Vertex degree — the series
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Nothing meets at three
Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.
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Six creases and the same straight line
The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.
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A knife edge nine decimals wide
Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.