A stack is a shared fate

Splitting insures only the failures that come alone priced two ways in which hinge failures fail to be independent. In one, a cause outside the mechanism — a cold soak, a hard launch — reaches every hinge at once. In the other, a tether between modules carries a jam from one module to the next. Both erode the case for cutting a deployable into separately driven pieces, and they erode it in opposite orders.

That essay ended by naming a third kind and leaving it unpriced. Hinges that lie on top of one another in the folded state share a fate because they share a place. A stack pressed together for months in stowage, a corner crushed in handling, a patch of film that cold-welds to the layer above it: whatever acts on a point of the stack acts on every hinge whose folded image passes through that point, and it does not act on the others.

A cause of that kind is neither global nor passed along. It is local, but local in the folded state rather than on the sheet, and the folded state is where a Miura’s geometry rearranges which hinges are neighbours. Two hinges at opposite ends of the flat sheet can be one millimetre apart in the stack. Two hinges sharing a vertex on the flat sheet can be a stack’s length apart.

So the question this essay answers is mechanical before it is statistical. Which hinges of a folded Miura land in the same place, and does a cut into modules keep each place inside one module or spread it across several?

Ten hinges, one segment #

The answer can be read off the solved folding. Fold the ten by ten Miura flat, map every hinge’s flat segment to its folded segment through the motion of the panel beside it, and group the hinges whose folded segments coincide. The 180 hinges of the sheet fall into 29 places.

Two kinds of place account for all of them, and both follow from a fact the rows are free established: every band between two straight creases folds exactly onto the band before it, the way the pleats of an accordion do, while every band between two zigzags walks the stack one step along itself, by the sine of the slant.

A zigzag place is a whole zigzag line. Every row of the sheet folds onto every other, so the segment of zigzag line three in the bottom row and the segment of the same line in the top row end up on the same piece of the folded plane. The figure marks one such place in red: ten hinges, one from each row, stacked one on top of the next. There are nine zigzag lines inside a ten-column sheet and so nine such places, each ten hinges deep.

A straight place is half a column. The straight creases are the accordion’s folds, and an accordion’s folds alternate sides. So the straight hinges of one column on the odd-numbered lines all end up on one side of the stack, and those on the even-numbered lines on the other. Each column gives two places, five hinges and four, and ten columns give twenty.

Nine and twenty make the 29.

The folded picture shows the two shapes a place takes. The zigzag place lies along the strip — a zigzag crease runs the stack’s length, which is why a column has to be a zigzag’s length long however narrow its paper. The straight place lies across it, from one long edge of the stack to the other.

Neither picture shows the depth. Every panel is drawn in the same flat colour, so a place holding ten hinges and a place holding four look the same. The ten hinges of the zigzag place are ten different layers of paper, one per row, and how far apart those layers sit is a question about thickness that a zero-thickness fold cannot answer. Three kinds of pile mapped how many layers lie over each point of these folded patterns; what is new here is not the count of layers but which hinges are in them.

Two cuts that look the same on paper #

A sheet split into modules is cut along crease lines. A Miura has two families of crease line, and a ten by ten sheet can be split into five modules either way: along four of its zigzags, giving five modules two columns wide, or along four of its straight lines, giving five modules two rows tall.

The two designs are as alike as a designer could ask. Each removes forty hinges and keeps 140. Each module has exactly 28. Each module folds on its own freedom with its own actuator, and splitting a sheet buys area, not certainty priced exactly this arrangement — modules driven separately so that a stuck hinge costs one module rather than the whole sheet. On a test bench that qualifies hinges one at a time, nothing tells the two cuts apart.

The places tell them apart completely.

Cut between rows, every place reaches every module. The cut removes four straight lines, and the zigzag places survive whole: each still holds one hinge from every row, which is now one or two hinges from every module. The straight places survive too, and they are no better. Modules two rows tall keep the straight lines between their own two rows — lines one, three, five, seven and nine, all odd — so every module’s straight hinges lie on the same side of the accordion, column by column, in the same ten places. Nineteen places in all, and all nineteen hold hinges from all five modules.

Cut between columns, no place reaches two. The cut removes four zigzag lines and leaves each module its own interior zigzag, whose place holds ten hinges from that module and nowhere else. A column’s straight places are that column’s, and a column belongs to one module. Twenty-five places, and every one lies wholly inside one module.

Both modules touch places, and the count each touches is the last column of the table: nineteen for a module of rows, five for a module of columns. A module of rows is exposed to every jam the sheet can have. A module of columns is exposed to the jams in its own two columns.

What a jam costs #

To price this, the failure model needs one more number than the earlier essays used, and it is chosen so that the hinges themselves do not change.

Every hinge works 999 times in a thousand on average, as in the crease count is a reliability budget and the essays after it, and every actuator works 99 times in a hundred. A share of each hinge’s failure is moved out of the hinge and into its place: the place jams with a small chance, and when it does every hinge in it is stuck; otherwise the hinge fails on its own with what is left. The two chances are set so that a hinge tested alone fails exactly one time in a thousand at every share, whether its failure is its own or its place’s. At a share of nought every failure is a hinge’s own business. At a share of one, every failure is a jam at a place.

