Origami & Folding
what one sheet can do
Designing a base

Which side arrives

A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

Assumes Bringing the other side to the front and Decided before the design.

19 min read 8 figures The pattern is the objectOne sheet, no cuts

Bringing the other side to the front treats a colour change as a technique, which is how the folding literature treats it: the designer wants the reverse of the sheet at a particular place and folds so that it arrives there. Decided before the design narrowed that considerably by measuring what is available — over most printed footprints, both sides of the paper lie somewhere over more than ninety-nine per cent of the area, so almost anywhere could show almost anything.

Available is not the same as showing. Which side a reader actually sees at a point is the side of the highest panel covering it, and until the panels could be put in an order there was no way to ask.

Which side of the paper is showing, once the pile is decidedFor every printed pattern with an ordering: the share of the folded footprint showing the side of the sheet that started face up. It is nothing like even, it is not a design choice, and on the preliminary base it is one part in a thousand.the bar is the share of the footprint showing the side that started face upthe rest of it shows the other side, and neither is chosen by anybodyThe preliminary base0.3%2 of 8 panels in viewThe square twist47.4%7 of 9 panels in viewThe hexagon twist90.8%10 of 13 panels in viewFold and cut — the triangle100.0%1 of 7 panels in view · 2 statesa panel out of view from above is not hidden — it is under the pile, and turning the sheet over shows a different set
Fig. 1 For every printed pattern whose panels can be ordered: the share of the folded footprint showing the side of the sheet that started face up. The note says how many of the pattern’s panels are in view from above at all.

Nobody chooses

Three of the four patterns with an ordering have exactly one, so there is nothing to decide. The pattern was drawn, the letters were put on it, and every question about which side shows where was answered at that moment — by arithmetic, and not by anybody.

pattern orderings shows the front panels in view
the preliminary base 1 0.1% 2 of 8
the square twist 1 48.6% 7 of 9
the hexagon twist 1 90.5% 10 of 13
fold and cut, the triangle 2 100% and 96.5% 1 or 2 of 7

The three values in the middle column are 0.1, 48.6 and 90.5 per cent. They are spread across the whole range, they belong to three patterns nobody drew with colour in mind, and not one of them is a decision.

The preliminary base is the extreme and it is the pattern everybody folds first. Fold one from paper that is white on one side and coloured on the other, and what a reader is looking at is the coloured side over 99.9% of the shape. That is why every traditional model built on the base — the crane, the lily — comes out in one colour and why the colour-change technique has to be invented rather than merely used: the default is not a mixture.

One ordering of the preliminary baseThe panels of a flat-folded pattern in one of the orders the non-crossing rules allow, drawn from the bottom of the pile to the top. Each panel is shown where it lands in the folded plane, with the outline of the whole footprint behind it, so the drawing is a stack seen from above rather than a diagram.the pile from the bottom upeach square is one panel where it lands, over the outline of the whole footprint8 panels · 28 pairs sharing ground · 12 rules12345678
Fig. 2 The preliminary base’s eight panels in the one order the rules allow. The two at the top of the pile are the two a reader sees; the other six are underneath them, and every point of the footprint has all eight over it.

How the share is computed

The procedure is three lines and each of them is a place the answer could go wrong, so they are worth naming.

Place the panels. Folding flat about a crease is reflection in that crease, so a walk over the panels places all of them from one starting panel by composing reflections. Whether a panel has been turned over is whether its motion reverses orientation, and that is the same quantity the crease rule uses to decide which of two panels is higher.

Order them. Search the orderings for one satisfying the crease rule and the two non-crossing rules, and either find one or exhaust the search. On these four patterns the answer is one ordering or two.

Look down. Sample the folded plane on a grid; at each sample find the panels covering it and take the highest; report whether that panel is turned. The share is over the samples that have paper over them, because a sample outside the footprint shows nothing and counting it would report the shape of the bounding box instead of the shape of the object.

The last step is where the sampling error lives, and it is checked against something known: the layer count integrated over the footprint has to come back as the area of the sheet. A grid fine enough for that is fine enough to say which panel is on top.

Two of eight

The second column is the more surprising one and it is a consequence of the first.

A folded object seen from above shows only its top layer, so the number of panels contributing anything to the view is at most the number of distinct panels that are highest somewhere. On the preliminary base that is two of eight. On the fold-and-cut triangle, in one of its two states, it is one of seven — a single panel covers the entire visible face of the folded object.

That is what a pattern which collapses everything over one point does. All eight panels of the preliminary base lie over every point of its footprint, so exactly one of them is on top everywhere, and the boundary between “the panel on top here” and “the panel on top there” can only occur where the footprint’s own outline changes — which on this pattern is almost nowhere.

