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 Penrose 5 fold tiling with kites (red) and darts (white) A quasicrystal

Tiling  is  another name for tessellations. There can be periodic tiling that has repeating patterns.  Periodic tilings can also include more complex shapes.  The Alhambra in Spain contains very complex and intricate tiles both with geometric shapes as well as different patterns.   Five fold symmetry was thought not to exist until Roger Penrose in 1974 discovered tow basic sets of tiles that can exhibit 5-fold symmetry.  They are called “dart” and “kite” shapes.  The shapes are composed of the isosceles triangles.  Penrose is the scientist that discovered that Einstein’s theory of general relativity predicts defects.    Penrose although an astrophysicist, engages in what is considered recreational math.  That is how he discovered the 5-fold symmetry tiling.

Tiles can also be aperiodic which refers to a non repeating pattern.  Ammann-Beenker Tile is a famous example of an aperiodic tilting.  In 1977 R. Ammann discovered an 8-fold non repeating tile.  In 1982, F. Beenker described the algebraic properties.  Mathematicians had described these tilings but had not yet found a physical example.  In 1984, quasicrystals were grown for the first time with repeating clusters which overlaps and shares atoms with their neighbor.  A mathematical prediction comes true!

 Ammann-beenker Tile
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