What a straightedge reaches on a receding line

Seven is not a power of two settled one route and left the map unfinished. Repeated bisection of a receding depth reaches the fractions with a power of two underneath and nothing else, so a third is out of its reach at any depth; a ladder along an auxiliary direction reaches every whole fraction in a fixed number of lines. Two methods, two reaches, and no statement about what the instrument can do.

This is that statement. A straightedge on a picture performs exactly two operations — join two marks, cross two lines — and the question is which marks on a given line the two of them generate. The answer is a set, it does not depend on which recipe is followed, and it is decided before any particular construction is chosen.

Three marks fix a projective coordinate, and everything the straightedge reaches from them is a ratio of whole numbers. Every rational is reached, at a cost that can be counted; no irrational is reached, at any depth, by any route. The measurements below are the density of the reachable set, the cost of the fractions a draughtsman actually asks for, and — the control that decides what kind of fact this is — the same construction carried out on a line that does not recede at all.

Rank is a construction count #

The word doing the work is rank, and it is worth defining carefully because it is the essay’s unit of cost and it is not a depth of recursion.

Start with three marks. Rank nought is those three. Rank one is rank nought together with the harmonic conjugate of every ordered triple that can be drawn from it. Rank two adds the conjugate of every triple among those, and so on. A mark’s rank is therefore the smallest number of complete quadrangles any construction of it needs — the length of its shortest proof, not of the first proof somebody found.

That is the definition that separates this essay from the one below it. Repeated bisection is a walk: it keeps one base pair fixed and moves the other end, so the marks it reaches are the ones reachable by that walk. The net keeps nothing fixed. Any two marks already made may serve as the base pair for the next conjugate, and the set of triples available grows as fast as the set of marks.

The three marks a receding line supplies are the two ends of the thing being divided and the vanishing point. That last is the one that makes the whole apparatus available on a picture at all, and it is available because parallel lines meet at a drawn point rather than at a metaphor. Nothing metric is used anywhere below; the horizon supplies the third mark and the straightedge does the rest.

How fast the net fills the line #

The counts are worth having exactly rather than as a rate, because the growth is the reason the enumeration stops where it does.

Two to three to eleven to sixteen hundred and fifty-five. The largest gap is the useful reading rather than the count, because it is what a draughtsman wanting a particular fraction cares about: at rank 2 the worst place on the segment is a fifth of its length from the nearest mark, and at rank 3 it is under four parts in a thousand.

What the figure cannot show is rank 4, and the reason belongs in the essay rather than in a footnote. Rank 4 is the harmonic conjugate of every triple among 4,962 points, which is six times ten to the tenth of them. That is not a limit of patience or of the drawing; the closure is defined for every rank and only the first four can be enumerated by taking every triple. A cleverer enumeration would reach further and is not needed for anything claimed here, because the claim is about the union over all ranks and the union is settled by algebra rather than by counting.

Why the picture crowds and the coordinate does not #

Under the net in the first figure there is a second rule carrying the same marks against a uniform scale, and the two together are worth a section, because the difference between them is the reason a reader’s eye is no guide at all to what has been reached.

On the receding line the marks pile up toward the vanishing point. Half of them sit in the last tenth of the drawn length, and the far end of the segment is a dense grey where the near end is comfortably spaced. On the uniform strip the same marks are spread over the interval in the pattern the density figure counts. Neither picture is a distortion of the other: the receding line is where the marks are on the paper, the strip is where they are in the coordinate, and the map between the two is the perspective divide.

So a draughtsman looking at a finished net cannot read its rank off the drawing. A crowded region is a region seen obliquely and not a region finely divided, and a sparse one may be either. That is the same confusion a ruler laid on the drawn side falls into when it halves an image instead of the image of a half, and it is why every reading in this essay goes through the cross-ratio rather than through a distance on the page.

It also decides where the construction is worth carrying out. The marks near the vanishing point are the ones whose quadrangles have nearly-parallel lines in them, so the arithmetic that is exact everywhere is drawable in the near half and increasingly not in the far one. That is a conditioning limit rather than a reach limit, and the two are separated by a test: a reach limit is unchanged by drawing larger, and a conditioning limit improves in proportion. Doubling the sheet buys marks near the horizon and buys not one new coordinate.

