Origami & Folding
what one sheet can do
Generator

Two packings, one radius — the contact graph generator

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.

contact-graph 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

Two packings, one radiusTwo arrangements of the same number of discs, each returned by a complete run of the same search, agreeing about the largest radius to four decimal places. The lines join discs that touch, and the crease pattern a designer builds is built from those lines. The two are not the same graph.6 flaps at radius 0.1875883 contact graphs among the runs that agree about itone run5 contacts · 5 against the paper's edgeanother run4 contacts · 5 against the paper's edge

view: "graphs", n: 5, tol: 0.0004

Two packings, one radiusTwo arrangements of the same number of discs, each returned by a complete run of the same search, agreeing about the largest radius to four decimal places. The lines join discs that touch, and the crease pattern a designer builds is built from those lines. The two are not the same graph.5 flaps at radius 0.2070532 contact graphs among the runs that agree about itone run4 contacts · 5 against the paper's edgeanother run4 contacts · 8 against the paper's edge

view: "graphs", n: 6, tol: 0.0004

Two packings, one radiusTwo arrangements of the same number of discs, each returned by a complete run of the same search, agreeing about the largest radius to four decimal places. The lines join discs that touch, and the crease pattern a designer builds is built from those lines. The two are not the same graph.6 flaps at radius 0.1875883 contact graphs among the runs that agree about itone run5 contacts · 5 against the paper's edgeanother run4 contacts · 5 against the paper's edge

view: "classes", counts: [4, 5, 6, 7, 8], tol: 0.0004

How often the radius fails to decideFor each number of flaps, how many provably different contact graphs the search returned among runs that agree about the best radius. At some flap counts it is one and at others it is several — and which it is at a given count is not something a designer can tell from the number the search reports.runs that agree about the radius, sorted by the graph they returnedthe count is a lower bound: graphs the label separates are certainly different4 flapsone graph at r = 0.250005 flaps2 graphs at r = 0.207056 flaps3 graphs at r = 0.187597 flapsone graph at r = 0.174368 flapsone graph at r = 0.17033

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.

Two packings, one radius

A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.

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