The square twist is the opposite case and it is why its number is near half. Seven of its nine panels are highest somewhere, its footprint is a shape with several distinct regions, and the front and back come out at 48.6 and 51.4 per cent. A pattern whose folded footprint has structure is a pattern whose visible face has structure, and the two come from the same place.

How deep the pile gets, and how deep it is on averageThe largest number of panels over any one point of each printed pattern's folded footprint, against the number of panels the sheet has. On three of the eight, every panel of the sheet lies over one point; on the fold-and-cut triangle the average is a little over one layer.the bar is the deepest point of the pile, as a share of the whole sheetThe preliminary base100%8 of 8 panels · 7.99 layers on averageThe Miura fold67%16 of 24 panels · 9.23 layers on averageThe square twist100%9 of 9 panels · 3.02 layers on averageThe hexagon twist54%7 of 13 panels · 3.25 layers on averageThe Yoshimura pattern92%60 of 65 panels · 60.00 layers on averageFold and cut — the triangle100%7 of 7 panels · 1.19 layers on averageThe tapered corrugation57%16 of 28 panels · 8.33 layers on averageThe waterbomb tessellation62%32 of 52 panels · 31.48 layers on averagea pattern that folds into a long thin object piles nearly all of itself in one place
Fig. 3 Where the piles are deep. A deep pile has few panels in view because one of them is on top of the whole of it; a shallow one has many, and its visible face is a patchwork.

The one that has a choice, and how little it is

The fold-and-cut triangle is the only printed pattern here with more than one ordering, so it is the only one where a designer could choose — and the choice is worth 3.5 per cent of the footprint.

Its two states show the front over 100% and 96.5% of the area respectively. They differ in the relative height of two panels that share ground, one of the two ends up on top in a small region, and that region is three and a half per cent of the shape. That is the entire design space available on this pattern, and it exists only because the pattern happens to admit two orderings rather than one.

Nothing on the printed shelf admits more than two. So on this site’s own patterns, the colour-change decision has one bit in it, once, and is absent everywhere else.

Against what was available

The comparison with the earlier measurement is the point of this rung, so it is worth putting the two numbers side by side.

Decided before the design reported that on most printed footprints, both sides of the paper lie somewhere over more than ninety-nine per cent of the area. That is a statement about the whole pile: at nearly every point, some panel showing the front and some panel showing the back are both present.

This reports that on the preliminary base, the front is what a reader sees over one part in a thousand.

Ninety-nine per cent available; nought point one per cent showing. The gap between those two numbers is the entire content of layer order, stated as a fraction. Everything the pile does is in there: the paper is all present, the choice looks wide open, and exactly one panel is on top.

That is also why the earlier measurement could not have found this. Availability is a question about the set of panels over a point, which needs no order at all. Showing is a question about the maximum of that set, which needs nothing but the order.

Which of the two rules holds each sheet downThe non-crossing rules a folded pattern generates, split by kind: a panel that a crease's folded image runs through, and two creases in the same place that must not interleave. Two of the eight patterns generate none of the first kind and are governed entirely by the second.the bar is every non-crossing rule the folded state generatesThe preliminary base120 through a fold · 12 interleavingThe Miura fold228144 through a fold · 84 interleavingThe square twist4836 through a fold · 12 interleavingThe hexagon twist9690 through a fold · 6 interleavingThe Yoshimura pattern11870 through a fold · 1187 interleavingFold and cut — the triangle1512 through a fold · 3 interleavingThe tapered corrugation351308 through a fold · 43 interleavingThe waterbomb tessellation654144 through a fold · 510 interleavinga pattern whose creases never land inside another panel generates none of the first kind
Fig. 4 The rules that fix the order, split by kind. Everything in this essay is downstream of them: they are what turns ninety-nine per cent available into one part in a thousand showing.

Where the technique actually lives

None of this says a colour change is impossible. It says the freedom is not where the technique’s description puts it.

A designer changing a colour is not reordering the layers of a fixed pattern — that is what has just been shown to be unavailable. They are changing the pattern: adding creases, moving a flap’s boundary, inserting a pleat so that a different piece of paper arrives at the top. The freedom is in the drawing, upstream of everything measured here, and the measurement is what the drawing then buys.

That is the same relocation decided before the design made once already, one step further back. Availability was found to be nearly total and therefore not the constraint. The ordering is now found to be nearly unique and therefore not the mechanism. What is left is the pattern.

Reading the other side

Turning the folded object over shows a different set of panels, and it is not the complement of the first set.

The bottom of the pile is one panel, the top is one panel, and between them are six on the preliminary base. From below, the visible face is the lowest panel covering each point — and on a pattern where every panel lies over every point, that is exactly one panel, which need not be the one that showed from above and need not be showing the same side of the paper.