Given which places jammed, the hinges are independent, and the chance that any set of modules opens is a product over hinges and over the places those modules touch. The chance that exactly so many modules open then follows by inclusion and exclusion over the thirty-two sets of five modules, with nothing sampled.

At a share of nought the two cuts cannot be told apart, and they should not be: with every failure independent, a module’s chance of opening depends on its 28 hinges and its actuator and on nothing else, and both cuts give every module the same. Either way 96.27 per cent of the area is expected to open, against 82.68 per cent for the same paper as one mechanism.

As the share rises, everything rises, because a jam takes several hinges at once and so the same average failure rate is spent in fewer events. That is the lift splitting insures only the failures that come alone found for a cause shared by every hinge, here arriving through places rather than through the whole sheet. The single mechanism gains most — it climbs to 96.17 per cent, because its 180 hinges now fail in 29 places rather than 180 ways — and the cut between rows gains least.

At a share of one the cut between columns expects 98.51 per cent of its area to open and the cut between rows 97.14. Measured as a lead over the single mechanism, that is 2.34 points against 0.97: the cut between rows has given back more than half of what splitting bought, because every one of its modules is exposed to all nineteen places and so a jam anywhere costs all of them.

Losing several at once #

The expected share hides how the losses arrive, and the arrival is where the two cuts differ most.

Cut between columns at a slant of 20.1 degrees, a jam stops one module and only one, so two modules are lost only when two separate things go wrong in the same deployment: 0.22 per cent of the time. Cut between rows, a jam anywhere stops every module, and that happens 1.88 per cent of the time; adding the rare deployments in which two modules fail for separate reasons gives 1.98 per cent. Nine times as often, from the same hinges, the same actuators, the same number of modules and the same area of paper in each.

The rows’ losses are nearly all total. Of the deployments in which the cut between rows loses anything beyond a single module, nearly all lose all five. A mission that can live with nine-tenths of its area — the antenna or reflector that the earlier essay found exposed to contagion — cannot live with nought, and the cut between rows offers it mostly the choice between everything and nothing.

There is one measure on which the cut between rows comes out ahead, and it is worth stating because it is the same reversal the shared cause produced. The chance that everything opens is 93.31 per cent for the cut between rows and 92.75 for the cut between columns. The cut between rows has nineteen places to jam and the cut between columns twenty-five, and “everything opens” needs every place clear. Sharing fates helps the all-or-nothing measure, because it packs the failures into fewer events, and harms every measure that counts partial success.

The slant that lines columns up #

The cut between columns kept every place inside one module at a slant of 20.1 degrees. It does not at every slant, and the slants at which it fails are not arbitrary.

At 45 degrees, the straight hinges of the third column on the odd-numbered lines and those of the fifth column on the even-numbered lines fold onto exactly the same segment. Nine hinges from two columns share one place. When the sheet is cut into modules two columns wide, the third and fifth columns are in different modules, so that place reaches across the cut — and so do seven others like it. Eight of the seventeen places at 45 degrees hold hinges from two neighbouring modules.

The arithmetic that predicts it is short. Fold a single column and its zigzag crease runs along the length of the stack, as the folded picture showed, so the column’s straight hinges on one side of the accordion sit a zigzag’s length — one over the cosine of the slant, $\sec a$ — further along the stack than its hinges on the other side. Each further column walks the stack on by $\sin a$, as the rows are free measured. So the hinges on one side of column $i$ arrive where the hinges on the other side of column $i + q$ are when

$$q \sin a = \sec a, \qquad q = \frac{1}{\sin a \cos a} = \frac{2}{\sin 2a}.$$

That is a whole number only at particular slants, and neighbouring columns lean opposite ways, so a segment lands on another rather than crossing it only when the two columns are an even number apart. Places cross columns exactly when $2 / \sin 2a$ is an even whole number: 45 degrees, where it is two; 15 degrees, where it is four; and on through six, eight and ten at ever shallower slants.

The table runs the measurement at eight slants. At 45 degrees, eight places cross the cut and two modules are lost together 0.93 per cent of the time; at 15 degrees, six places and 0.75 per cent; at 9.7 degrees, where $2 / \sin 2a$ is six, four and 0.57; at 7.2 degrees, two and 0.39. As the offset grows the shared places run off the end of a ten-column sheet, and fewer of them have both ends on paper. At the odd values, three and five, and at the slants in between, nothing crosses at all and the figure sits at 0.22 per cent.

Through all of it the expected share does not move. A module of two columns touches five places at every slant in the table; what changes is whether two neighbouring modules touch the same one. So the slant decides nothing about the average and everything about whether two modules go down together.

The slant that piles evenly #

Those slants have been met before. The slant that stacks in one depth found that a folded Miura’s pile at any point is the number of rows times the number of column images over it — on average $2 / \sin 2a$ columns’ worth — and that when that number is an even whole number the pile is one depth everywhere but the ends. Two columns deep at 45 degrees; four at exactly 15. When it is odd, the pile stays mixed, because neighbouring columns lean opposite ways.

