Origami & Folding
what one sheet can do
Generator

Which of a pattern's symmetries its letterings keep

A generator in the designing a base library, called 6 times across 1 essay. 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.

pattern-symmetry 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

Which of a pattern's symmetries its letterings keepEvery symmetry that carries a crease pattern to itself, and how many of the mountain-and-valley letterings that fold are carried to themselves by it. A symmetry no lettering keeps is a symmetry the drawing has and no folded object made from it does.The preliminary base: 8 symmetries, 112 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turnnone of 112 — this symmetry cannot be foldeda half turnnone of 112 — this symmetry cannot be foldedthree quarters of a turnnone of 112 — this symmetry cannot be foldeda mirror across the sheet12 of 112a mirror up the sheet12 of 112a mirror in one diagonal12 of 112a mirror in the other diagonal12 of 112

view: "kept", pattern: "preliminary-base"

Which of a pattern's symmetries its letterings keepEvery symmetry that carries a crease pattern to itself, and how many of the mountain-and-valley letterings that fold are carried to themselves by it. A symmetry no lettering keeps is a symmetry the drawing has and no folded object made from it does.The preliminary base: 8 symmetries, 112 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turnnone of 112 — this symmetry cannot be foldeda half turnnone of 112 — this symmetry cannot be foldedthree quarters of a turnnone of 112 — this symmetry cannot be foldeda mirror across the sheet12 of 112a mirror up the sheet12 of 112a mirror in one diagonal12 of 112a mirror in the other diagonal12 of 112

view: "kept", pattern: "square-twist"

Which of a pattern's symmetries its letterings keepEvery symmetry that carries a crease pattern to itself, and how many of the mountain-and-valley letterings that fold are carried to themselves by it. A symmetry no lettering keeps is a symmetry the drawing has and no folded object made from it does.The square twist: 8 symmetries, 256 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turn4 of 256a half turn16 of 256three quarters of a turn4 of 256a mirror across the sheetnone of 256 — this symmetry cannot be foldeda mirror up the sheetnone of 256 — this symmetry cannot be foldeda mirror in one diagonal32 of 256a mirror in the other diagonal32 of 256

view: "halves"

Two eight-fold drawings that lose different halvesThe preliminary base and the square twist are both carried to themselves by all eight symmetries of the square. Their letterings keep different ones: the base keeps every mirror and no rotation, and the twist keeps the diagonal mirrors and neither of the others.how many letterings each symmetry carries to itselfa dash is none at allquarter turnhalf turnthree quartersmirror acrossmirror updiagonalanti-diagonalThe preliminary base112 letterings———12121212The square twist256 letterings4164——3232

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 symmetry the letters cannot keep

Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.

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