So a folded object has two visible faces and both are computed from the same ordering. What it does not have is a way to swap them: turning the pile over is a relabelling and not a rearrangement, and the object a reader holds is the same object either way up.

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.12 panels, two coloursno crease has the same colour on both sidesall 6 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge
Fig. 5 The two-colouring of the panels, which is what decides which side each panel would show if it were on top. It is a property of the pattern alone and knows nothing about the order; the order is what selects among it.

What is not settled

The undecided patterns are undecided. Four of the eight printed patterns have too many panels for an exhaustive ordering search, so no share is reported for them at all. The Miura is one of them, and a Miura folded from duo paper is a thing people photograph; this essay says nothing about what it shows.

The share is sampled. The visible face is read off a grid over the footprint, so the percentages carry the grid’s resolution as their error, and a feature narrower than a sample is invisible. The three-and-a-half per cent difference between the fold-and-cut triangle’s two states is well above it; a difference of a tenth of a per cent would not be.

And the front is a label, not a colour. “The side that started face up” is a fact about the sheet before folding. Duo paper makes it visible, plain paper does not, and nothing here is about ink.

Nor is any of it about a real model. These are flat folded states of flat patterns with zero-thickness panels. A folded object made of paper has a crease with a radius and layers that do not lie perfectly, and at the boundary of two regions in the visible face a reader sees a rounded edge rather than a line. The shares here are shares of an idealisation, and the idealisation is the same one every other measurement on this site rests on.

Sharing ground is not lying over everything

There is a trap between this measurement and the census of which panels lie over which, and it is worth disarming because the two results look as though they should contradict each other.

That census reports the square twist as complete: all thirty-six pairs of its nine panels share ground, meaning every panel overlaps every other somewhere. Read carelessly, that says the whole sheet is one pile and one panel is on top of all of it — which would give one panel in view, and the table above says seven.

Both are right, because overlapping in pairs and overlapping all together are different things. Nine convex patches can meet each other two at a time and have no point common to all nine; the classical result on convex sets in the plane says that to conclude a common point one needs every three of them to meet, not every two. The square twist is exactly the case that separates the two conditions: pairwise complete, and with no point carrying all nine.

So the number of panels in view is not predicted by pair-sharing at all. It is decided by how the footprint breaks into regions with different panels on top, and a pattern can be pairwise complete and still break into several.

Which is the same split, seen from above

Put the two tables together and the pattern in the second column becomes legible.

The preliminary base shows two panels of eight, a quarter of them. The fold-and-cut triangle shows one or two of seven. The square twist shows seven of nine and the hexagon twist ten of thirteen, better than three quarters each.

Those are the same two groups the footprint census separates, and by the same mechanism. A pattern that collapses toward a point piles everything over a small region, so one panel is on top of nearly all of it and the visible face is nearly uniform. A pattern that collapses toward a shape — the twists keep a footprint with distinct areas in it — has a different panel on top in each area, and its visible face is a patchwork.

That gives the useful rule, and it needs no ordering search to apply. How much of a folded object a reader sees is decided by the shape it folds into, not by how many panels it has. A fifty-panel model that collapses toward a point shows one or two of them; a nine-panel one that keeps its shape shows seven.

It also explains why the colour shares run the way they do rather than being scattered. The preliminary base is 0.1 per cent because a single panel covers nearly the whole face and it happens to be turned; the square twist is 48.6 per cent because seven panels each cover a piece of the face and the pieces are of comparable size, so the average behaves like an average. A number near a half is evidence of a patchwork rather than of balance, and a number near nought or a hundred is evidence of a single panel — which is a thing a reader can diagnose from the percentage alone, without seeing the model.

The number that is not a half

There is an expectation the table quietly refutes and it is worth making explicit, because it is the reason the measurement is interesting rather than a formality.

A folded pattern’s panels split evenly into two colours — that is what two-colourability means, and on a pattern built by tiling a unit the split is close to exact. Bringing the other side to the front found the square twist off balance by seven parts in a thousand and every other printed pattern closer than that.

So half the panels show the front and half show the back, and a reader might reasonably expect the visible face to come out near half as well. It does on exactly one pattern.

The reason is that the visible face is not an average over panels; it is a maximum. Eight panels lie over a point of the preliminary base, four of each colour, and one of them is on top — and which one is decided by an ordering that has nothing to do with the colouring. A balanced pattern can have a completely unbalanced face, and the quantity that decides is the order.

That is the same distinction between a set and its maximum that runs through everything in these essays, and it is why the ordering was worth computing at all: nearly every statement about a folded object that a reader can check by looking is a statement about the top of a pile.

