann, Casey, Jennifer McLoud-Mann, and David Von Derau. "Convex pentagons that admit i-block transitive tilings." Geometriae Dedicata 194, no. 1 (2018): 141-167.
• Rao, Michael. "Exhaustive search of convex pentagons which tile the plane." arXiv preprint arXiv:1708.00274 (2017).

Miscellaneous

• Eppstein, David. "Diamond-kite adaptive quadrilateral meshing." Engineering with Computers 30 (2014): 223-235.
• Bašić, Bojan, Aleksa Džuklevski, and Anna Slivková. "Solutions to seven and a half problems on tilings." The Electronic Journal of Combinatorics (2023): P2-50.
• Li, Zhi, and Latham Boyle. "The Penrose Tiling is a Quantum Error-Correcting Code." arXiv preprint arXiv:2311.13040 (2023).
• Adams, Jadie, Gabriel Lopez, Casey Mann, and Nhi Tran. "Your friendly neighborhood Voderberg tile." Mathematics Magazine 93, no. 2 (2020): 83-90.
• Klaassen, Bernhard. "How to define a spiral tiling?." Mathematics Magazine 90, no. 1 (2017): 26-38.
• Matsushima, Toshiaki, Yoshihiro Mizoguchi, and Alexandre Derouet-Jourdan. "Verification of a brick Wang tiling algorithm." In 7th International Symposium on Symbolic Computation in Software Science, pp. 107-116. 2016.