CHAPTER 2
Periodic patterns
This chapter firstly reviews the ordering matrices of local patterns and trace
operators [2, 3]. It then derives rotational matrices Rn and Rn, and studies their
properties. The Rn-symmetry of the trace operator is also discussed. Finally, some
properties of periodic patterns on
Z2
are investigated. In particular, the
n l
0 k
-
periodic pattern is proven to be
n 0
0
nk
(n,l)
-periodic.
For clarity, two symbols on the unit square lattice Z2×2 are initially examined.
Chapter 4 will address more general situations.
2.1. Ordering matrices and trace operators
For given positive integers N1 and N2, the rectangular lattice ZN1×N2 is defined
by
ZN1×N2 = {(n1,n2)|1 n1 N1 and 1 n2 N2} .
In particular, Z2×2 = {(1, 1), (2, 1), (1, 2), (2, 2)}. Define the set of all global pat-
terns on Z2 with two symbols {0, 1} by
Σ2
2
= {0,
1}Z2
= U|U :
Z2
{0, 1} .
Here,
Z2
= {(n1,n2)|n1,n2 Z}, the set of all planar lattice points (vertices). The
set of all local patterns on ZN1×N2 is defined by
ΣN1×N2 = {U|ZN1×N2 : U
Σ2}.2
Now, for any given B Σ2×2, B is called a basic set of admissible local patterns.
In short, B is a basic set. An N1 × N2 pattern U is called B-admissible if for any
vertex (lattice point) (n1,n2) with 0 n1 N1 2 and 0 n2 N2 2, there
exists a 2 × 2 admissible pattern (βk1,k2 )1≤k1,k2≤2 B such that
Un1+k1,n2+k2 = βk1,k2 ,
for 1 k1,k2 2. Denote by ΣN1×N2 (B) the set of all B-admissible patterns on
ZN1×N2 . As presented elsewhere [2], the ordering matrices X2×2 and Y2×2 are
introduced to arrange systematically all local patterns in Σ2×2.
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