Origami & Folding
what one sheet can do
Concept

The one-layer simple fold — where it appears

A simple fold that turns a single layer of the stack. It sounds like the most permissive of the three machines and is not, because folding one layer drags everything attached to it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

evenly spaced — 4 creasesany flat folding16 of 16some-layers16 of 16all-layers16 of 16one-layer2 of 16crimping only6 of 16uneven — 4 creasesany flat folding8 of 16some-layers8 of 16all-layers0 of 16one-layer2 of 16crimping only0 of 16a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move

The patient machine is the weak one

A machine that folds one layer at a time sounds like a machine with more freedom, not less. It has less, and the reason is the most ordinary fact about paper there is: it is joined, so whatever a machine declines to hold it also cannot move.

complexity · Simple foldability 34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

What a dashed line can say

Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

history · Notation the pale bar is every folded state the strip has; the dark one is what the machine reachesthree machines on the same strips, and none of them is the flat-folding theorem3 equal stamps · takes every layerall 123 equal stamps · takes one layer4 of 123 equal stamps · takes any block reaching an edgeall 124 equal stamps · takes every layerall 324 equal stamps · takes one layer4 of 324 equal stamps · takes any block reaching an edgeall 325 equal stamps · takes every layerall 1005 equal stamps · takes one layer4 of 1005 equal stamps · takes any block reaching an edgeall 1006 equal stamps · takes every layerall 2886 equal stamps · takes one layer4 of 2886 equal stamps · takes any block reaching an edgeall 288counted over every marking of the strip that folds flat at all

Deciding is not making

Four earlier essays here ask which machines can flatten a strip at all, and the answer sorts them into a lattice with one column full and three with holes in it. Asked instead what each machine can produce, the three sort completely differently: the machine that may choose its block reaches every folded state of every strip tried, the machine that takes one layer reaches exactly four whatever the strip is and however long, and the machine that takes the whole pile is the only one whose answer depends on the spacing at all.

complexity · Simple foldability

Named alongside it

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

The all-layers simple foldThe machine modelSimple foldabilityConnectivityLayer orderingNotationReachabilityStamp foldingYoshizawa

All concepts