A growing clump prints like the screen with as many dots

Brightened office paper glows: optical brighteners turn the ultraviolet in daylight into visible blue, which is why some paper is brighter than white, and a halftone tint printed on the paper carries that glow from the bare paper between its dots and, more faintly, from under them. A thin dot edge lets the glow back in found the ink film thinning at each dot’s rim and letting the ultraviolet through to the brightener, so the share of the ink that is rim sets how much the tint’s hue is turned for a reader fully adapted to the sheet — the residue, the part of the turn adapting to a brightened sheet leaves the glow found surviving the reader’s adaptation. A real reader, adapted partly to the surround as well, sees more than the residue, as a reader sees the reversal until almost adapted found for newsprint; the residue is the floor. Stochastic dots hold the glow at every tint found a screen of small separate dots paying it at every coverage, and merged spots give the mid-tones half the glow back found merging returning half of that cost in the mid-tones and none in the pale tints.

Squares of four print a pale tint like a 150-line screen switched the stochastic screen’s spots on in clusters. Squares of four took a five-per-cent tint to within a tenth of a degree of a 150-line screen for every pigment, and below a 300-line screen; but from ten per cent the 150-line screen pulled ahead, by up to three quarters of a degree, because a conventional dot grows as the tint darkens and a fixed cluster only multiplies.

Its closing section proposed the obvious hybrid, which commercial screening offers in various forms: lay clusters in the pale tints and grow each one as the tint darkens, holding the number of clumps, so that the mid-tones print as a fine conventional screen with irregularly placed dots. The prediction was that the growing clusters keep the squares’ five-per-cent residue, reach within a tenth of a degree of the 150-line screen by thirty per cent, and stay slightly above it through the mid-tones, because their clumps are placed by a dispersed order rather than on a lattice and so begin merging earlier.

Thirteen growing clumps follow the 150-line screen #

On the census’s tile of 32 by 32 twenty-micrometre sites, thirteen clumps started as the squares’ thirteen clusters of four and grown spot by spot from their centres leave every pigment’s five-per-cent residue exactly where the squares do. Through the ramp the orange’s residue follows the 150-line screen’s within two tenths of a degree at every coverage: 7.93 degrees against 7.77 at ten per cent, 7.42 against 7.40 at twenty, 7.22 against 7.09 at thirty, 6.30 against 6.27 at fifty, and 5.65 against 5.67 at sixty. Every pigment follows its 150-line residue within a tenth of itself. With twenty-six clumps on the tile the orange follows the 200-line screen, and with fifty-two the 300-line screen, each within a quarter of a degree from ten per cent on and closer to its own ruling than to either other.

• The prediction holds, and its reason does not. The clumps follow the 150-line screen through the ramp; they sit just below it at sixty per cent, not above it.
• The number of clumps is a ruling. Thirteen on the tile are 143 to the inch, and the growing screen prints like the conventional screen with as many dots.
• Irregular placement costs nothing measurable. A clump’s position on the tile does not show in the glow; only how many clumps share the ink.

The four curves start in two pairs and end in two. At five per cent the single spots sit near nine degrees and the squares, the growing clumps and the 150-line screen together near eight. From ten per cent the squares climb away from the other two, which descend together along the ramp; by sixty per cent the single spots have merged their way down to 6.90 and the squares to 6.00, while the growing clumps and the 150-line screen sit together at 5.65 and 5.67. The axis starts at five degrees.

What growing does to the ink #

Each clump grows into the cell of sites nearer its own centre than any other’s — thirteen irregular cells on the torus — switching on its spots in order of distance from its centre, one clump at a time in turn. At five per cent the tile holds thirteen squares; at thirty, thirteen rounded blobs about a hundred micrometres across; at sixty, thirteen large clumps that have begun to touch across the cells’ boundaries. That is a conventional screen’s growth pattern on an irregular lattice: a dot that starts small and round and grows until it meets its neighbours. The sizes match too. A clump of the same area as a circle is 45 micrometres across at five per cent, 110 at thirty and 155 at sixty; a 150-line screen’s round dot is 43, 105 and 148. The growing clumps are, to within a few per cent, the 150-line screen’s dots, scattered instead of ruled — which is what holding thirteen clumps on a tile the width of a 143-line screen’s thirteen cells should give.

