Origami & Folding
what one sheet can do
Generator

A third of the face, and half the sheet

A generator in the flat-folding library, called 7 times across 2 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

colour-cost is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

A third of the face, and half the sheetHow much of the folded sheet shows its front and how much shows its reverse, as the flap that makes the colour change widens. The reverse rises as fast as the flap; the front falls twice as fast, because the flap covers as much paper as it is.00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

flaps: [0.1, 0.2, 0.3, 0.4], show: 0.3

A third of the face, and half the sheetHow much of the folded sheet shows its front and how much shows its reverse, as the flap that makes the colour change widens. The reverse rises as fast as the flap; the front falls twice as fast, because the flap covers as much paper as it is.00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

view: "buried", samples: 120

The other side is there, and it is underneathFor each printed pattern: the share of its folded footprint that has panels of both colours lying over it, and how deep the stack is where they do. The reverse side is almost never missing; it is almost always buried.patternfootprint with both colours over itlayers thereThe preliminary base8 panels, 4 one way up and 4 the other100%8.0The Miura fold24 panels, 12 one way up and 12 the other100%9.2The square twist9 panels, 5 one way up and 4 the other65%4.1The hexagon twist13 panels, 7 one way up and 6 the other72%4.1The Yoshimura pattern65 panels, 32 one way up and 33 the other100%60.0Fold and cut — the triangle7 panels, 4 one way up and 3 the other4%6.4The tapered corrugation28 panels, 14 one way up and 14 the other100%8.4The waterbomb tessellation52 panels, 26 one way up and 26 the other100%31.1

flaps: [0.08, 0.16, 0.24, 0.32], show: 0.16

A third of the face, and half the sheetHow much of the folded sheet shows its front and how much shows its reverse, as the flap that makes the colour change widens. The reverse rises as fast as the flap; the front falls twice as fast, because the flap covers as much paper as it is.00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

Decided before the design

A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.

Every generator · The flat-folding field · The patterns a reader can fold