## Hofstetter-4fold (plain)

### Info

The Viennese Artist Hofstetter Kurt found a new iteration method that produces aperiodic
tilings. The idea is illustrated in the figure below: one starts with a single square tile
of edge length one. A second square tile of the same size is put below the first one,
such that its SW corner is located at the midpoint of the first one. (The idea is that
the first tile covers a part of the second tile.) Then a third tile is put below the second
one, such that it’s NW corner is located at the centre of the second tile. A fourth tile
is then put (partly) below the others as indicated.

Now we repeat the construction with copies of this first generation patch on a larger scale:
place four copies of the first patch in a similar manner as above, where every distance
is now doubled. This yields a second generation patch with 16 squares altogether. Four copies
of this 2nd generation patch yield the 3rd generation patch and so on.

This construction can be applied to undecorated squares, but the squares can as well be
equipped with a decoration.
It was shown in [FH15] that a decorated version of the resulting
tilings, the Hofstetter arrow tilings,
are mld to a certain proper substitution
tiling. Using this it was shown that the arrow tilings are indeed
aperiodic, as well as
limitperiodic. We believe (but haven’t proven
yet) that the same is true for the undecorated tiling shown here.

Hofstetter Kurt generated several artistic versions of these tilings, using distinct
decorations of the prototile. One is shown under Hofstetter 4fold (decorated).

There are also 3-fold and 6-fold variants of this construction. Artistic versions
of these are Hofstetter 3fold tilings and
Hofstetter 6fold tilings.

### Substitution Rule

### Patch

### References

[FH]

FrettlĂ¶h, D and Hofstetter, K

**Inductive rotation tilings**

*Proc. Steklov Inst. Math.*
2015,
288,
pp. 247-258,