The bays that are not equal

The bay repeated by a straightedge established that the diagonal method does not drift: draw one bay, repeat it twelve times with a straightedge and nothing else, and the last constructed corner is a picosecond of a pixel from the corner the camera projects. The construction is a homology of the picture, an exact map iterated is exact, and there is nothing to accumulate.

That is a statement about error, and it left a different question untouched. A colonnade whose bays are not all the same — a nave of two wide bays and a narrow one, an arcade of 1 : 1.5 : 2, a facade whose openings are set out on a golden section — is a perfectly ordinary thing to want to draw. The diagonal is exact. Is it enough?

It is not, and the reason is not accuracy but arithmetic. What the diagonal reaches from a single given bay is a set, the set is the whole multiples of that bay, and a boundary that is not a whole multiple is either brought into reach by halving the bay first or it is not in reach at any depth whatever. The figures below measure which is which, and the honest limit turns out to be reachability rather than residual — a distinction the collection has not had to make before, because until now every construction it measured either landed on its target or missed it by an amount.

What the diagonal actually computes #

Naming the operation precisely is what makes the reach visible, and the naming costs one sentence.

Given three points on the receding line — two consecutive bay edges and the vanishing point — the diagonal construction returns the harmonic conjugate of the earlier edge with respect to the later one and the vanishing point. In the affine coordinate in which the vanishing point is the point at infinity and the bay is the unit, the harmonic conjugate of $a$ with respect to $b$ and infinity is $2b - a$. So the operation is reflect the last but one in the last, and applied to 0 and 1 it produces 2, then 3, then 4.

The vanishing point is doing the work, and it is doing it as a fixed point. The map a row of posts is classifies the projectivity that advances a bay: it is parabolic, its one fixed point counted twice is the vanishing point, and the drawn posts crowd toward it and never arrive. Iterating the diagonal is iterating that map, and a group generated by one parabolic element acting on a coordinate in which its fixed point is at infinity is the group of integer translations.

Which is the whole answer. The reachable set is $\{0, 1, 2, 3, \ldots\}$ in units of whatever bay was given, and the arithmetic floor of 1e-14 of a bay in the figure above is not a tolerance around those integers. It is the distance between the number the drawing produced and the integer it is, and the integers have nothing between them.

The map being iterated is a homology of the picture with the horizon as its axis, which is the object the census of this site’s plane maps found under a shadow, a floor anamorph and a mirror as well. Naming it that way is what licenses the iteration: a homology composed with itself is a homology, so twelve applications are as exact as one. Naming it as a translation of the world would license nothing, because the picture is not the world and the drawn gaps are not equal.

And the midpoint the construction needs on the way is the same object one rung down. The diagonals find the middle establishes that crossing a rectangle’s diagonals gives the image of its centre exactly, because “which lines meet where” is the one kind of statement a projection cannot damage. Every mark in every figure of this essay is built out of that one operation and nothing else.

The control the diagonal passes is worth borrowing from where it was measured rather than redrawing, and it is the hero of the bay repeated by a straightedge one rung below this one.

A reach is a stronger claim than an accuracy #

The two figures above make claims of different kinds and it is worth separating them, because the second kind is rare in this collection and is the one that cannot be improved by care.

An accuracy claim says a construction lands within some distance of its target, and it invites the questions a residual invites — at what camera, over how many iterations, with what conditioning. Every one of those has an answer and most of the answers are reassuring.

A reach claim says a target is not in the set the construction produces. There is no distance in it and no tolerance to tighten. A draughtsman with a perfect hand, an infinitely large sheet and unlimited patience reaches exactly the same set as one with a blunt pencil, and the only thing patience buys is more of the same integers.

So the question an unequal arcade puts to the diagonal is not how close but whether, and the answer depends on one number per boundary.

Halving buys a denominator, and which denominator decides everything #

The bay’s own centre is free. Crossing the diagonals of the first bay gives it, and it is the same complete quadrangle run inwards rather than outwards. Once the centre is in hand the unit is a half rather than a whole bay, and the reachable set is the multiples of a half.

Repeat and the unit is a quarter, then an eighth. After $k$ halvings the reach is

$$\left\{ \frac{m}{2^{k}} : m = 0, 1, 2, \ldots \right\}$$

in bays — the dyadic rationals with denominator up to $2^k$, which is exactly the set seven is not a power of two identified for repeated bisection and for the same reason. This essay is that result read as a reach on a colonnade rather than as a division of a single depth.

