Triangular · Square · Hexagonal · Penrose kite–dart · Pan & zoom
A tessellation (or tiling) is a covering of the plane by geometric shapes with no gaps or overlaps. Only three regular polygons tile the plane by themselves: equilateral triangles, squares and regular hexagons. The Penrose tiling, discovered by Roger Penrose in the 1970s, uses two shapes (a "kite" and a "dart" derived from the golden ratio) to create an aperiodic tiling — one that never repeats yet fills the plane completely, exhibiting 5-fold symmetry forbidden in periodic crystals.
In 2023 a single shape called the "einstein" (German for "one stone") was discovered — a 13-sided polygon that tiles the plane aperiodically all by itself, solving a 60-year-old open problem. Penrose tilings appear in nature as quasicrystals, first observed in 1982 by Dan Shechtman (Nobel Prize in Chemistry, 2011). Medieval Islamic architecture at the Darb-i Imam shrine in Isfahan, Iran (1453) contains near-perfect Penrose-like patterns, predating Penrose's work by five centuries.
This simulator draws tessellations — coverings of the plane by polygons that leave no gaps or overlaps. It renders the three regular tilings (triangular, square and hexagonal), generated row by row on a 2D canvas, and the famous Penrose kite–dart tiling, built by repeatedly subdividing triangles using the golden ratio φ ≈ 1.618. The regular tilings are periodic, while the Penrose pattern is aperiodic: it fills the plane forever yet never repeats, displaying the 5-fold symmetry forbidden in ordinary crystals.
Four tilings rendered to a canvas. Triangular, square and hexagonal tilings are laid out on a regular lattice. The Penrose tiling starts from ten kite-half triangles arranged in a star, then applies a deflation rule that splits each triangle using divisions of 1/φ, so the kite-and-dart pattern grows finer at each subdivision step.
Pick a tiling from the Tiling Type dropdown. The Tile Size slider (15–100) scales the regular tilings; Penrose Depth (2–8) sets how many subdivision passes the Penrose tiling runs. Toggle tile outlines, colour fill and tile labels (showing Kite/Dart names). Drag the canvas to pan, press Redraw to regenerate, or Reset View to recentre.
In 2023 a single 13-sided shape nicknamed the "einstein" (German for "one stone") was found to tile the plane aperiodically by itself, settling a 60-year-old open problem about whether one such tile could exist.
A tessellation, or tiling, is a way of covering a flat plane completely with one or more shapes so there are no gaps and no overlaps. Among regular polygons, only equilateral triangles, squares and regular hexagons can tile the plane on their own, because their interior angles divide evenly into 360 degrees.
It is built by deflation. The simulation begins with ten thin triangles (kite halves) arranged in a star, then runs a subdivision rule that splits every triangle into smaller kite and dart pieces, placing new vertices at distances scaled by 1/φ where φ is the golden ratio. The Penrose Depth slider controls how many times this subdivision is applied.
Tiling Type chooses between the triangular, square, hexagonal and Penrose patterns. Tile Size scales the regular tilings, while Penrose Depth sets the recursion count for the Penrose pattern. The checkboxes toggle outlines, colour fill and Kite/Dart labels, and dragging the canvas pans the view.
An aperiodic tiling fills the plane without any repeating translational pattern, so you can never shift it to land exactly on itself. The kite-and-dart pieces fit together according to matching rules that force this non-repeating arrangement, while still showing local 5-fold symmetry that periodic crystals cannot have.
The regular tilings are geometrically exact lattices. The Penrose tiling uses the genuine deflation method based on the golden ratio, so its structure is faithful, although at low depths only a finite, approximate patch is shown. Real quasicrystals, discovered by Dan Shechtman in 1982, share this same forbidden symmetry.