Origami & Folding
what one sheet can do
Concept

Tapered panel — where it appears

A panel whose edges are cut back at an angle so that a thick fold can close further than its own depth would otherwise allow. It is one of the standard thickness accommodations, and it trades paper for travel at a rate the geometry fixes rather than the maker.

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

0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does

Getting thickness round a corner

There are half a dozen ways to build a fold in a panel that has depth, and the useful way to arrange them is not by what the cross-section looks like. It is by what each one gives away.

rigid · Thickness grown symmetricallyoffset to one sidepanels overlap over 4% of their areawhich is the jam every thick-panel design meetspanels do not overlap at alland the hinge axes have not movedwhich is not free: fold it tighter and this offset runs out toosheet 0.16 panel-lengths thick, folded to 100° — the overlap is measured from the geometry

Panels with somewhere to go

Every way of giving a folded panel real thickness costs something. Tachi's offset-panel technique costs the least interesting thing there is — it stops the panels being a surface, and leaves the hinges exactly where the zero-thickness pattern put them.

rigid · Thickness the bar is the proportion of the sheet the pattern asks for2 columns, 2 rows1.183(2 + tan 20°) / 2 = 1.1833 columns, 3 rows1.122(3 + tan 20°) / 3 = 1.1224 columns, 3 rows1.455(4 + tan 20°) / 3 = 1.4556 columns, 4 rows1.591(6 + tan 20°) / 4 = 1.5916 columns, 6 rows1.061(6 + tan 20°) / 6 = 1.0618 columns, 6 rows1.394(8 + tan 20°) / 6 = 1.394a square sheet needs a proportion of exactly one, which the counts and the slant have to be chosen for

The paper a pattern asks for

A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.

tessellation · Miura 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

A leaf ends its pattern

A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.

biology · Leaf folding the bar is the share of the footprint at least half as deep as its deepest pilethe note is how deep that pile is and how much of the footprint it coversThe preliminary base100.0%uniform · deepest 8 layers over 99.7%The Yoshimura pattern100.0%uniform · deepest 60 layers over 100.0%The waterbomb tessellation96.6%uniform · deepest 32 layers over 96.6%The tapered corrugation87.9%graded · deepest 16 layers over 0.9%The Miura fold75.5%graded · deepest 16 layers over 11.9%The hexagon twist24.5%island · deepest 7 layers over 24.5%The square twist17.5%island · deepest 9 layers over 17.4%Fold and cut — the triangle3.0%island · deepest 7 layers over 2.9%uniform: the pile is the pattern · island: the deep region is a patch · graded: deepest on a sliver, half as deep nearly everywhere

Three kinds of pile

A thick-panel technique is priced at the deepest pile a pattern has, and the depth of that pile says nothing about where it is. Mapped over the folded footprint, the printed patterns fall into three kinds. On a uniform pile the deepest count is the whole footprint — sixty layers everywhere on the Yoshimura. On an island it is a patch and the rest is shallow. On a graded pile it is a sliver — under one per cent of the tapered corrugation — while nearly nine tenths is at least half as deep, and the Miura, the pattern that gets built, is graded.

rigid · Thickness the bar is the share of the paper in panels that lie over more than one deptheach panel's folded image is laid on the depth map and the depths under it are countedThe preliminary base0.0%uniform · 8 panels · 1 depth under eachThe Yoshimura pattern0.0%uniform · 65 panels · 1 depth under eachThe waterbomb tessellation0.0%uniform · 52 panels · 1 depth under eachFold and cut — the triangle90.0%island · 7 panels · 1 to 3 depths under eachThe hexagon twist92.5%island · 13 panels · 1 to 4 depths under eachThe square twist94.2%island · 9 panels · 1 to 4 depths under eachThe Miura fold100.0%graded · 24 panels · 3 to 4 depths under eachThe tapered corrugation100.0%graded · 28 panels · 4 depths under eacha panel over one depth can be given one thickness; a panel over several cannot

A panel is not the unit of depth

A thick-panel design gives each panel a thickness, an offset or a taper, so a graded pile could be met panel by panel only if every panel's folded image lay over one depth. On the Miura none does. Every one of its twenty-four panels lies over three or four of the four depths its pile takes, and on the tapered corrugation every panel lies over all four. The uniform piles are the opposite — every panel of the Yoshimura, the waterbomb and the preliminary base lies over exactly one depth — which is why panel-by-panel techniques look adequate on the patterns they are drawn for. On the Miura the steps between depths cross the middle of panels, and they run parallel to the panels' own sides.

rigid · Thickness

Named alongside it

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

Thickness accommodationCrease patternLayer countThe offset-panel techniqueStackingThicknessUnit cellBoundary vertexHinge axisKinematicsLeaf foldingManufacturing

All concepts