Aperiodicity of Penrose tilings

The text presents an argument to show that Penrose tilings must be aperodic which depends on inflation of the tiling. However any periodic tiling is built on a lattice of polygons. In particular, the semiregular tessellations are built on lattices of  rhombs or hexagons. All of these lattices have symmetry of scale - that is, for example, the rhombs can be grouped 4 at a time to form larger, similar rhombs, which can serve as a framework for an inflated version of the original tiling, and this process can be repeated to produce larger and larger versions of the original tiling.  Thus if P and P' are minumum distance d apart in the original tiling, they will eventually be contained in the same tile in a large enough inflation.
        Thus according to the text argument, this shows the tiling cannot be periodic, But we already know that the examples we are considering are periodic, so this argument is insufficient, and cannot constitute a complete proof that Penrose tilings are not periodic.

Tessellation using midpoint rotation on all 3 sides of triangle

parallelogram grid tiling

The parallelogram tile is modified by parallel translation in both directions.

Further information on tilings

This site gives a more thorough study of tilings using regular polygons.
http://www.probabilitysports.com/tilings.html

Totally Tessellated

This is a good source for information about plane tilings.


Totally tessellated