Tiles make great decorations. But when a surface is to be covered by regular polygons [sides and angles all equal], then only when all are shaped as equilateral triangles [3 sides], squares [4 sides] or regular hexagons [6 sides ] do they fill all the spaces. As regular pentagons [5 sides], heptagons [7 sides], octagons [8 sides], spaces remain. See proof here [ great site – thanks]
A surface covered with just one of the three regular polygons [3, 4, 6 sides] will leave no spaces, and the pattern will repeat infinitely.
A pattern repeating infinitely is relatively straightforward to understand, from a mathematical point of view. But for centuries, enquiring minds have considered what is the minimum number of tile shapes that are needed so that the pattern is NEVER repeated. Amongst those who have addressed the puzzle are astronomer Johannes Kepler [1571 to 1630][ his most important work was done while in Prague]. It was Kepler who first proved that the Earth and the other planets orbit the sun in slight ellipses and not in Copernicus’s precise concentric circles. [Nice planetary diagram here and again here]. Since the 1960s there have been other proposals for this minimum number of tiles : Hao Woa [“never possible to avoid repetition”], Robert Berger [20,426 tiles , later reduced to 104, then 92], Donald Knuth [92 tiles ].
But in 1974 Professor Sir Roger Penrose of Oxford University, and winner of the 2020 Nobel Prize in Physics, achieved this in as few as 2 tiles ! These included :
Kite and Dart.
2 Rombus: each with sides of equal length, but with different angles.
Kite and Dart tiles must be arranged so that the red dots shown line up when the tiles are placed next to each other. So, for example, a dart cannot be placed directly above a kite.
More explanation here.
Cut out your own Penrose tiles here
Examples of Penrose tiling in the public arena are sadly too rare.
In London : At Queen Mary’s University of London, School of Mathematical Sciences.
In Oxford : The Andrew Wiles Building, Institute of Mathematics, Oxford.
In Helsinki, Finland :
You can draw your own Penrose tilng here – fun. You can even design a tiling project here – thanks.
And Penrose tiles make great fashion items. Lots here.
There are notable scientific consequences of the mathematics supporting Penrose tiling, as mentioned here.
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John's World View 