Penrose Tiling Growth Screensaver
This screensaver builds a real Penrose tiling live, using kite and dart tiles combined through genuine substitution rules. Unlike a regular grid, a true Penrose tiling has real five-fold symmetry and — provably — never repeats itself no matter how far it's extended.
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How the Penrose Tiling Growth screensaver works
The tiling is built from two real tile shapes, kites and darts, matched edge to edge under strict real Penrose matching rules that forbid ordinary periodic tilings.
Growth proceeds by real substitution: each tile is subdivided into smaller kites and darts following Penrose's inflation rules, then the whole pattern is scaled back up.
Repeating this substitution process is what produces the tiling's real aperiodicity — mathematically, the pattern is proven to never exactly repeat, however far it extends.
The result carries genuine five-fold rotational symmetry — a kind of order regular square or triangular grids can't achieve, since only 2, 3, 4, and 6-fold tilings can repeat periodically.
Roger Penrose and tilings with no repeat
In the 1970s the mathematician Roger Penrose found sets of shapes that tile the plane only in patterns that never repeat. This followed Robert Berger’s earlier aperiodic set of 20,426 Wang dominoes; Penrose’s first tiling, from a 1974 paper, used six prototiles, and he later reduced the set to two shapes, the kite and the dart, with matching rules that force the non-repeating arrangement.
Martin Gardner publicised the tilings in his January 1977 Scientific American column. Their wider importance came with quasicrystals: in 1982 Dan Shechtman saw an electron diffraction pattern with a symmetry thought impossible in crystals, and the discovery won the 2011 Nobel Prize in Chemistry. Penrose tilings are often cited as having helped explain how an ordered but non-periodic structure can exist.
Look for the golden ratio. Inflating the tiling rescales the tiles by φ, and in a large kite-and-dart tiling the ratio of kites to darts approaches φ. Any finite patch you see occurs infinitely often elsewhere in the full tiling, which is why a region can look familiar without the whole ever repeating.
A worked example
Zoom your attention on any local cluster of kites and darts and you'll never find another region elsewhere in the tiling that matches it exactly — a real, provable property of Penrose tilings, unlike an ordinary repeating floor pattern.
Settings & tips
- Kite color and dart color recolour the two real tile shapes independently.
- Growth restarts from a fresh seed once the tiling fills the visible area.
- Look for the real five-pointed star clusters that appear naturally at several scales.
Frequently asked questions
- Is it really mathematically proven never to repeat?
- Yes — Penrose tilings are a well-known real mathematical result: no true Penrose tiling can be periodic, however far it's extended.
- What are kites and darts?
- The two real rhombus-derived tile shapes Roger Penrose used to build his classic aperiodic tiling in the 1970s.
- Is a key needed?
- No — it's a pure local computation.
- Is it free?
- Yes — free, no download, in your browser.
- What is a quasicrystal?
- A quasicrystal is a solid whose atoms are ordered but not periodic, which allows rotational symmetries that ordinary crystals cannot have. Dan Shechtman received the 2011 Nobel Prize in Chemistry for discovering quasicrystals.
- Why is the golden ratio involved?
- Each inflation step rescales the tiles by the golden ratio, and in large kite-and-dart tilings the ratio of kites to darts approaches it.