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What other shapes will fit together to cover a surface? Surprisingly, pentagons will (but only across a limited area). Despite what your eyes may tell you, there are no curves in Wheels Within Wheels.

Pentagons are 5-sided shapes and fall into three groups:

•  regular have all angles equal and all sides equal
•  semi-regular have a line of symmetry
•  irregular.

Regular pentagons will not fit together to cover a surface. There will always be gaps. Semi-regular pentagons will fit together and are very common in Islamic tiling. The patterns they form are known as Cairo tessellations.

It used to be thought that irregular pentagons would not fit together then five types were found, by K Reinhardt, in 1918. In 1967 Richard Kerschner found three more and Richard E James found another in 1975.

In 1976 Marjorie Rice, who was a housewife with no mathematical background, found four more.  A fourteenth was found by Rolf Stein in 1985. It is not known whether this is a complete list.

The pentagons used in this afghan are special because they have all sides equal but all five angles are different. The number of ways they will fit together is very limited. The angles are 100, 80, 140, 60 and 160 degrees.

The afghan shown in the photo consists of 90 identical pieces which are stitched together. The stitching is very easy because they are all in garter stitch and the edges are easy to match together. The pieces are exactly alike except that some have been turned over. They are all made in the same way and turned over as required.