🔲 Tessellations — Regular & Penrose Tilings

🔲 What It Demonstrates

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.

How to Use

• Select a Tiling Type from the dropdown.
• Adjust Tile Size (for regular tilings) or Penrose Depth (subdivision depth).
• Toggle Colour fill and Tile outlines for visual style.
• Enable Tile labels to see shape names on each tile (useful for Penrose kite/dart identification).
• Use Redraw to regenerate with current parameters.

Did You Know?

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.