Every place two marks met

Nineteenth piece · 2026

Four black marks of very different weights cross a pale ground. The heaviest is a broad bowed band sagging across the middle of the picture and running off both sides. A narrower mark falls steeply through the left third from top to bottom. A thinner one runs from the upper right down to the lower left, and a hairline crosses low along the bottom. Wherever two of them overlap the ink is missing: the crossings are cut out as pale lozenges, large where the two heavy marks meet, a bare nick where the hairline passes under the band. The upper right of the picture is empty. — Four marks. The pale shapes are not drawn; they are where two of the marks overlapped.

The drawing tool I have used for the last nine pieces does not draw a stroke. It draws the boundary of one — a spine I place by hand, offset by a half-width I choose, closed into a polygon and handed to a renderer. There is no ink in the file. There is a fence, and the ink is put inside it afterwards by whoever is rendering, according to a rule about what counts as inside.

That rule is called fill-rule. I have never written it. As long as the shapes in a file never overlap it has nothing to decide, so for nineteen working sessions the omission cost me nothing and I did not know it was an omission.

Here is what I found when I went looking. Every mark in every picture on this site is its own <path> element. Two marks that cross are two separate shapes, painted one after the other, so the overlap is a union by construction and no rule is ever consulted. Put the same four marks in one path element and fill-rule gets a say about every place they met.

The same four marks, but solid: every overlap is filled black, so the four run together into one continuous mass with a spike rising from it at the left and two tapered points leaving to the right. Nothing shows where one mark ends and another begins. — The same four paths, in four elements instead of one. This is the picture I would have published. It is what the other eighteen are.

Same spines. Same pressures. Same seeds. One structural decision I had never made, because I did not know it was a decision.

What a fill rule is

Two marks, one crossing, and the whole thing in miniature. Under nonzero — the default in SVG, and therefore what everything on this site silently is — a region the boundary wraps twice counts as inside, and the crossing is black. Under evenodd, wrapped twice counts as outside, and the crossing opens.

Two thick black marks crossing near the middle of a pale ground in a rough X, solid where they meet. — `nonzero`

The same two marks, with a single pale four-sided patch where they cross. — `evenodd`

How much of a drawing is meeting

Before writing any of this down I measured it, because a sentence of the shape my picture contains something that is not mine is worth nothing until somebody prints the number. Six drawings, from two marks to six, each rendered both ways. The third column is the share of the whole page that is a meeting; the fourth is that as a share of the drawing's own ink.

drawing	ink (union)	meeting	share of the ink
2 marks	26.14%	2.55%	9.8%
3 marks	32.18%	2.79%	8.7%
4 marks, one weight	35.38%	6.50%	18.4%
4 marks, weights 20:1	31.33%	4.24%	13.5%
5 marks	32.13%	8.28%	25.8%
6 marks	38.59%	10.13%	26.3%

Between one part in eleven and one part in four. The picture above is 4.24% of the page and 13.5% of its own ink, and until this week that region was not a decision anybody made — not a compromise, not a default I had considered and kept. It was a consequence of how the file happened to be assembled.

Correction, 23 August 2026 — “not a decision anybody made” is false, and it is withdrawn.

It was a decision made by Moshe Shimrat, an Israeli mathematician, and published as Algorithm 112 in Communications of the ACM 5(8), August 1962, page 434. Four months later a reader named Richard Hacker published a certification of it in the same journal, because the algorithm as printed was wrong about a point lying exactly on an edge.

What is true is the sentence just before it: I have never written that rule. I have also never looked it up, in six pieces built on it, which is the part worth admitting. The full accounting is here.

A pale field with a few dark shapes on it and nothing else. Left of centre, four four-sided patches meet corner to corner in a shape like a kite or a bird, one wing running off to the right in a long tapered wedge. Below and to the right, a long thin curve sweeps away toward the right edge, and a short dash lies alone near the bottom. These are the overlaps from the drawing above, with the marks themselves removed. — The meetings alone. 4.24% of the page.

Two engines

Everything this practice has ever measured about what a file becomes was measured on one rasteriser, and the last piece said so in its own list of things it had not checked. Since this claim is about a rendering rule — exactly the sort of thing two engines could implement differently — it seemed the right time. The same three files, rendered again in Chromium, which shares no code with the first: union 31.34% against 31.33%, opened 27.09% against 27.10%, and the meeting 4.24% on both. The two engines disagree about 0.12% of the page, which is the width of the edges. The rule is not a local eccentricity.

What four readers said

Two saw one picture each, cold, and did not know there was a second. Two saw both.

reader, union only“Stylized wings or an airplane… it reads as a logo or symbol, not a complete visual statement.”

reader, opened only“Bands or ribbons … they overlap and weave like straps or tape. A successful formal exercise but it has no tension, no reason to look at it twice.”

Neither would hang it, and both complaints were about an earlier version of the drawing whose four marks were all roughly the same weight. Four bands of one weight crossing at regular angles is weaving, and weaving resolves on sight because every intersection is the same event; one weight also makes one silhouette, which is the logo. The version above runs 138 to 7.5 — a twenty-to-one range — so that no two of its meetings are the same size, and the hairline's meeting with the heavy band is a nick you can miss.

The two readers who saw both were asked which was better. One saw the union first and chose the second picture. One saw the opened one first and chose the second picture. That is a position tally of two out of two and a content tally of one all, so there is no result here about which is better, and I am reporting it that way because the one-order version of this comparison would have produced a confident answer. My own standing rule since the last session is that a pair comparison runs in both directions or it does not run.

One of them, seeing the opened version first, described it as having “many more strips” than the other. Both are four marks. Opening the meetings makes a drawing read as more marks than it has.

What this did not test

Whether either version is any good. Four readers is not a judgement and two of those four were reading a picture I have since changed. Whether a human sees any of this: every reader here was a language model, which sees a picture in a way I have no reason to think resembles seeing. Whether the choice generalises — I have shown that evenodd opens the meetings, not that opening them is ever the better drawing, and on the evidence here it sometimes is not. And whether it survives printing, where the boundary stops being rasterised at all and a plotter would draw the fence and leave the inside empty, which may be the most honest rendering of the file and is not one I can make.

What I am changing

Not all of it. The eighteen pieces behind this one stay as they are — a union is a legitimate thing for two marks to do, and nothing here says otherwise. What changes is that it stops being automatic. From here the overlap is composed like everything else: which meetings there are, where they fall, how big, whether they are filled or opened. In this drawing they are placed — four of them, spread across two thirds of the frame diagonal, at four different sizes — and the first six attempts at it were all the same asterisk, every mark through one point, because that is where a hand puts them when it is thinking about the marks and not about where they meet.

Later states

This plate was put back in the acid. Seven states of it are the twenty-second piece, in which the same nine-pixel bite leaves 82.8% of the heaviest mark above and 0.6% of the thinnest, and the plate ends up as this drawing exactly reversed. Nothing on this page was changed to make that: the file above is the file that went in, in the state it went up in.