The rim falls as the clumps grow: 52 per cent of the ink at five per cent, 28 at twenty, 19 at fifty and 16 at sixty, against the 150-line screen’s 48, 26, 17 and 16. The two curves lie on top of one another from twenty per cent. The squares’ rim stays near half until the clusters merge, which is why they fall behind. A rim share is a perimeter over an area, and a clump of a given area has a perimeter set by its size, so a screen whose dots grow at the conventional screen’s rate carries the conventional screen’s rim whatever lattice they sit on.

The number of clumps is the ruling #

If the rim is set by how big each dot is at a coverage, and that by how many dots share the ink, then a growing screen should follow whichever conventional screen has the same number of dots per area. The census’s tile is 640 micrometres across; thirteen clumps on it are one per 177 micrometres, as many as a 143-line screen; twenty-six are as many as a 202-line screen, and fifty-two as a 286-line screen. With twenty-six clumps the orange’s residue follows the 200-line screen from ten per cent — 8.07 against 7.99 at ten per cent, 7.43 against 7.26 at thirty, 6.88 against 6.90 at forty, 5.84 against 5.86 at sixty — and with fifty-two the 300-line screen, 8.53 against 8.42 at ten per cent and 6.10 against 6.22 at sixty. Each count is closer to its own ruling than to either other.

At five per cent the larger counts start partway: twenty-six clumps of about two spots each and fifty-two single spots, which is where those counts’ palest tints have to begin. The fifty-two-clump screen’s five-per-cent tint leaves the single spots’ 8.94 degrees, and the 300-line screen’s leaves 8.95. A 300-line screen’s palest dot is a twenty-micrometre spot, as the squares essay found; a stochastic screen with as many clumps as a 300-line screen has dots is a 300-line screen to the glow.

One curve in dots per inch #

The handle compares one count with three rulings; the same numbers can be laid on one axis. A conventional screen’s ruling is a count of dots per inch, and a growing screen’s clump count can be written the same way: thirteen clumps on a 640-micrometre tile are 143 to the inch, twenty-six are 202, fifty-two are 286. Placed by that number, the growing screens sit on the conventional screens’ own curve. At thirty per cent coverage the conventional rulings give 6.91 degrees at 100 lines, 7.09 at 150, 7.26 at 200 and 7.59 at 300; the growing clumps give 7.22 at 143, 7.43 at 202 and 7.60 at 286. At sixty per cent the conventional curve runs from 5.49 to 6.22 and the clumps give 5.65, 5.84 and 6.10. The clumps sit on the curve or a little above it, never more than two tenths of a degree away.

That is a stronger statement than the handle makes. It says the residue a screen leaves is, to within a few per cent, a single function of how many dots it has per area and how much ink it lays — a curve along which a 150-line screen and thirteen irregular clumps are the same point. The lattice, the dots’ shape, the irregular cells and the order in which the spots were switched on all drop out. What the curve does not include is the first-order stochastic screen, whose spots never grow: it has 1,024 dots on the tile at every coverage until they merge, and in dots-per-inch terms it is off the chart to the right, which is why it pays the highest residue of any screen the census has.

The cells are unequal, and the smallest fill first #

The growing screen’s thirteen cells are not equal. Thirteen centres placed by a dispersed order on a torus divide its 1,024 sites into cells of 46 to 100, and each clump grows into its own cell. At sixty per cent coverage each clump has been handed 47 spots, and the smallest cell, 46 sites, is full; its clump has reached every boundary of its cell and touches its neighbours, while the clump in the largest cell is still a round blob with paper all round it. A production screen would more likely balance its cells, giving each clump the same territory.

Unequal cells might have been expected to cost the growing screen its match with the conventional one, because a full cell’s clump has lost most of its rim and a half-full cell’s has not, and the average of the two is not the rim of two equal clumps. The census says the cost is small. At sixty per cent the growing screen’s rim is 16 per cent of the ink and the 150-line screen’s 16 per cent; its residue is 5.65 degrees against 5.67. Whatever the unequal cells do to individual clumps, the tile as a whole carries the rim its clump count implies.

Every pigment follows #

The pigments carry very different residues, from the yellow’s degree and a half to the red’s eleven, and the growing clumps follow each one’s 150-line residue within a tenth of its size at every coverage. At thirty per cent the largest gaps are the red’s 0.16 degrees and the cyan’s 0.13 — under two per cent of the red’s residue and under six per cent of the cyan’s. As in the earlier screening essays, the pigment sets the size of the residue and the screen’s geometry sets its share, and a geometry that matches a conventional screen’s rim matches its residue for every pigment.

