Periodicity blocks notes

In an “excursion” from A gentle introduction to Fokker periodicity blocks, Paul Ehrlich discusses non-parallelogram periodicity blocks. He doesn’t seem to mention the following (though I’m sure others have).

If we define a periodicity block as a set of points in an n-dimensional lattice of JI intervals having the property that, for some set of n independent commas, no two points in the set are comma-equivalent (i.e., can be connected by any linear combination of those commas), while every point outside the set is comma-equivalent to a point in the set, then it follows that we can construct a periodicity block as follows:

  1. Pick a point and color it green
  2. Find all points comma-equivalent to that point and color them red
  3. Repeat steps 1-2 until all points are colored

For instance: On the 5-limit lattice, two commas are 25/24 and 81/80 corresponding to vectors (-1 2) and (4 -1). Start with a lattice colored black:

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

color one point green

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

and color the comma-equivalent points red:

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

Do the same with a second point

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

and another

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

and three more

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

Now we can pick one more, to get the same periodicity block Ehrlich shows with a parallelogram boundary:

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

But such a block need not necessarily fit inside a parallelogram boundary, or a hexagon. We could pick another nearby point instead:

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

It need not even be simply connected (if you understand what I mean by “simply connected” when we’re talking about individual points); for example:

* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *

though I don’t know why you’d want to. Any of these will give a periodicity block as defined above; any will tile the plane.