One consequence is worth drawing out because it runs against how colour-change designs are usually described. If the visible face is a maximum over a pile, then a design that wants a particular piece of paper to show has to get that piece to the top, not merely into the region. Getting it into the region is a question about where the paper lands, which the pattern controls directly and which the availability measurement showed is nearly always satisfiable. Getting it to the top is a question about the order, which the pattern controls only through the letters and which the search here shows is usually settled outright. That is why the technique is expensive: it is buying a place in a queue, not a place on the sheet.

What a folder should take from it

The colour of a traditional base is not a coincidence and not a choice. The preliminary base shows one side of the paper over almost the whole of its face, because all eight of its panels lie over every point and one of them is on top.

Count the panels in view, not the panels. A folded object with fifty panels may show three. The rest are under them, doing structural work and no visual work.

And if the colour has to change, change the pattern. The pile will not be argued with; it has one arrangement and no legal rearrangement, and the place to put a decision is the drawing.

One ordering of the square twistThe panels of a flat-folded pattern in one of the orders the non-crossing rules allow, drawn from the bottom of the pile to the top. Each panel is shown where it lands in the folded plane, with the outline of the whole footprint behind it, so the drawing is a stack seen from above rather than a diagram.the pile from the bottom upeach square is one panel where it lands, over the outline of the whole footprint9 panels · 36 pairs sharing ground · 48 rules123456789
Fig. 6 The square twist, whose nine panels give the most balanced visible face on the shelf at 48.6 per cent. Seven of the nine are in view somewhere, which is why it has one at all.
A panel at the edge of the paper lies over fewer of the othersFor every printed pattern with panels away from the sheet's edge: the average number of other panels one panel shares ground with, taken separately over the panels carrying a raw edge and the panels that do not. The rim is lower on every pattern measured.the upper bar is the rim, the lower is the middlethe value is how many other panels an average panel of that kind lies overThe Miura fold18.0 · 21.016 at the rim, 8 away from itThe square twist8.0 · 8.08 at the rim, 1 away from itThe hexagon twist10.0 · 12.012 at the rim, 1 away from itThe Yoshimura pattern61.6 · 64.021 at the rim, 44 away from itThe tapered corrugation19.0 · 22.218 at the rim, 10 away from itThe waterbomb tessellation31.0 · 37.716 at the rim, 36 away from itthe difference is small and it has the same sign every time
Fig. 7 The panels least likely to be in view are the ones carrying a raw edge of the sheet, and they are also the ones lying over fewest of the others. Both facts have the same cause: a rim panel is a cut-off cell, and a smaller panel covers less.

None of this would say anything about the drawing if the pattern admitted many piles, because then a share of the visible face would be a choice somebody made in the folding rather than a property of the crease pattern.

What is left of the square twist's letterings when the layers are askedEvery mountain-valley labelling of one printed pattern, sieved three times: by the conditions at each vertex, by whether the letters can be ordered among themselves at all, and by whether an ordering of the panels exists. The middle number is the one every gate on this site used to measure.the square twist, sieved three timesevery lettering4,0962 to the 12passes every vertex2566.3% of themletters are consistent2524 force a loop of panelshas a folded state80.20% of themthe bars are on one scale, so the last one is the size of the answer against the size of the question
Fig. 8 And the reason there is one ordering to look down at rather than a hundred. Eight of the square twist’s 256 admitted letterings have a folded state at all, and the one printed here has exactly one.

Where this goes next

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 Bringing the other side to the front Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows. patternfootprint with both colours over itlayers thereThe preliminary base8 panels, 4 one way up and 4 the other100%8.0The Miura fold24 panels, 12 one way up and 12 the other100%9.2The square twist9 panels, 5 one way up and 4 the other65%4.1The hexagon twist13 panels, 7 one way up and 6 the other72%4.1The Yoshimura pattern65 panels, 32 one way up and 33 the other100%60.0Fold and cut — the triangle7 panels, 4 one way up and 3 the other4%6.4The tapered corrugation28 panels, 14 one way up and 14 the other100%8.4The waterbomb tessellation52 panels, 26 one way up and 26 the other100%31.1 Decided before the design A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper. the bar is the pairs of panels that lie over one anotherThe preliminary base288 panels · 12 rules · an ordering existsThe Miura fold22824 panels · 228 rules · not decidedThe square twist369 panels · 48 rules · an ordering existsThe hexagon twist6613 panels · 96 rules · an ordering existsThe Yoshimura pattern205565 panels · 1187 rules · not decidedFold and cut — the triangle217 panels · 15 rules · an ordering existsThe tapered corrugation28228 panels · 351 rules · not decidedThe waterbomb tessellation92652 panels · 654 rules · not decideda pattern with no bar has no two panels over one another, and its order is not a question No height to swap A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

Colour changeDesign techniqueFolded stateLayer countLayer orderTwo-colouring