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 Rnsymmetry 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
= UU :
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 = {UZN1×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 Badmissible 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 Badmissible 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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