Ternary parallelogram scales are MOS substitution: Difference between revisions
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After rotating ''w'', we may assume that (0, 0) is labeled 0. The labeling now extends to a surjective homomorphism <math>\varphi: \mathbb{Z}^2\langle \mathbf{v},\mathbf{w}\rangle \to \mathbb{Z}/mn\mathbb{Z},</math> where {{nowrap|φ('''v''') {{=}} ''k''<sub>'''v'''</sub>}} and {{nowrap|φ('''w''') {{=}} ''k''<sub>'''w'''</sub>.}} φ has {{nowrap|[0 : ''m''] × [0 : ''n'']}} as a fundamental domain. | After rotating ''w'', we may assume that (0, 0) is labeled 0. The labeling now extends to a surjective homomorphism <math>\varphi: \mathbb{Z}^2\langle \mathbf{v},\mathbf{w}\rangle \to \mathbb{Z}/mn\mathbb{Z},</math> where {{nowrap|φ('''v''') {{=}} ''k''<sub>'''v'''</sub>}} and {{nowrap|φ('''w''') {{=}} ''k''<sub>'''w'''</sub>.}} φ has {{nowrap|[0 : ''m''] × [0 : ''n'']}} as a fundamental domain. | ||
=== Lemma: Any ''m'' × ''n'' window in ℤ<sup>2</sup> has the same cyclic ordering of elements === | |||
=== Step 2: By ternarity, exactly one of the 1-step vectors is parallel to a coordinate axis === | === Step 2: By ternarity, exactly one of the 1-step vectors is parallel to a coordinate axis === | ||