I am interested in Cairo pentagonal tiling.
In following link of wikipedia: https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling
It claims "The Cairo pentagonal tiling has two lower symmetry forms given as monohedral pentagonal tilings types 4 and 8".
I am totally confused it. Well, I think Cairo pentagonal tiling should be pentagonal tilings types 2 and 4, but not 8.
Can someone give me a right answer?
Thanks
I agree with you.
The type 8 pentagon tiling has one degree of freedom, and although you can choose it so that clusters of four tiles form a large hexagonal shape similar to that seen in the Cairo tiling, the tilings are never the same, and the type 8 tile can never be the same shape as the Cairo tile shape. It is not possible for the type 8 tile to be two 90 degree angles at the same time like the Cairo tile has.
The type 8 tile only becomes symmetric when the degree of freedom is chosen such that all sides are equal, and with that tile you can make a cairo-like tiling, but that is not the same arrangement as the general type 8 tiling.
The Cairo tile is certainly of type 4 (with the added restriction that it has 4 equal sides) and also of type 2 (the two 90 degree vertices are the ones that add to 180).