Origami & Folding
what one sheet can do
Generator

The excess fixes the family, not the member

A generator in the folding nobody designed library, called 10 times across 2 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

wave-family is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

The excess fixes the family, not the memberA strip whose rim has grown longer than its span has to put the extra length somewhere. Spending it in two waves and in eight are the same metric — identical arc length, identical excess — and the geometry has nothing to say about which. Ranking them needs a rule about the material, which is not a question this site answers.one excess of length, 4 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 276amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

view: "confined", depths: [0.04, 0.02, 0.01]

The container picks the memberEvery member of a wave family carries the same excess of length, and the geometry is indifferent between them. Amplitude times wave count is constant across the family, so a container of a stated depth is a floor on the number of waves — and a tighter container is a higher floor, exactly inversely. Bending cannot make this choice: it rises at every step, so it always prefers the fewest waves.wavesamplitudefits inside2∫κ² = 170.03980.043∫κ² = 390.02660.045∫κ² = 1080.01590.04, 0.028∫κ² = 2760.01000.04, 0.02, 0.0112∫κ² = 6210.00660.04, 0.02, 0.0120∫κ² = 17240.00400.04, 0.02, 0.01depth 0.04 needs 2 waves or more · depth 0.02 needs 4 waves or more · depth 0.01 needs 8 waves or moreexcess 6.0% · amplitude × waves = 0.0797 throughout · floor = that constant ⁄ depth

excess: 0.06, waves: [2, 3, 5, 8, 12, 20]

The excess fixes the family, not the memberA strip whose rim has grown longer than its span has to put the extra length somewhere. Spending it in two waves and in eight are the same metric — identical arc length, identical excess — and the geometry has nothing to say about which. Ranking them needs a rule about the material, which is not a question this site answers.one excess of length, 6 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 27612 wavesheight 0.0066∫κ² ds = 62120 wavesheight 0.0040∫κ² ds = 1724amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

view: "confined", depths: [0.03, 0.012, 0.005]

The container picks the memberEvery member of a wave family carries the same excess of length, and the geometry is indifferent between them. Amplitude times wave count is constant across the family, so a container of a stated depth is a floor on the number of waves — and a tighter container is a higher floor, exactly inversely. Bending cannot make this choice: it rises at every step, so it always prefers the fewest waves.wavesamplitudefits inside2∫κ² = 170.0398none of them3∫κ² = 390.02660.035∫κ² = 1080.01590.038∫κ² = 2760.01000.03, 0.01212∫κ² = 6210.00660.03, 0.01220∫κ² = 17240.00400.03, 0.012, 0.005depth 0.03 needs 3 waves or more · depth 0.012 needs 7 waves or more · depth 0.005 needs 16 waves or moreexcess 6.0% · amplitude × waves = 0.0797 throughout · floor = that constant ⁄ depth

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

The container picks the member

Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.

The excess does not choose its waves

A rim that has grown longer than its span has to put the extra length somewhere, and two large waves and eight small ones are the same metric — identical arc length, identical excess. Geometry fixes the family and is completely indifferent about the member.

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