Rounding in the cheap direction

Spelling a tree on a grid put a subject onto a box-pleating grid by rounding each limb to a whole number of units and measuring how far the proportions drift. The rounding was chosen to keep the shape: either each limb to its nearest unit, or the set of whole numbers whose worst ratio error is smallest. Nothing in that choice asked what the rounding did to the size of the model.

A price holds until the arrangement moves then measured what moving length between a subject’s edges does to the scale its sheet supports, and found the prices good for a move of about a tenth of a unit on the bird before the arrangement shifted. A rounding is a move of that size. Rounding the head down and the tail up takes length from an edge the arrangement prices highly and gives it to one it prices low, and the prices say which way that should go.

So the question the grid had not asked can be put directly: round every edge in its cheap direction instead of its nearest, and see what the model gains and the proportions lose.

What a rounding moves #

The bird’s tree has a body of 0.5, wings of 1.6, a head of 0.9, legs of 0.8 and a tail of 1.3. On a grid of eight units across the wings, a unit is 0.2 and the limbs come to 2.5, 8, 4.5, 4 and 6.5 units. The wings and legs are whole already. The body, the head and the tail each have to go one way or the other, and each way is half a unit of length, a tenth of the bird’s own measure.

Rounding to the nearest unit settles the three by the fractional part, and all three are exactly a half: conventional rounding sends them all up, to 3, 5 and 7. That is a move of length onto the body and the head, the two edges the price of a limb is not its length found dearest, and onto the tail, which was nearly free.

The prices would have sent the dear two the other way. Round the head and the body down and the tail up — 2, 4 and 7 — and the move takes a half-unit off each of the two dearest edges and puts a half-unit on the cheapest.

Measuring the size of a model #

Comparing two roundings needs a measure of size that does not reward simply rounding everything up. A rounded tree in grid units has its own scale on the sheet, but a tree with longer limbs is a bigger subject, and its scale is smaller for that reason alone.

The measure used is the sheet length that one unit of the subject’s own length receives. Take the rounded limbs, find the factor that best matches them to the subject’s true proportions — the least-squares ratio of units to lengths — and multiply the arrangement’s scale by the unit and by that factor. Rounding every limb up by the same proportion leaves this unchanged, because the scale falls exactly as the factor rises; rounding only the cheap limbs up does not, because the scale hardly falls. For the unrounded bird it is simply the scale, 0.2651.

Eight units #

On eight units the nearest rounding, 3, 8, 5, 4, 7, gives a model of size 0.2631 — smaller than the unrounded bird. The rounding has put length onto both of the dear edges, and the arrangement has paid for it at their prices.

Rounding in the cheap direction, 2, 8, 4, 4, 7, gives 0.2782: 5.7 per cent larger than the nearest rounding and 4.9 per cent larger than the bird before it was rounded at all. Taking a half-unit off the head and the body and putting one on the tail is a move of exactly the kind the spending measured, and it gains about what the first steps of the spending gained.

The proportions pay. The nearest rounding’s worst limb is 15 per cent from its true ratio; the cheap-direction rounding’s is 20, because a body of two units against wings of eight is a body a fifth short. Five points of shape error for six per cent of model is a trade a designer might well take and might well refuse, and the point is that it was being made in one direction or the other either way.

Every rounding, not just two #

Two roundings compared are two points, and the claim needs more than that. On eight units there are three edges that can round either way, so eight roundings; on four and six units there are also eight, since three of the five groups fall between whole numbers on each grid.

On every grid, the rounding that sends the dearer edges down and the rest up is the largest model of all the roundings tried. That was checked by solving all of them, not by trusting the prices. On four units it is 1, 4, 2, 2, 4 at size 0.2807; on six units it is 1, 6, 3, 3, 5 at 0.2784; on eight, 2, 8, 4, 4, 7 at 0.2782. Each is larger than the unrounded bird’s 0.2651.

The nearest roundings land wherever their fractions put them: 0.2675 on four units, 0.2724 on six, and 0.2631 on eight — twice larger than the unrounded bird by accident of which way three fractions fell, and once smaller.

Six units, and a trade not worth taking #

The six-unit grid is where the rule’s price shows. A unit there is 0.267, and the body of 0.5 is 1.875 units — very nearly two. Rounding it in the cheap direction takes it to one, which is a body almost half as long as the subject’s.

The model gains: 0.2784 against the nearest rounding’s 0.2724, 2.2 per cent. The shape loses badly: the worst limb’s error goes from 11 per cent to 45. A bird with half a body for two per cent more model is not a design anybody would choose, and the rule cannot see that, because it knows the prices and not the fractions.

The lesson is a narrow one and worth stating exactly. The prices say which direction gains size; the fractional parts say how much each direction costs in shape; and a rounding that ignores either is choosing blind on one axis. On eight units the fractions were all a half, so the shape cost of either direction was the same and the prices decided everything. On six they were not, and the body’s fraction was so near a whole that rounding it down was a large error bought for a small gain.

A grid that fits the prices #

The shape error by itself falls slowly and unevenly as the grid gets finer, and a designer choosing a grid is choosing where on that curve to sit. The prices add a second reason to prefer some grids over others: a grid on which the dear edges fall just above a whole number and the cheap ones just below lets the cheap-direction rounding cost almost nothing in shape, because every rounding is small.

On the bird that would be a grid where the head and the body round down by a little and the tail rounds up by a little. Eight units is close to that — all three are halves — and it is the grid where the cheap-direction rounding gains most for the least shape. Six is the opposite: the body sits just below a whole unit, so the cheap direction for it is the long way.

