Ternary parallelogram scales are MOS substitution: Difference between revisions
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# the two other 1-step vectors are not found on the axis orthogonal to the first vector, as that would imply order ''mn'' for ''k''<sub>'''w'''</sub>, nor on {{nowrap|[-''m'' + 1 : 0] × {0}}} as that contradicts ''k''<sub>'''v'''</sub> having order ''mn'' | # the two other 1-step vectors are not found on the axis orthogonal to the first vector, as that would imply order ''mn'' for ''k''<sub>'''w'''</sub>, nor on {{nowrap|[-''m'' + 1 : 0] × {0}}} as that contradicts ''k''<sub>'''v'''</sub> having order ''mn'' | ||
# φ is minimally periodic under translation by ''n'''''w'''. Hence the two non-axial 1's on the grid from the 0 at the origin are translated by ''n'''''w'''. | # φ is minimally periodic under translation by ''n'''''w'''. Hence the two non-axial 1's on the grid from the 0 at the origin are translated by ''n'''''w'''. | ||
# Since ''m'' is of order ''n'', ''m'' must be found on the '''w'''-axis as well. | # Since ''m'' is of order ''n'', and the cyclic group {{nowrap|ℤ/''mn''ℤ}} has at most one subgroup of any given order, ''m'' must be found on the '''w'''-axis as well. | ||
==== Template word is MOS ==== | ==== Template word is MOS ==== | ||