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A complete solution to the biased Alon-Krivelevich-Spencer-Szabó criterion problem for the discrepancy game Scaffolds for Higher Tropical Grassmannians: Foundations Krahn-Szegő type inequalities for graphs $\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs Modular inequalities and Alexander polynomials of pencil type conic-line arrangements Counterexamples regarding elementary symmetric partitions Bounds on Multiple $b$-Burst Deletion-Correcting Codes The mex statistic on combinatorial structures Proper Additive Edge Colorings of Regular Graphs Off-diagonal Ramsey numbers Semi-inducibility of 4-vertex graphs Word-Representable Co-Bipartite Graphs: Vertex Ordering, Representation Number, Speed, and Entropy Borel Kernels in Borel Directed Graphs Large induced subgraphs with prescribed degree parity Maya-Tupi graphs: a generalization of split graphs Recursions, Trains, Trees, and Combinatorial Rod Set Algebra Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs On the characterizations of $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ Ramsey numbers for 1-degenerate 3-graphs Bipartite graphs with minimum degree at least 15 are antimagic A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs Bounds on the game isolation number and exact values for paths and cycles Minors of non-hamiltonian polyhedra and the Herschel family Triangle-free subsets of the $r$-distance graph of the hypercube Notes on sum-free sets in abelian groups When are Hopf algebras determined by integer sequences? A note on the irrationality of $ζ_2(5)$ Support-sensitive bounds for shortest zero-sum subsequences Weighted Random Dot Product Graphs More efficient sifting for grid norms, and applications to multiparty communication complexity Ordered Yao graphs: maximum degree, edge density, and clique numbers Refined enumeration of planar Eulerian orientations Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent Bubble sort and Howe duality for staircase matrices Self-affine quadrangles On zero-sum Ramsey numbers modulo 3 Covering half-grids with lines and planes Borel Local Lemma: arbitrary random variables and limited exponential growth Dynamic capacity allocation of hybrid transportation units for cargo-hitching in urban public transportation systems An Efficient Genus Algorithm Based on Graph Rotations Neural Networks and (Virtual) Extended Formulations Some conjectures on $r$-graphs and equivalences Pointwise order of generalized Hofstadter functions G, H and beyond On the periodic decompositions of multidimensional configurations A q-analogue of graph independence polynomials with a group-theoretic interpretation Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic Higher-rank trees arising from polyhedral graphs $L^p$ averages of the Fourier transform in finite fields A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes Open problems on relations of numerical semigroups Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups A Mathematical Reconstruction of Endothelial Cell Networks Matchings in the hypercube with specified edges Geometric planted matchings beyond the Gaussian model Extensions and Deletions of matroid classes closed under flats String attractors and bi-infinite words Progressive and Rushed Dyck Paths Some results on Kleitman's conjecture Symmetric noncrossing partitions of an annulus with double points On two families of Nekrasov-Okounkov type formulas A remark on certain restricted plane partitions and crystal melting model Lecture hall graphs and the Askey scheme The Partial Partition Complex Generating functions of substitutions Standard Monomials for Positroid Varieties Tiling dense hypergraphs A structural duality for path-decompositions into parts of small radius The grid-minor theorem revisited Sumsets and entropy revisited Structure of cell decompositions in Extremal Szemerédi-Trotter examples Coloring, List Coloring, and Painting Squares of Graphs (and other related problems) Determining a Points Configuration from a Subset of the Pairwise Distances Improved Bounds for Multicovering Hypergraphs Totally Nonnegative Tropical Flags and the Totally Nonnegative Flag Dressian Enumeration of Odd-Dimensional Partitions modulo 4 Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups Multidimensional Manhattan Preferences Balancing permuted copies of multigraphs and integer matrices Mixing times of one-sided $k$-transposition shuffles Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices The pre-Pieri rules Synchronization of strongly connected partial DFAs and prefix codes Large rainbow matchings in edge-colored graphs Surface cluster algebra expansion formulae via loop graphs Lattice structure of Weyl groups via representation theory of preprojective algebras A Cambrian framework for the oriented cycle Strong Backdoors to Nested Satisfiability e-Valuate: A Two-player Game on Arithmetic Expressions -- An Update Combinatorial frameworks for cluster algebras Backdoors to Satisfaction Backdoors to Acyclic SAT Confidence-based Reasoning in Stochastic Constraint Programming A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives Palette-colouring: a belief-propagation approach Restructuring in Combinatorial Optimization Speeding up SAT solver by exploring CNF symmetries : Revisited Input Parameters Optimization in Swarm DS-CDMA Multiuser Detectors Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function Sortable Elements for Quivers with Cycles
The Path to Aperiodic Monotiles
Craig S. Kaplan · 2025-09-02 · via math.CO updates on arXiv.org

This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I contributed to the solution of the last of these problems in 2023, but many related questions remain open and worthy of study. My goal is to get more students excited about studying tiling theory.

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