Origami & Folding
what one sheet can do
Generator

Packing the flaps

A generator in the designing a base library, called 6 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.

circle-packing 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

Packing the flapsOne circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together.leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

r: 0.22

Packing the flapsOne circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together.leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

r: 0.2

Packing the flapsOne circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together.leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 66% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

r: 0.24

Packing the flapsOne circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together.leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 78% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

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.

Packing is the hard part

Once a subject is a set of circles, designing the model is fitting them into a square. That step has no general algorithm, no known optimum, and it is where every remaining difficulty in origami design now sits.

The flap nobody holds

An optimal packing is presented as an answer: here are the circles, here is where they go. For some numbers of flaps that is not what it is. The best arrangement of seven discs in a square leaves one of them free to wander over an eleventh of the sheet without changing the answer at all — and the algorithm reports one point of that region and stops.

All the figures · The designing a base field · The patterns a reader can fold