Which is the second time this cluster has had to hold the distinction apart, and both times it was the picture that suggested the wrong one. A vanishing point that has left the paper looks like an accuracy problem and is a drawability one; a net that looks solid at its far end looks like a finer division and is the same eleven marks seen edge-on.

What a third costs, and why it is not never #

The single sharpest comparison this collection can draw sits in one bar chart.

Repeated bisection never reaches a third, at any depth whatever. The full net reaches it in two quadrangles — eight lines and six crossings, which is an afternoon’s work by hand and rather less than the ladder construction costs for the same fraction.

The difference is not effort, and it is not that one method is more accurate. Bisection keeps one base pair fixed and the net may take any two marks already made. That single permission is the whole of it: with the base pair frozen the group generated is the dyadic translations, and with it free the group is everything. A recipe’s reach is decided by which of its inputs are allowed to move, and that is a question a reader can ask of any straightedge recipe in any manual without executing it.

It also explains why the cost stops climbing. An eleventh is reached at rank 3 exactly as a sixth is, because by rank 2 the segment already carries marks whose conjugates land on small denominators, and the shortest construction of a fraction is a fact about how its denominator factors through the marks available rather than about its size.

The chart, and the other chart #

Reading a mark’s coordinate back is where the machinery nearly went wrong, and the failure is instructive because both answers were correct.

A coordinate on a line is a cross-ratio of four points — the mark, and the three that make the frame. But four points admit several orderings, and two of them are perfectly good projective coordinates for the same line. Taking the frame’s two finite marks with the vanishing point and the mark being read in one order gives the coordinate this construction is in; swapping the last two gives its reciprocal, which is an equally valid chart on the same projective line and reads every mark as one over what was wanted.

Nothing about the second ordering is wrong. It is a chart, it is projective, it is invariant under every projectivity, and it would have made every claim in this essay false while passing every test that asks whether a cross-ratio survives a projection. What caught it was requiring two independent readings — this construction’s own chart and the coordinate reader the collection already had — to agree, which they do to a few parts in $10^{15}$ over six samples, and which the reciprocal ordering fails immediately and loudly.

The operation being read is not new here, and the figure below is borrowed from the diagonals find the middle, where the collection first measured it.

That is the lesson worth carrying past the bug. A quantity that is invariant is not thereby the right invariant, and a test that only asks whether something survives a projection cannot tell two charts apart. The eight-point algorithm’s basis is the same shape of trap met in a different field, and both are caught the same way — by a second implementation that had no opportunity to inherit the first one’s convention.

The number the net cannot reach #

Everything above is a positive statement, and the negative one needs a different kind of evidence, because the honest version is not that the approach is slow.

The argument is not about either curve. A join of two marks is a rational operation on their coordinates and so is a crossing of two lines, so the set of coordinates reachable from 0, 1 and infinity is closed under addition, subtraction, multiplication and division and contains 0 and 1. That is a field, and the smallest field with those two elements is the rationals. Every mark a straightedge makes has a rational coordinate, and $\sqrt{2} - 1$ does not.

So the gap at every rank is a specific positive number and not a residual against a threshold. At rank 3 the nearest mark of the net is 7.22e-5 away in coordinate; at rank 4 there is a nearer one and it is still at a positive distance, because the nearest rational to an irrational is never the irrational. No further rank closes it, and the reason is not that the ranks run out but that the whole union over all ranks is a set that does not contain the point. A construction reported as landing within some tolerance of this target would be reporting the tolerance.

That is the case where the distinction earns its keep rather than being pedantry. The circle-in-the-square rule wants exactly this coordinate along a diagonal, and the taught version quietly substitutes a ratio measured on the paper — a measurement, imported, and no longer a straightedge construction — for one the instrument cannot make.

The control, on a line that does not recede #

Every measurement above was made on a receding line with a vanishing point in it, and there is an obvious reading of the result that the essay has to refuse: that the reachable set is somehow a fact about perspective, or about foreshortening, or about the way a picture crowds its marks toward the horizon.

