A sheet with two edges

The square is a choice, and this collection has taken the choice apart in several directions: rectangles, hexagons, sheets with holes, sheets with notches. Every one of those is an outline — a different shape of boundary on a flat piece of paper.

A cylinder is not an outline. It is a rectangle whose left edge and right edge are the same edge, and no shape of flat paper is that.

What a designer’s sheet is for #

The boundary does three things in design and it is worth separating them.

It is where flaps end. A flap costs a circle, and the circle is a disc of paper centred on the flap’s attachment; the flap’s tip comes from the boundary of the region it claims. A sheet’s edge is where the paper stops and therefore where the points of a base come from.

It is where constructions start. Every reference point begins from a corner or an edge — a fold is made by bringing one marked thing onto another, and the sheet’s own boundary is the only thing marked to begin with.

It is where the folding is loose. A panel at the rim has creases on fewer sides, so it is constrained less, and that shows up as free choices in a lettering and as the panels a folded stack can rest on.

A cylinder halves all three.

What the halving costs, measured #

The freedom the boundary supplies is countable, and the count is the creases the glued edges divide.

On a two-period cell of the square twist tessellation: forty free letters cut out of the plane, thirty-six with one pair of edges glued, thirty-two with both. Panels: twenty-five, twenty, sixteen.

A tenth of the freedom for each glued pair, on that cell. On a one-period cell it is a sixth apiece, because a small patch is mostly boundary.

What the halving buys #

The sheet is the one that gets built.

Almost every folded object in engineering is a tube or a shell: a deployable boom, a stent, an airbag, a packed antenna, a corrugated bellows. Each of them is a sheet joined to itself along a seam, and none of them is a flat square.

So a designer working on a cylinder is working on the object rather than on a specimen of it, and every count above is a count about the thing that will be manufactured rather than about a rectangle that will be rolled up afterwards.

The corner, which disappears #

A square has four corners and a corner is worth four times the middle: the paper there is claimed by fewer flaps, so it goes further.

A cylinder has none. Its two circles of boundary have no corners at all, and the corner premium has nothing to attach to.

That is a separate argument and it is the sharpest single change: a designer’s most valuable paper simply does not exist on a closed sheet, and the accounting that ranks sheet shapes by how much of their boundary is corner has nothing to rank.

The sheets a designer can actually choose #

It is worth listing what is available, because the list is longer than square or rectangle and shorter than one might hope.

Flat shapes. A square, a rectangle of any proportion, a hexagon, a triangle, a disc, a shape with holes, a shape with notches. Every one is a flat piece of paper with an outline, all of them are cuttable, and this collection has priced several of them.

A cylinder. A rectangle with one pair of edges joined. Physical, manufactured, and the sheet most folded hardware has.

A cylinder with a twist — a Möbius band. Physical, and it inverts the parity condition rather than merely having one. Nobody designs on one and it is a genuinely available sheet.

A torus. Not physical.

That is the whole list of sheets this collection can build. It is worth noticing that only the first category has ever been used for design, and that the second is what everything gets turned into afterwards.

Where the freedom actually goes #

The measured loss is free letters, and a designer might reasonably ask what a free letter is worth to them.

A free letter is a crease whose assignment is unconstrained: nothing forces it, and setting it either way leaves everything else satisfiable. On a flat sheet the boundary supplies them, because a crease running to the edge has a vertex on one side only and the propagation stops there.

For a designer that is slack in the pattern. A pattern with free letters can be adjusted — one crease reversed, a fold made the other way for convenience — without anything else having to change. A pattern with none cannot: every change propagates.

So the loss on a cylinder is a loss of local adjustability. The pattern still folds; it is just less forgiving to modify, and a modification that would have been absorbed near the edge now has to be paid for somewhere else.

That is a real design cost and it is not one that appears in any of the usual accounting, which is about paper efficiency rather than about how much a pattern can be nudged.

The seam, from a designer’s point of view #

A designer joining a sheet into a tube has one decision the mathematics above does not see: where to put the seam.

Mathematically it does not matter. The cylinder is the same sheet wherever the rectangle was cut open, and every count is unchanged.

Practically it matters a great deal. The seam is a line of adhesive or a fold-over or a stitched joint, it is stiffer than the paper, it is thicker, and it is the most likely place for the object to fail. Putting it in the middle of a panel is different from putting it along a crease, and both are different from putting it at a vertex.

None of that is modelled here. The gluing treats a seam as a mathematical identification with no thickness, no stiffness and no width, which is the same idealisation the collection makes about creases and has the same character: exactly right for the questions being asked and wrong for the ones a manufacturer asks.

Worth stating plainly, because a reader coming to these results from the hardware side will want to know whether the seam has been modelled, and it has not.

Two ends, and what they are for #

A cylinder’s remaining boundary is two circles, and it is worth asking what a designer can do with them.

Flaps. Every point of a base comes from the boundary, so a cylinder’s base can have points only at its two ends. That is a fringe rather than a tree, and it rules out the whole uniaxial method, which assumes a sheet whose boundary is a closed curve enclosing everything.

References. A construction can start from either circle. The two are the only marked things on the sheet, so every reference point is built from them, and a cylinder is much poorer in references than a square — which has four edges and four corners and therefore a great many.

Attachment. The two ends are where a tube joins something else: a hub, a plate, another tube. That is the practical use and it is not a folding question.

So the boundary that remains is doing a different job from the boundary a square has. It is an interface rather than a source of material, which is a reasonable description of what a tube’s ends are for.

What cannot be done at all #

Two things, and they are worth stating because they are absolute rather than expensive.