That turns the choice of grid from a question about resolution into a question about alignment. Spelling a tree on a grid found the shape error a noisy function of the grid; with prices attached, the noise has a direction, and a designer can look for the grids where it points the useful way.

Where box pleating’s trade was being made #

Box pleating has always been defended as a trade. Designing on a grid puts it as giving up some of a free packing’s efficiency to buy creases that fall on a square grid, and what the grid settles measures the other side of the same trade: how much of the design’s freedom the grid removes. Both treat the loss of efficiency as a price paid once, at the moment the grid is adopted.

The roundings say the price is not paid once. It is paid per limb, in a direction chosen by the fractions, and it can be negative. The cheap-direction rounding on every grid tried gives a larger model than the unrounded tree, which is to say that for this subject the grid, spelled well, costs nothing in size and something in shape; spelled by nearest units on eight, it costs a little in both.

That reframes the familiar comparison between a free design and a gridded one. A free design is optimised for the proportions given; a gridded one has proportions that have been nudged anyway, and the nudges can be aimed. The efficiency box pleating gives up is partly a consequence of aiming them at the shape alone.

The continuous spending and the roundings agree about how much there is to gain. Moving length from dear edges to cheap ones continuously reaches a scale 6.3 per cent above the drawn bird and stops there, where the prices come together; the best rounding on each grid reaches between 4.9 and 5.9 per cent above it. A rounding is a coarse, one-shot spending, constrained to whole units and to one move per edge, and it gets most of what the fine spending gets because the prices it reads are good for about the size of move it makes.

Why the prices still hold here #

A rounding moves each edge by at most half a unit, and on eight units that is a tenth of the bird’s own measure — about the length the bird’s prices were found to hold for. The prices were measured on the unrounded bird and used on roundings that move three edges at once, which is a larger move than any single step of the spending, and the fact that the prediction is right on all three grids is a sign that the moves stay mostly inside the region where the arrangement’s binding pairs are the same.

On four units, where a unit is 0.4 and a rounding moves an edge by up to 0.2, that is less obviously so, and the cheap-direction rounding still wins. There the nearest rounding sends all three fractional limbs down, since each is a quarter past a whole number, and the cheap direction differs from it only at the tail — so the whole of the 4.9 per cent between them is three quarters of a unit of tail — 0.3 of the bird’s own length, rounded up where the nearest rounding had taken it down — placed on the edge that what the condition does not decide would have called slack. That the entire difference sits on the one edge with room to absorb length is the clearest single confirmation in the table that the prices are reading the arrangement correctly. It is also a larger move than the spending found the tail could take for free, and the model still grew: the tail’s slack was used up and then some, and the extra length on it cost less than the scale it bought through the proportions. The measurement does not say by how much margin; it says only that among the eight roundings on each grid the rule picked the largest, and a larger grid unit or a subject with prices nearer together could make it pick the second.

What the comparison assumes #

Size is measured per unit of the subject’s own length. That makes rounding everything up in proportion neutral, which is the property the comparison needs, and it treats a subject whose proportions have drifted as the same subject at a different size — which is exactly the part a designer might dispute.

The arrangement is the best a search finds. Each rounding’s scale comes from the same hill-climbing search with sixty starts, as every scale in this account does, and a difference of a per cent between two roundings is near the edge of what the search resolves. The largest differences here are five per cent and more.

The prices are fixed at the unrounded tree. The rule reads them once and applies them to every grid; a rule that re-priced after each edge was rounded would follow the arrangement more closely, and a price holds until the arrangement moves suggests it would sometimes choose differently.

What the table does not show #

It does not show the crease pattern. A rounding decides the grid lengths of the tree’s edges and the base follows; whether the cheap-direction rounding’s base is harder to fold, or has flaps at awkward places, is not measured here, and every flap on one axis is the reminder that the base is where the tree becomes paper.

It does not decide what counts as the subject. Twenty per cent of shape error on a body is a stubby bird and forty-five is a different animal, and the line between the two is a matter of design rather than of geometry.

It does not account for the other constraints a grid imposes. The designer’s grid is the dearest thing here finds the grid expensive to check for a consistent lettering, and a rounding that changes which limbs are long changes where the pleats run; the size comparison treats both roundings as equally foldable, which neither measurement establishes.

And it covers one subject. The bird has a clear price order with a large gap between its dearest and cheapest edges, which is the case where direction matters most. A subject whose edges all cost about the same would gain little from rounding either way, and the table would show two nearly equal sizes.

Still open: rounding on both axes at once #

The rule here used the prices alone and got the size right and the shape wrong on one grid of three. A rounding that weighs both — gain in size from the prices, loss in shape from the fractions — is a small two-objective problem with eight candidates per grid, and its answer is a frontier: for each amount of shape error a designer will accept, the largest model available. The nearest rounding and the cheap-direction rounding are two points on or near that frontier, and on six units neither may be the point anybody wants.

The other direction is the one the price of a limb is not its length opened with: a subject with latitude in its proportions. If a head may be anywhere between 0.7 and 1.1, the grid can be chosen so that the head’s latitude lands on a whole number in its cheap direction, and the rounding error on that limb is not error at all but a proportion the subject allowed. Combining latitude, prices and grid alignment is a search over grids rather than over roundings, and it is the version of this question a designer at a box-pleating grid actually faces.

The habit worth carrying is about rounding in general. Every rounding is a small reallocation, and a reallocation has prices. Rounding to the nearest value chooses a direction by the fractional part alone, which is the one thing about the quantity that has nothing to do with what it costs; when the costs are known, the direction of each rounding is a decision, and leaving it to the fraction is choosing by coin.