Recently, Prof Roger Penrose has come up with some tilings that exhibit five-fold symmetry yet which do not repeat themselves for which the technical term is aperiodic or quasiperiodic. When they appear in nature in crystals, they are called quasicrystals. They were thought to be impossible until fairly recently. There is a lot in common between Penrose's tilings and the Fibonacci numbers.
The picture above is made up of two shapes of rhombus or rhombs - that is, "pushed over squares" where each shape has all sides of the same length. The two rhombs are made from glueing two of the flat pentagon triangles together along their long sides and the other from glueing two of the sharp pentagon triangles together along their short sides.
This picture is part of the HyperCard stack developed by me (Ron Knott) available from this site. [Download 156K binhex file.] The tiling picture was made with Quasitiler 3.0 which is a web-based tool and its link mentions more references to Penrose tilings.
A floor has been tiled with Penrose Rhombs at Wadham College at Oxford University.
I plan more to follow here, but in the meantime, here are some interesting links to the Penrose tilings at other sites.
• The Golden section and Penrose Tilings .
• Here are some ready-to-photocopy Penrose tiles for you to photocopy and cut-out and experiment with making tiling patterns.
• Penrose's rhombs (a fat and a thin diamond) tilings.
1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 . .More. .
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