It is the same number for the same reason. A pile one depth everywhere is a stack in which columns that many apart lie exactly over one another; a place crossing columns is two of their hinges lying exactly on one another. The slant a thick-panel designer would choose, because a pile of one depth can be met with one panel thickness, is the slant at which a cut between columns stops keeping fates apart.

Neither consideration is wrong. They are two consequences of one coincidence in the folded geometry, and a designer who chooses 45 degrees for the first has, without asking, chosen the second. That is the kind of trade nobody makes deliberately because the two halves of it are priced in different essays, by different specialists, and only meet in the folded stack.

Separation, which a nuclear plant learned by fire #

None of this is particular to folding, and the field that learned it learned it expensively.

In March 1975 a worker at the Browns Ferry nuclear plant in Alabama used a candle to look for air leaks in a cable room, and set the foam sealant alight. The fire burned for hours through cable trays where, it turned out, the control cables of supposedly redundant safety systems ran side by side. Systems that were independent on the schematic shared a fate because they shared a place, and the place was where the fire was. The fire-protection rules that followed require redundant safety trains to be kept apart physically — by distance or by barrier — rather than only electrically, because the reliability arithmetic that multiplies independent failure chances is only as good as the separation behind it.

The reliability literature calls this a common-cause failure, and the general lesson is the one splitting insures only the failures that come alone drew from portfolios: diversification removes what belongs to each part and none of what the parts share. The folded version adds the part a cable tray does not have. A deployable’s “places” are not chosen when it is routed; they are decided by its crease pattern, they are invisible on the flat sheet, and the same crease pattern gives different sharing depending on which way it is cut. A schematic of the five modules is identical for both cuts. The stack is not.

What the model assumes #

Zero thickness. A place is an exact coincidence of folded segments in a fold with no thickness. Real panels have depth, the ten hinges of a zigzag place sit at ten heights, and a cause acting on the stack — pressure, a crushed corner — reaches all of them only if it reaches through the depth. A cause that acts on one face of the stack, such as contamination of the top layer, would reach a few hinges of a place rather than all, and the shared fate would be partial.

Every place equally likely to jam. A place holding ten hinges and a place holding four are given the same chance. A deeper stack might well jam more often, which would make the zigzag places — every one of them shared by all five modules of rows — still more dangerous to the cut between rows than they are here.

Exact coincidence, not nearness. Two places a hair apart in the stack are counted as two places, and at a slant a hundredth of a degree from 45 the shared places disappear from the count entirely. A real jam has a size, and anything within it shares the fate. The even slants are the case where the coincidence is exact; slants near them are cases where hinges from two modules lie near one another, which a model with a jam radius would treat as sharing.

The pattern. Everything here is the Miura, as Koryo Miura published it in 1970 and as the solar array that flew in 1995 used it. Its rows folding onto one another is what makes the zigzag places whole lines, and a pattern whose rows walk as well — a waterbomb, a twist — would distribute its places differently and would need the census run again.

How the places were found and the chances computed #

The places are read off the folding, not predicted. The flat folding is solved panel by panel, each hinge’s flat segment is carried into the folded plane by the motion of a panel beside it, and two hinges share a place when their folded segments agree end for end to a ten-millionth of a sheet width. The claims checked on every drawing are that a zigzag line’s place holds one hinge per row and nothing else; that a column’s straight place holds that column alone except at the even slants, where it holds exactly two columns the predicted distance apart; that a cut between rows leaves every surviving place in all five modules; and that across eight slants a place crosses the column cut exactly when $2 / \sin 2a$ is an even whole number.

The chances are exact. For each set of modules the chance that all of them open is the product of their actuators, their hinges’ own chances and the chance that every place they touch stays clear; the chance of exactly those modules opening follows by inclusion and exclusion. Checked: that both cuts give the same expected share when no failure comes from a place, to twelve decimal places; that the cut between columns stays ahead of the cut between rows at every share; and that the slant leaves the column cut’s expected share unchanged while changing how often it loses two modules together.

Still open: whether a cut can follow the stack #

The cuts compared here follow crease lines of the flat sheet, because that is where a sheet can be cut and still fold. A cut that followed the stack instead — keeping together every hinge that shares a place, whatever its position on the flat sheet — would put no place across any boundary at any slant, and on a Miura it would be exactly the cut between columns, except at the even slants, where it would have to join columns that many apart into one module. Whether a module made of non-adjacent columns can be driven and tethered at all is a question about mechanisms rather than about places, and a cut along the zigzag needs a tether is where the tether’s own price was set.

The model with a jam radius is the other direction, and the one that would make the slant a continuous dial rather than a set of special values. Places near one another would share partially, the even slants would become the centres of bands rather than points, and the width of those bands — how far from 45 degrees a slant must be before two columns stop sharing a fate — would be measured in the thickness of the panels. A panel is not the unit of depth found that a single panel of a folded Miura lies over several depths of its pile; the hinges at those depths are where a jam with a size would reach.

The habit worth carrying is about diagrams. Before trusting that two parts are independent because a diagram draws them apart, ask where they are when the thing is packed. A schematic of modules is a flat drawing; a deployable spends its stowage folded, and the folded state decides which parts are neighbours.