Planar tiling
\(A_2\) Online Tiler
Grow Hat, Turtle, and custom tile patches, or test whether they fill a chosen boundary.
A collection of mathematical investigations
Geometrically Constrained
Tree Search
Building structure, learning constraints, and checking what follows. Explore the projects through three connected perspectives.
Start with the essay & interactive Turtle demoPure Math
08Explore tilings, packings, and topology through interactive examples.
Aperiodic order
Penrose Model-Set Tiler
Explore thin and thick rhombs on a cut-and-project point set, with live search and backtracking.
Spatial tiling
3D Lattice Tiler
Build with lattice polyhedra and custom polycubes. Inspect periodic tilings, frontier growth, and the research catalogue.
Substitution
3D Reptiles
Inspect the Chair44 tile, its markings, and its assembly. Explore rotating orientations in the quaquaversal construction.
Obstruction certificates
Nine-Cube Non-Tiler
Follow the nine-cube cross investigation through geometric constraints, search trees, and checkable certificates.
Finite coronas
Heesch Problems
Explore finite layers around a tile and the local obstructions that stop a patch from extending.
Hyperbolic geometry
Hyperbolic Tiling Lab
See the Böröczky family in hyperbolic space, then inspect the branches and backtracks of a live search.
Topology
Knot Movies
Follow changing strands through a verified sequence of Reidemeister moves, one diagram at a time.
Applied Math
04Explore geometric toy models, constrained packings, and models you can build.
Geometric toy models
Crystallography Toy Models
Explore basic 3D lattice tilings as geometric toy models for crystallography.
Constrained packing
Held Circle Packings
Search for circle arrangements under geometric constraints and inspect how local choices shape the packing.
Packing & structure
Tetrahedron Atlas
Compare published tetrahedron quasicrystals, periodic approximants, and dense packings in an interactive three-dimensional viewer.
Physical models
Turtle & Trefoil Models
Rotate and animate tile assemblies, then explore their model descriptions and physical construction details.
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