So the arcade of 1 : 1.5 : 2 is settled by arithmetic before anything is drawn. Its cumulative boundaries are 1, 2.5 and 4.5 bays; the halves have denominator two; one halving suffices and the run is then exact, at a cost of a few more quadrangles and no measurement of any kind. The figure above reaches them at zero halvings by 250 millimetres and at four exactly, and the interesting placement is the one between, where one halving does it.

That is a real capability and it should be said plainly, because the essay’s title invites the opposite conclusion. A great many unequal arcades are inside the diagonal’s reach. Any pattern whose cumulative depths are halves, quarters or eighths of a common unit is reachable, and that includes most of the rhythms a facade is actually set out on. What it excludes is a pattern with a three, or a five, or a seven underneath it.

The measuring point, which does not care #

The construction that has no such limit is the one that imports a measurement, and the contrast is exact rather than approximate.

Three lines and three ruler marks, for a pattern the diagonal cannot state without first being told to halve. And the reason is not that the measuring point is cleverer. It is that the two constructions are answering different questions: the diagonal is asked to generate a spacing from a spacing already drawn, and the measuring point is asked to transfer a spacing that arrives from outside the picture, on a ruler, along a line where a ruler is legitimate.

The measuring point sets out how that transfer works and where the point lives; a measuring point for a ramp carries the same construction onto a plane that is not the ground, by taking the vanishing point of the difference of two directions. What matters here is the accounting. The diagonal needs no focal length, no eye and no scale, and its reach is a group. The measuring point needs the eye’s distance, and its reach is everything.

That is the same division of labour the whole construction field runs on, and this is the sharpest instance of it: metric information enters once, and the amount of arcade it buys is the difference between a lattice and a line.

The control, which is the equal arcade #

A measurement that only ever confirms its thesis has not been tested, so the same construction is run on the case the diagonal can do.

The residual moves from 6e-14 px to 1e-13 px between the two arcades, which is arithmetic noise changing its mind about the last bit and not a signal. That is the point of running it: had the equal arcade come back dramatically better, the measuring point’s indifference to the ratio would be a claim rather than a measurement, and the essay’s whole comparison would be resting on an untested assertion.

The other half of the control is the one the collection already holds. Twelve equal bays by the diagonal, exact, in the figure borrowed above. Two constructions, one target, both exact — and then one target more, where only one of them arrives.

Two and a third is not reached at any depth #

The pattern the halvings never rescue is worth taking slowly, because never is the kind of word a tolerance can quietly convert into not yet.

An arcade of 1 : 1.333 : 2 has its middle boundary at 7/3 bays. After $k$ halvings the reachable marks are $m/2^{k}$, so a mark lands on 7/3 exactly when $3m = 7 \cdot 2^{k}$ for some whole $m$ — which needs 3 to divide $2^{k}$, and three is odd. There is no $k$. The nearest reachable mark is always at least $1/(3 \cdot 2^{k})$ of a bay away, which shrinks and is never zero, and the shrinking is what makes the claim look like an approach when it is a partition.

Two curves, and the gap between them is the essay’s honest limit. The upper one is a statement about a set and it goes on for ever. The lower one is a statement about a drawing carried out in double precision, and it stops — not because the target moved but because each quadrangle is built on the last one’s output and the bracket being bisected is by then a ten-thousandth of a pixel.

And the rule drawn across the plot is the number that matters to anybody holding a pencil. The bracket falls under a pencil line’s width after four halvings. So on paper the reach is the sixteenths of a bay, and the difference between four halvings and twenty-three is the difference between a claim about a drawing and a claim about a method — the same separation carrying a height across the room draws between exactness and drawability, arriving here from the arithmetic side rather than the geometric one.

The golden bay, which is out of reach for a second reason #

The arcade in the hero above is set out on a golden section — bays of 1 : 1.618 : 2, with its middle boundary at $1 + \varphi$ bays. The measuring point lays it out to 6e-14 px like everything else, because a ruler along the ground line does not know what kind of number it is being asked for.

