Mathematical Characteristics of Periodicity Blocks
The periodicity blocks form a secondary, oblique lattice, superimposed on the first one. This lattice may be given by a function φ:
which is really a linear combination:
where point (x0, y0) can be any point, preferably not a node of the primary lattice, and preferably so that points φ(0,1), φ(1,0) and φ(1,1) are not any nodes either.
Then membership of primary nodes within periodicity blocks may be tested analytically through the inverse φ function:
Let
then let the pitch B(x,y) belong to the scale MB iff i.e.
For the one-dimensional case:
where L is the length of the unison vector,
For the three-dimensional case,
where is the determinant of the matrix of unison vectors.
Read more about this topic: Fokker Periodicity Blocks
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