Origami & Folding
what one sheet can do
Concept

Underdetermination — where it appears

Having less information than is needed to decide between possibilities. A folded object underdetermines its crease pattern, and adding everything a photograph carries does not fix it.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

one excess of length, 4 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 276amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

The excess does not choose its waves

A rim that has grown longer than its span has to put the extra length somewhere, and two large waves and eight small ones are the same metric — identical arc length, identical excess. Geometry fixes the family and is completely indifferent about the member.

biology · Growth as metric 4 patterns, one folded profile1/122/124/127/1225311/123/126/128/1225312/124/125/127/1225312/125/127/128/122531foldedthe layer counts under each band are the same in every row, and so are the widths

The shadow does not name the pattern

A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough. Crease patterns in genuinely different places fold to identical outlines with identical layer counts, and nearly a third of the folded objects a short strip can reach are reached by more than one pattern.

flat-folding · Inverse problem 7 flaps, packed as tight as they will goradius0.174457630loose flaps1room to move0.1127the inner ring is where that flap's centre may sit; the algorithm reports one point of it and stops

The flap nobody holds

An optimal packing is presented as an answer: here are the circles, here is where they go. For some numbers of flaps that is not what it is. The best arrangement of seven discs in a square leaves one of them free to wander over an eleventh of the sheet without changing the answer at all — and the algorithm reports one point of that region and stops.

design · Circle packing the 16 repeating rules that fold, written outrows first, then the two column classes — and every one of them alternates down the columnrows · columns above|below20VV · MV|MV21MV · MV|MV22VM · MV|MV23MM · MV|MV24VV · VM|MV25MV · VM|MV26VM · VM|MV27MM · VM|MV36VV · MV|VM37MV · MV|VM38VM · MV|VM39MM · MV|VM40VV · VM|VM41MV · VM|VM42VM · VM|VM43MM · VM|VMfour ways of writing the rows times four ways of alternating the columns is sixteen, and there is nothing else

Half the recipe is decoration

Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

history · Pedagogy one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

The drawing does not say what is glued

One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.

flat-folding · As drawn 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

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

history · Notation wavesamplitudefits inside2∫κ² = 170.03980.043∫κ² = 390.02660.045∫κ² = 1080.01590.04, 0.028∫κ² = 2760.01000.04, 0.02, 0.0112∫κ² = 6210.00660.04, 0.02, 0.0120∫κ² = 17240.00400.04, 0.02, 0.01depth 0.04 needs 2 waves or more · depth 0.02 needs 4 waves or more · depth 0.01 needs 8 waves or moreexcess 6.0% · amplitude × waves = 0.0797 throughout · floor = that constant ⁄ depth

The container picks the member

Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.

biology · Growth as metric 00.20.40.60.81-11234radius of the three marks, on the flat sheetmiss against flat (%)grown by the same factor everywhere · −0.00%grown more at the rim · +4.27%the growth of a spherical cap · −1.71%grown so that the curvature cancels · +3.52%a centre mark and three a third of a turn apart · on a flat sheet the three sit √3 times their radius apart

Three marks see nothing

The measurement a rim cannot make is an interior one, and the obvious interior measurement — two marks a known distance apart, measured again after growth — cannot detect curvature at all, because a uniformly enlarged sheet changes that distance and stays flat. Three marks cannot either: any three distances obeying the triangle inequality are the sides of a flat triangle. Four marks give six distances, and six distances are not free on a flat sheet. The growth profile a rim measurement reads as flat misses by three and a half per cent with four marks at the rim.

biology · Growth as metric

Named alongside it

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

Crease patternGrowthMetricArc lengthBoundaryBucklingDifferential growthFold formatGluingNotationAssignmentCircle packing

All concepts