Origami & Folding
what one sheet can do
Generator

The last free parameter — the universal molecule generator

A generator in the designing a base library, called 20 times across 4 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.

universal-molecule 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

The last free parameterA region left between the flaps, filled by shrinking its outline inward at a uniform rate. Every corner traces a straight line as it goes, and those traces are the creases. The packing fixed everything else about the design; how far this shrink runs before the outline collapses is the one number still free, and it is computed here rather than chosen.the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

poly: [0.12,0.14, 0.88,0.2, 0.8,0.86, 0.3,0.9], steps: 5, view: "molecule"

The last free parameterA region left between the flaps, filled by shrinking its outline inward at a uniform rate. Every corner traces a straight line as it goes, and those traces are the creases. The packing fixed everything else about the design; how far this shrink runs before the outline collapses is the one number still free, and it is computed here rather than chosen.the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

steps: 5

The last free parameterA region left between the flaps, filled by shrinking its outline inward at a uniform rate. Every corner traces a straight line as it goes, and those traces are the creases. The packing fixed everything else about the design; how far this shrink runs before the outline collapses is the one number still free, and it is computed here rather than chosen.the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

poly: [0.18,0.16, 0.84,0.14, 0.9,0.72, 0.5,0.92, 0.14,0.66], steps: 6

The last free parameterA region left between the flaps, filled by shrinking its outline inward at a uniform rate. Every corner traces a straight line as it goes, and those traces are the creases. The packing fixed everything else about the design; how far this shrink runs before the outline collapses is the one number still free, and it is computed here rather than chosen.the shrink6 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.6115 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

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.

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

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