Origami & Folding
what one sheet can do
Concept

Self-contact — where it appears

Two pieces of one sheet arriving at the same place in a folded state. Whether that is allowed is decided by the layer order rather than by the geometry, which is why a collision is really a question about an ordering.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

panels with nothing below them, and where they aresquare ×1125 panels, 16 of them touching the edge · all 1 at the edgesquare ×2281 panels, 32 of them touching the edge · all 2 at the edgesquare ×33169 panels, 48 of them touching the edge · all 3 at the edgetriangular ×1369 panels, 39 of them touching the edge · all 3 at the edgetriangular ×25233 panels, 79 of them touching the edge · all 5 at the edgehexagonal ×1469 panels, 39 of them touching the edge · all 4 at the edgehexagonal ×27233 panels, 79 of them touching the edge · all 7 at the edgehexagonal ×310493 panels, 119 of them touching the edge · all 10 at the edgeelongated ×12105 panels, 48 of them touching the edge · all 2 at the edgeelongated ×23369 panels, 96 of them touching the edge · all 3 at the edgethe sheet these letters belong to has no such panel at all

An order with no least element

Enumerating every way a folded pattern can be stacked works by building upward from a panel with nothing below it. The smallest square twist patch has exactly one such stacking and takes eleven thousand steps to find it. The pattern that patch was cut from has no panel with nothing below it at all, so the enumeration has nothing to start from — and the sheet is perfectly well stacked anyway.

rigid · Self-contact one period of the square grid's twist tessellationa ring is where a crease leaves and returns on the far side40 crease pieces → 32 creases25 drawn panels → 16 panels16 vertices, every one interiorV − E + F = 0mountainvalleyraw edge

Two panels that are one panel

Paper cannot pass through paper, and every test for it compares pairs of panels. On a glued sheet two pieces of the drawing can be the same piece of paper — so a test that does not know the identification either reports a collision between a panel and itself, or misses one where the sheet meets itself round the loop.

rigid · Self-contact

Named alongside it

The objects these essays reach for when they reach for this one.

BoundaryLayer orderPanelConstraintExhaustive searchGluingLayer countLayer orderingNon-crossing conditionPeriodicitySelf-intersectionStacking

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