For the diagonal it is out of reach, and the reason is one step stronger than the reason 7/3 is. Two and a third is a rational number that no power of two divides, so it is missed by an amount that halves with each quadrangle and never reaches zero. The golden ratio is not rational at all, so it is not merely outside the dyadics — it is outside the whole set a straightedge generates from three marks, at any depth, by any route, including the full harmonic net that reaches every ordinary fraction. The gap does not merely fail to close; there is no sequence of joins and meets whose limit it is the value of, because every join and every meet is a rational operation on the coordinates it is given.

That distinction matters to a draughtsman rather than only to an argument. A rational target missed by one part in a thousand is a target a shorter run and a sharper pencil bring within a line’s width; the diagonal at four halvings puts the boundary of a 1 : 1.5 : 2 arcade exactly where it belongs, and puts a seven-thirds boundary a sixteenth of a bay out and closing. An irrational target is a different species of impossible, and the honest response is the one the circle in the square records: the taught rule substitutes a ratio measured on the paper for one that lives in the room, and calls the substitution a construction.

Which is why the machinery refuses rather than approximating. Asked for a reach plot at a target the halvings do land on, it declines — a curve closing on something it actually reaches is a picture of nothing, and a routine that returned one would be reporting success on the one input where its own claim is empty.

The overflow that reported itself as a degeneracy #

One thing the machinery found while the reach plot was being built is worth recording, because it is the failure mode this collection’s habit of iterating is most exposed to.

A point of the picture is held as three homogeneous numbers, which have no natural scale — multiply all three by anything and the point is unchanged. An iterated construction that never renormalises them squares their magnitudes at every rung, so a ladder of five came back with components of order $10^{293}$ and the sixth was infinity.

What made it instructive is the message. The meet of two lines through infinite components is not infinite; it is NaN, so the assertion that fired was the one checking that the third diagonal of the quadrangle actually cuts the line. A perfectly well-conditioned construction on a perfectly ordinary arcade reported itself as degenerate, and the first five rungs had been exact to eight figures, so nothing before the overflow gave any sign of it.

The lesson generalises past the arithmetic. An assertion that fires is evidence that something is wrong and is not evidence about what, and a geometric assertion firing on a numerical fault is the shape of bug that sends a session looking for a degeneracy that is not there. The construction now renormalises at every rung and the check that would have caught it — running the naive loop on purpose and requiring it to overflow within eight rungs — is part of the machinery rather than a note.

What this does not settle #

Three limits, and the first is about the word reach.

Everything above is a statement about which marks a construction produces, given a bay. It says nothing about whether a draughtsman can tell, looking at a finished arcade, which construction made it. Two arcades with identical marks are identical pictures, and a 1 : 1.5 : 2 run reached by one halving of the diagonal and one reached by a measuring point are the same set of lines on the paper. The distinction is in the procedure, which is exactly why nothing in the drawing reports it.

Second, the reach is stated in units of the given bay, and the given bay is not free. It arrives from outside — a measured depth, a distance point, Alberti’s section through the eye — and the whole lattice is anchored to it. Getting the first bay wrong moves every subsequent mark, exactly, and the diagonal’s exactness is what guarantees the error is carried rather than corrected.

Third, the plot’s lower curve is a fact about double precision and not about paper. The four-halving rule on it is an estimate that assumes a pencil line’s width and a fixed drawn length, and a draughtsman working larger gets more. What does not change with the size of the sheet is which targets are in the set at all, and that is the half of the finding worth carrying.

Which constructions have a reach #

The general shape is worth stating because it sorts the collection’s constructions into two kinds, and the sorting is not the one their difficulty suggests.

A construction made only of joins and meets, iterated from a fixed starting configuration, generates a set — and the set is decided by the group the iterated map belongs to. The diagonal generates the integers. Halving first generates the dyadics. The full harmonic net, which is allowed to take any two marks already made rather than keeping one base pair fixed, generates the whole rationals and reaches a third in two quadrangles rather than never. The difference between those three is not effort; it is which pairs the operation is permitted to use.

A construction that imports a measurement has no reach in this sense at all. It has a precision, a conditioning and a point somewhere off the paper, and the ratio it is asked for is simply a number it is handed. Dividing to a point off the board is one of those, and it divides a wall into any number of courses for the same three operations whatever the number is.

So the question to ask of a taught straightedge recipe is not how accurate it is. It is which pairs it is allowed to reuse, because that decides the group, and the group decides everything the recipe will ever draw.