Why do Pentagon tilings not solve the Einstein Problem?

They key is that such a tile can only tile the plane aperiodically. The other tiles you cite do not satisfy this criterion because while they can tile the plane in an aperiodic fashion, there exist other tilings of that shape that are periodic.


To add to heropup's answer, notice that even a square can tile the plane non-periodically (assuming by non-periodic we need a full rank sublattice of translational symmetries); just shift a subsequence of the columns in the usual integer lattice displaced tiling by some non-integer amount.

Take on the other hand the set of Robinson tiles. The shapes are important here, the decoration with lines on their interior just helps to prove their aperiodicity:

enter image description here

These tiles can tile the plane but the only way they can is in a non-periodic fashion. It is impossible for these tiles to tile periodically. These two criteria are necessary to be satisfied in order to be called an 'aperiodic' set of prototiles.

The full tiling looks as follows, where only the decoration has been left intact for the majority of the image:

enter image description here

Tags:

Geometry

Tiling

Discrete Geometry

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