Origami & Folding
what one sheet can do
Concept

The all-layers simple fold — where it appears

A simple fold in which every layer under the fold line is folded at once. It is the most restrictive of the three simple-fold machines, and there are strips that fold flat which no all-layers machine can reach.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

creases at 0.40, 0.50 — assignment MVflat1 layerno sequence of all-layers folds finishes this pattern — the search exhausted 5 states

The fold a machine can make

A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.

complexity · Simple foldability 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 50% of the stack, 9 foldsthe whole stack, 9 folds13 interior vertices of odd degree13 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

material · Crumpling the pale bar is every folded state; the dark one is the states the machine reachescounted over every marking of the strip that folds at all3 equal stamps12 of 12 reached4 equal stamps32 of 32 reached5 equal stamps100 of 100 reached6 equal stamps288 of 288 reachedcreases at .13 .31 .62 .780 of 24 reached — 24 missedcreases at .08 .24 .28 .35 .720 of 48 reached — 48 missed

Where the machine catches up

The weakest machine in the subject folds every layer at once and is stopped by a strip with two creases in it. On a strip of equal stamps it is stopped by almost nothing: every one of the 288 folded states a six-stamp strip has is reachable by a sequence of all-layers folds, and on every unevenly creased strip tried it reaches none of them. At seven stamps the completeness ends, and finding out where it ended is what checking it past six was for.

complexity · Simple foldability 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 machine modelSimple foldabilityThe one-layer simple foldLayer orderingReachabilityStamp foldingConnectivityCrease patternCrumplingThe decision problemIdealisationKawasaki's theorem

All concepts