Origami & Folding
what one sheet can do
Concept

River — where it appears

A path of paper in a circle packing that joins two flaps at a stated distance apart. It is the general case a circle is the special case of, and it is what makes packing hard rather than merely constrained.

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

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

Packing is the hard part

Once a subject is a set of circles, designing the model is fitting them into a square. That step has no general algorithm, no known optimum, and it is where every remaining difficulty in origami design now sits.

design · Circle packing riverwidth 0.3legarmheadlegtailthe check10 pairs testedtightest by 0.0800(leg and arm)the ruledistance on the sheetat leastdistance through the treetwo nodes, and an edge between themthe extra width is thebody the flaps hang fromthe discs are what each flap costs; the strip is what joins the two halves of the subjectand both are the same condition, read off different pairs of leaves

What joins the flaps

Circles are the rule for flaps that all hang from the same point. As soon as two groups of flaps hang from different places, the paper between them has to be paid for too — and the payment is a strip whose width is the distance between them.

design · Circle packing quadrilateralconvexmolecule builtpentagonconvexmolecule builtL, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkingdart, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkinga convex polygon shrinks inward and stays a polygon; a reflex corner is a wall the shrink runs intoso a non-convex region is split into convex pieces first, and choosing the split is a searchwhich is where a construction that always works hands the difficulty to whatever comes before it

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

design · Molecules the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

Paying in paper

A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.

design · Grafting one graft, then a second across itstrips 0.3 and 0.3 wideas designedone stripand one across itthe first strip adds 0.300the second adds 0.390two separate bills would be 0.600the sheet gained 0.690the excess is 0.090and the strips cross over 0.090the crossing rectangle is 13.0 per cent of everything the two features cost

The second term

A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

design · Grafting the lines a graft could useevery gap between vertex columns, and every gap between vertex rows, triedpatternvertical lineshorizontal linesgraftsThe preliminary base0 of 20 of 2nowhereThe Miura fold6 of 130 of 4one wayThe square twist0 of 60 of 6nowhereThe hexagon twist0 of 42 of 11one wayThe Yoshimura pattern0 of 120 of 5nowhereFold and cut — the triangle2 of 82 of 8both, in margin onlyThe tapered corrugation7 of 150 of 4one wayThe waterbomb tessellation0 of 80 of 8nowherea 4 by 4 grid4 of 44 of 4both waysthe grid, one square creased diagonally3 of 43 of 4both waysa line is admissible when every crease it crosses is square to it; the entry is admissible of the lines between vertex columns or rows

A graft needs a square line

A strip can be slid into a finished crease pattern only along a line every crossed crease meets square, because only such a crease continues across the strip as itself. Tried on every line between the vertex columns and rows of eight printed patterns, four take no strip in either direction and three take one in a single direction. The eighth, the fold-and-cut triangle, appears to take strips both ways, and every line it admits runs through blank margin. No printed pattern takes a strip across its creases in both directions — and one diagonal crease in a plain grid removes exactly the row and the column it sits in.

design · Grafting

Named alongside it

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

GraftingBox pleatingConservationCrease patternDesign techniqueOptimisationTree methodUniaxial baseAreaAxial polygonCircle packingCompleteness

All concepts