A field guide to aperiodic order
Known tilings, in motion.
Change the shapes. Move the section. Explore how different patterns inhabit the same number field.
Penrose rhombs
What you are seeing
Tile inspector
Click a tile to see its shape and its exact cyclotomic address.
Construction, matching rules, and numerical scope
This atlas visualizes published constructions. Reveal is an animation of a computed patch, not a tile-by-tile search or a demonstration of learned markings. The phase control changes the affine section in the higher-dimensional construction; tiles switch locally as it crosses boundaries. Highlights identify tiles that have just appeared.
Rhomb examples come from multigrid duality. Tübingen triangles and their sevenfold relatives come from sections of root-lattice Voronoi cells and projection of the dual Delaunay triangles. Penrose kite-and-dart and Robinson triangles are local transformations of the same Penrose rhomb patch. The original pentagonal Penrose example is a finite reference patch with reveal and inspection.
Vertex addresses are integer coefficients reduced by the cyclotomic polynomial. Grid intersections and section membership use floating-point calculations. Numerically ambiguous section crossings are reported rather than silently omitted. These are visual construction controls, not exact infinite-tiling certificates or the GCTS point–candidate reference algorithm.
The second panel shows one other field embedding of the displayed vertices. In higher-degree fields it is only one projection of the internal coordinates. It does not claim that the point cloud itself is the complete acceptance window.
Tile colors distinguish shapes, except for the three pentagon classes in the original Penrose reference. They are not matching constraints. Only the optional Penrose arrows display an actual matching rule here; the dedicated search pages show additional published markings.