Ternary parallelogram scales are MOS substitution: Difference between revisions
| Line 28: | Line 28: | ||
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 under the φ-labeling ==== | ==== Lemma 1: Any ''m'' × ''n'' window in ℤ<sup>2</sup> has the same cyclic ordering of elements under the φ-labeling ==== | ||
For any integers ''i''<sub>0</sub> and ''j''<sub>0</sub>, if {{nowrap|φ((''i''<sub>0</sub>, ''j''<sub>0</sub>)) {{=}} ''a'',}} then for any (''i'', ''j'') in [0 : ''m''] × [0 : ''n''], we have | For any integers ''i''<sub>0</sub> and ''j''<sub>0</sub>, if {{nowrap|φ((''i''<sub>0</sub>, ''j''<sub>0</sub>)) {{=}} ''a'',}} then for any (''i'', ''j'') in [0 : ''m''] × [0 : ''n''], we have | ||