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 at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

circle-packing 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

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 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.

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.

Every generator · The designing a base field · The patterns a reader can fold