A uniaxial base needs boundary. Every flap’s tip comes from the sheet’s edge; a sheet with no edge has nowhere for a point to come from. On a cylinder there are two circles of edge and flaps can only come from them, which restricts a base to something like a fringe at each end.

A reference construction needs a mark. The axioms are about bringing marked things onto marked things, and a cylinder’s seam is not a mark — it is an ordinary line of paper, with no crease on it and nothing to align to.

And one condition arrives #

Designing on a closed sheet adds a constraint that does not exist on a flat one.

A path running once round a cylinder cannot be shrunk to a point, so the number of creases it crosses has a parity that is not forced by anything local — and a sheet where that parity is odd has no flat folded state at all.

For a designer that is a rule about how many creases may run along a tube: an even number, or the tube does not flatten. It is checked by counting and it is the first constraint in this subject that a designer can verify before drawing anything.

Why nobody designs on a cylinder #

The sheet exists, it is manufactured, and no design method targets it. The reasons are worth naming because none of them is that it would not work.

The methods are flat. Circle packing, box pleating, the tree method: all of them place things on a flat sheet and none of them has a version that places them on a closed one. Extending any of them means redoing the geometry on a surface where a straight line comes back to itself.

Paper is flat. A designer works with paper, and paper is a flat sheet. Making a cylinder means taping first and folding afterwards, which is awkward and which loses the ability to open the model out and look at the pattern.

The applications tape last. An engineer designing a folded tube designs a flat pattern, manufactures it flat, and joins it as the final step. That is how sheet material is handled, and it means the flat pattern is the working object even when the tube is the product.

So the flat sheet is the design surface for reasons of tooling and habit, and the cylinder is the product. That division is stable and there is no obvious reason it should change.

What changes is that the analysis can now be done on the product. Whether the tube flattens, how many configurations it has, how much freedom the pattern retains: those are questions about the cylinder, and they were previously answered on the rectangle by default.

The efficiency question, which does not transfer #

Sheet shape is normally compared on efficiency: how much of the paper a design uses, which for a uniaxial base is the fraction covered by the flaps’ circles.

That measure does not survive the move to a cylinder, and the reason is not that it is hard to compute.

A circle-packing efficiency compares the area the flaps claim against the area of the sheet. On a cylinder the flaps can only come from the two end circles, so the middle of the sheet is claimed by nothing — and an efficiency computed that way is close to nought regardless of the design, which is not a useful number.

What a cylinder is efficient at is something else entirely: enclosing volume, packing along its own axis, deploying to a length. Those are the quantities the applications care about, and none of them is a paper-efficiency in the design sense.

So the comparison between sheet shapes cannot be extended to closed sheets by computing the same statistic. It would need a different statistic, chosen for what a closed sheet is for, and nothing here proposes one.

An outline and an identification #

The distinction the essay opens with is worth restating at the end, because it is the whole of what makes a cylinder a new kind of choice.

Every sheet a designer has ever chosen is specified by an outline: a closed curve on the plane, with the paper inside it. A square, a hexagon, an A-series rectangle, a disc, a shape with a hole — each is a curve and the paper it bounds.

A cylinder needs a second piece of data: an identification, saying which points of the outline are the same point. That is not a curve and it cannot be drawn, and it is the thing no crease pattern and no file format records.

So choosing a sheet has always been a one-parameter decision and it is a two-parameter one. The second parameter has been empty for the whole history of the subject, which is why it has never been named, and it has exactly four values for a rectangle: nothing, one pair, the other pair, both.

Four values is not many. It is more than one, which is what it had been.

What was actually measured #

Since the essay is mostly argument, the measurements behind it deserve to be separated out.

Free letters and panels on four sheets, for three drawings at two or three sizes each. Those are counts, they are exact, and they are what the claims about lost freedom rest on.

Search cost per panel, same objects, one variable order. Those are measurements of a procedure and they support the claim that the cylinder sits between the other two.

Verdicts, same objects. Those are what support the parity claim, and they are checked by two computations that share no code.

Nothing about flaps, references, efficiency or manufacturing. Those sections are argument from what the boundary does, and the arguments are short and are not computations. A reader wanting to know how much a cylinder’s base can hold will not find it here, because nothing here computes bases at all.

The one number a designer should take #

If a single thing is worth carrying out of this, it is the parity.

A pattern rolled into a tube flattens only if an even number of creases run along the tube. Count them, including the seam if the seam becomes a fold. Odd, and there is no flat state to reach; the tube will collapse partway and then buckle.

That is checkable before anything is cut, it costs an addition, and it is the only condition in this subject that a designer can verify by counting rather than by folding.

The accounting, side by side #

A square. Four edges, four corners, the most freedom, the most flaps, the most references, and it is not the object anybody builds.

A cylinder. Two edges, no corners, less freedom, flaps only at the two ends, no corner premium, a parity condition — and it is the object.

A torus. No edges, no corners, no flaps at all, no references, two parity conditions, and it cannot be made from paper.

Each step down that list removes something a designer uses and gets closer to a manufactured shape, until the last step, which gets closer to nothing and is a mathematical control.

What this changes #

Very little about existing design method and one thing about how a result should be stated.

Design methods operate on a flat sheet, produce a crease pattern, and the pattern is then rolled and joined if the application wants a tube. That is a perfectly good workflow and nothing above says otherwise.

What the four sheets add is that the properties of the result — how many configurations it has, whether it folds flat, how much freedom the pattern retains — are properties of the joined object, and measuring them on the flat pattern measures a different sheet.

For most design work the difference does not arise, because the questions asked are about flaps and lengths rather than about counts. Where it does arise, the correction is available and is a gluing.