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:
- Pick a point and color it green
- Find all points comma-equivalent to that point and color them red
- 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:
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color one point green
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and color the comma-equivalent points red:
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Do the same with a second point
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and another
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and three more
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Now we can pick one more, to get the same periodicity block Ehrlich shows with a parallelogram boundary:
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But such a block need not necessarily fit inside a parallelogram boundary, or a hexagon. We could pick another nearby point instead:
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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:
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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.