Two lines, two entirely different pictures, one set of coordinates. The receding line’s marks pile up toward the horizon and the fronto-parallel line’s are evenly spread; the half-way mark of one is nearly a canvas width from the half-way mark of the other; and the coordinates come back the same to the arithmetic floor.

Which settles what kind of fact this is. The reach is a property of the projective line, and the picture is where it happens to be observed. A fronto-parallel line is the affine case, its third mark is a real point at infinity rather than a drawn stand-in, and the same complete quadrangles produce the same rationals on it. Perspective is not the source of the result; it is the setting in which the result is surprising, because a reader who has watched the marks crowd expects the crowding to be in the answer.

The refusal is the part that makes this a control rather than a demonstration. Asked for a vanishing point of a direction parallel to the picture plane, the camera declines — there is no such point, the third homogeneous component of the mark is exactly zero, and the machinery holds that as an ordinary value rather than as a large number standing in for one. A construction that had quietly returned a distant point instead would have produced a picture indistinguishable from this one and a claim that was not being tested.

The same object, read along a different line #

One neighbouring result is worth naming, because it is this essay’s machinery pointed at a line nobody thinks of as divisible.

The stair that turns has a vanishing point that moves measures the twelve tread-edge vanishing points of a spiral stair and finds them to be a projectivity of the horizon, with four consecutive reading a constant cross-ratio and the fitted map coming out elliptic — no real fixed point at all.

Put beside the account above, that is the same object under a different classification. A straightedge executes a perspectivity of one line onto another, and iterating a perspectivity gives a projectivity; which projectivity it is decides what the iteration reaches. The diagonal on a receding line is parabolic and its one fixed point is the vanishing point, so it generates translations and reaches a lattice. A map with no real fixed point generates no lattice at all, and its orbit walks the whole line without ever settling anywhere — which is why the stair’s vanishing points do not converge on anything and why nothing about them can be reached by counting bays.

So the reachable set is decided by two facts together: that the instrument executes a perspectivity, and which projectivity the chosen iteration turns out to be. Neither alone says anything. The three kinds of map a line admits is the classification, and this essay is what the classification is for.

What this does not settle #

Three limits, and the first is the one that separates every number above from a drawing board.

Rank counts complete quadrangles and nothing else. It does not count how well two lines cross, and a construction of rank 3 whose third quadrangle is built on two nearly-parallel lines is worse on paper than one of rank 4 whose crossings are square. The bar chart is a lower bound on labour and says nothing about which construction a draughtsman should choose, which is a conditioning question and is decided by where the auxiliary points are put.

Second, the density figure’s rank 3 is an enumeration and not a proof of density. That the net’s closure is dense in the line follows from its being the rationals; the 0.0038 is a measured gap at one finite rank, and the two facts support each other without either being the other.

Third, none of this reaches beyond incidence. The net needs three marks and a straightedge; it does not need the eye, a focal length, or a scale, and it accordingly gives back no length. What one picture of a plane determines sets out the ladder this sits on: incidence buys the cross-ratio, the vanishing line buys midpoints and equal steps, and a circle buys the metric. Everything in this essay is paid for on the first rung, which is why it holds on a photograph of a scene nobody surveyed.

What the reach is #

The statement, once, in the form worth carrying.

Given three marks on a line and an instrument that only joins and crosses, the reachable set is exactly the points whose coordinate in the frame those three marks define is a ratio of whole numbers. Every such point is reached at a finite rank, the small denominators are cheap, and the cost does not climb with the denominator. No other point is reached at any rank.

That is a complete answer to a question the manuals put in pieces. A manual gives one recipe for halving, another for dividing into $n$, another for repeating a bay, and treats their reaches as three practical facts about three procedures. They are one fact about one instrument, and the procedures differ only in which of the marks already on the paper they are willing to reuse — which is what the bays that are not equal measures on a colonnade, where freezing one pair costs an arcade that free pairs would have drawn.

And the boundary is sharp in a way almost nothing else in this collection is. Most of its findings are quantities: how far off, how much it costs, at what conditioning. This one is a partition of the line into points that are in and points that are out, with no border region, no tolerance and no improvement available. A draughtsman asked for a rational fraction of a receding depth can always do it and can be told the price in advance. A draughtsman asked for an irrational one is being asked to import a number, and the only honest constructions are the ones that say so.