What this means for a printer on brightened stock #

A stochastic screen with growing clusters is, to the glow on brightened office paper, the conventional screen with as many dots per area — at every coverage, not just the palest. The squares essay found a clustered screen matching a 150-line screen in the highlights and falling behind it in the mid-tones; holding the clumps’ number and growing them closes that gap entirely. A printer choosing a hybrid screen for brightened stock can choose its clump density as a conventional ruling: thirteen clumps to the 640-micrometre tile behaves like 150 lines, twenty-six like 200.

That also says what the hybrid cannot do. It cannot beat the conventional screen of the same density, because it is that screen as far as the rim is concerned. The glow’s cost is a perimeter, and the only way to lower it at a given coverage is fewer, larger dots — a coarser screen — which trades the glow against the visibility of the pattern, as every halftone choice does. Along the dots-per-inch curve, going from 150 lines to 100 saves about two tenths of a degree at thirty per cent coverage and costs a screen pattern half again as coarse; going from 300 to 150 saves half a degree. A printer who wants the glow’s cost low on brightened stock should choose the coarsest pattern the job’s viewing distance hides, and then may lay it stochastically or on a lattice as other considerations decide.

How the screens were built #

The tile, spots, rim, film and glow are the merged census’s: 32 by 32 sites twenty micrometres apart, square spots twenty micrometres on a side, the film thinning over the last six micrometres to bare paper, read on a one-micrometre raster; the residue is the turn in Oklab a reader fully adapted to the sheet still sees, less the turn on a sheet dyed to its colour. A growing screen of N clumps takes the centres of the first N two-by-two blocks of the squares’ dispersed block order, gives every site to the centre nearest it on the torus, and orders each cell’s sites by distance from its centre. A tint at coverage a switches on a × 1,024 spots, the next site of each clump in turn. With N = 13 the five-per-cent tint is exactly the squares’ tint. The conventional screens are the dot-rim census’s round dots at 150, 200 and 300 lines.

What the census does not reach #

Cells of unequal size. Thirteen dispersed centres on a torus make cells of 46 to 100 sites; the smallest clumps fill their cells first, and a real hybrid screen balances its cells, which would make it still more like a conventional screen.

The pattern. A growing-clump screen’s irregular lattice is chosen to avoid moiré and rosettes; the census measures only the glow. The equivalence it finds is an equivalence in rim, and so in glow, and in nothing else: two screens on the same point of the dots-per-inch curve can look quite different, one a regular rosette and the other a fine irregular stipple, and which a reader prefers is a question about texture that the glow does not answer.

Dot gain and one rim. Every spot prints at its nominal size; on press, clumps gain less ink than separate spots, which would move every comparison towards the growing screen. Optical gain — the light printing does not change the sign followed from the paper between the dots to under them — also scales with the dots’ perimeter, so it would order the screens as the rim does.

Still open: whether a coarser clump count beats the 150-line screen in the highlights #

The growing screen is a conventional screen to the glow, so the way to lower the glow is fewer clumps. A clumped screen can start coarser in the highlights than a conventional screen would dare, where its pattern is faint, and grow finer as it darkens, splitting clumps as the tint rises.

The calculation is a growing screen with six clumps on the tile in the pale tints, doubling to thirteen at twenty per cent and to twenty-six at forty, its residue along the ramp beside the 100-, 150- and 200-line screens. The prediction is that it beats the 150-line screen by more than half a degree below twenty per cent, sitting with the 100-line screen there, and matches the 150- and 200-line screens where its count matches theirs; so that a screen’s clump count can be scheduled along the ramp, coarse where the glow costs most and the pattern least. If the splits cost more rim than the coarse highlights save, a screen should hold its clump count, and the choice of count is the whole of what screening can do for the glow. The rim was never most of the glow makes that calculation.

A count can stand in for a shape #

The habit here is about recognising when two designs are the same design.

The growing-clump screen looks nothing like a conventional screen: its dots sit on an irregular lattice chosen by a dispersed order, and its cells differ in size by a factor of two. The prediction, reasoning from the look, expected the irregularity to cost something in the mid-tones. Measured, the screen matched the conventional screen with as many dots almost exactly, and matched the 200- and 300-line screens when its count matched theirs.

The failure mode is to compare designs by their appearance rather than by the quantity the effect depends on. The test is to name that quantity — here the rim, which is a perimeter, which is set by how many dots share the ink — and match it across the designs before predicting a difference. Matched by count, the two screens were one.