Origami & Folding
what one sheet can do
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The figure library — page 11

Every picture here is generated from code at build time. This is every generator, ordered by how many essays call it, each rendered at its defaults.

same-outline

3 essays — The shadow does not name the pattern · The order does not name it either · Which side is showing · 3 claims checked

Different patterns, one folded objectCrease patterns that a photograph of the folded strip cannot tell apart. The creases are in different places and the folded profiles — the outline, and the number of layers over every point of it — are identical.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

sat-gadget

3 essays — The gadgets that make it hard · Hardness is about the worst one · A short reason to say no · 3 claims checked

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions

sheet-shape

3 essays — The square is a choice · The rectangle that keeps its shape · One member of a family · 3 claims checked

Which shape of sheet the flaps wantHow much of the sheet the flaps can claim, as the sheet is stretched from a square to three to one at constant area. The number of flaps decides where the peak is, and for some counts it is not at the square.11.522.5360%65%70%75%sheet, longer side to shorterfraction of the sheet claimedbest at 1.8 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

wedge-and-cut

3 essays — The star that was cut before it was proved · A tree cannot argue · How many wedges the paper allows · 7 claims checked

Fold into 10, cut onceThe traditional method, which is not the theorem. The sheet is folded into equal wedges about a point, one straight cut is made, and the shape that falls out has the symmetry the folding imposed. It gives a regular star for nothing and it gives nothing at all for a shape without that symmetry.10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed

buckling-first

2 essays — Found before it was designed · Almost every pattern fails · 9 claims checked

Found before it was designedThe diamond pattern a thin cylinder falls into under axial load, drawn as a crease pattern and put past the theorems. It satisfies them everywhere, its vertices are all alike, and its mountain-and-valley assignment carries Maekawa's split — none of which anybody chose. The physics produced the colouring as well as the creases.what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

circle-packing

2 essays — Packing is the hard part · The flap nobody holds · 1 claim checked

Packing the flapsOne circle per flap, radius equal to that flap's length, packed into the square without overlapping. The packing determines the crease pattern; the leftover paper between circles becomes the structure that joins the flaps together.leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs

colour-cost

2 essays — Bringing the other side to the front · Decided before the design · 6 claims checked

A third of the face, and half the sheetHow much of the folded sheet shows its front and how much shows its reverse, as the flap that makes the colour change widens. The reverse rises as fast as the flap; the front falls twice as fast, because the flap covers as much paper as it is.00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

crimp-reduction

2 essays — A machine that can only crimp · The cost is in the coincidences · 2 claims checked

A crimper reducing a strip to nothingCrimping folds one segment back between its two neighbours. It needs the creases at either end to turn opposite ways and the middle segment to be no longer than either neighbour, and it consumes two creases at a time — which is why a strip with an odd number of creases can never be crimped away entirely.4 creases, assignment MVMVthe strip0.200.200.200.200.20MVMV3 availableafter crimp 10.200.200.20MV1 availableafter crimp 20.20nothing left2 crimps, each removing two creases4 creases is an even number, and that is not a coincidencethe merged segment measures outer minus middle plus outer

curved-vertex

2 essays — One curve and one number · Where curved creases meet · 6 claims checked

The same vertex, drawn three waysOne vertex with four creases, drawn with the creases straight and then with them curved by two different amounts. The tangents at the vertex are identical in all of them and so are the sectors between them, which is where the conditions live.straight creasescurvature 1.4curvature 2.6the tangents are the same in every panelsectors 70.0°, 110.0°, 110.0°, 70.0°they sum to 360.0°, and alternately to 180.0° and 180.0°which is Kawasaki, on tangents rather than on lines

degree-census

2 essays — Nothing meets at three · A cut is not local · 3 claims checked

Every interior vertex in the library, by degreeHow many creases meet at each interior vertex of every pattern in the library, counted together. The odd columns are empty and the column at two is not drawn at all: a vertex where creases stopped rather than passed through would be one of those, and the paper cannot fold there.interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases

error-direction

2 essays — A tolerance is a direction · The allowance is spent at the end · 2 claims checked

The same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.00.20.40.60.81-7-6-5-4-3-2-1error in every length, millimetres on a 150 mm sheetclosure mismatch (powers of ten)across the surface of solutionsthe direction the closure's own derivative points ina direction chosen without regard to the surfacewhich has a component along bothalong the surface of solutionsone of the twelve directions the equations do not seethe same step costs 5,130 times as much one way as the other

folded-outline

2 essays — The outline is mostly crease · The shadow does not name the pattern · 2 claims checked

The outline is mostly creaseFor four folded patterns, the total length of edge with paper on one side and nothing on the other, split into the sheet's own raw edge and the creases. The raw edge is the minority everywhere: what a folded model shows the world is mostly fold, and the boundary of the paper has gone inside.patternraw edge against crease, by lengthpreliminary base8 panels29.3% rawMiura, 5 by 420 panels20.8% rawsquare twist9 panels26.7% rawYoshimura, 6 by 565 panels6.7% rawthe shaded part is the sheet's own edge; the rest of the outline is creasemeasured with a step of 0.001 of the sheet, and checked across a tenfold sweep of itevery length here is summed over the layers, so a buried edge counts for nothing