Seven is not a power of two

A receding line runs into the picture and has to be divided into equal parts. Seven bays of a colonnade, five sleepers, three windows in a wall.

The construction every book gives divides it into two. Draw the rectangle whose far edge is the far end, cross its diagonals, and the crossing is the image of its centre. The diagonals find the middle establishes that this is exact — not approximately right, exact, because “the diagonals of a rectangle cross at its centre” is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.

Do it again on each half and there are quarters. Again, eighths.

There is no number of times it can be done that produces a third.

Why halving stops where it stops #

The construction that finds a midpoint on a receding line is the harmonic conjugate: given the two ends and the vanishing point of the line, the fourth point that makes the cross-ratio −1 is the image of the midpoint. It is built with a complete quadrangle — pick a point off the line, draw two rays, cross them, read where the remaining diagonal cuts — and it uses nothing but lines and their crossings.

Give the line an affine coordinate in which the near end is 0, the far end is 1 and the vanishing point is the point at infinity. The midpoint construction sends any pair (a, b) to (a + b) / 2.

Start from 0 and 1. Everything reachable by repeated midpoints is a fraction whose denominator is a power of two — a dyadic fraction. That is not a limitation of the drawing; it is arithmetic. One third has denominator three, three does not divide any power of two, and so a third is never reached.

What halving can do is get close. Bisecting toward a third and keeping the half the target is in, the marks close on it geometrically, and after seven halvings the nearest is about a third of a pixel away on the picture drawn here. After twelve it is about a hundredth of a pixel.

That last part was not expected and is the better half of the finding. Each halving builds a quadrangle on the previous halving’s output, so the constructions compound. In double precision the dyadic structure — the fact that the parameter reached after k halvings is a whole number over 2^k — is intact for a dozen halvings and unreadable by the seventeenth. The approach to a third stalls two orders of magnitude short of where the construction below lands, and it stalls because of the arithmetic of the halving rather than because of the target.

A pencil compounds error very much faster than a double does. Fifteen complete quadrangles is fifteen auxiliary points, sixty lines and sixty crossings, each of them drawn by hand on the previous one’s answer.

The construction that reaches all of them #

Take the near end A, the far end B, and pick any third point C on the ground that is not on the line AB. Then:

1. Step out from A toward C — mark A + k(C − A) for k up to n, using nothing but the parallelogram or the harmonic step. This is a ladder of n equal intervals along a direction of the draughtsman’s choosing.
2. Join the last rung to B.
3. Draw the parallels. Through each rung of the ladder, run the line to the vanishing point of that join. Where each meets AB is the k/n point.

Measured against the camera, sevenths land on the projected divisions to about two parts in ten trillion of a pixel. So do thirds, fifths, ninths and elevenths. The construction has no preference among denominators; it costs the same for eight as for eleven.

That is the same “exact” as the transfer in carrying a height across the room and the repetition in the bay repeated by a straightedge, and it is exact for the same reason: every step is a join or a meet.

What is doing the work #

The ladder in step 1 is a repeated translation along the ground. In the picture, a translation is not a translation — it is a projective map — and the constructed rungs crowd toward their own vanishing point exactly as the bays of a colonnade do.

What makes it constructible is the horizon. Given the horizon, the vanishing point of a ground direction is available by joining two points of that direction and cutting the horizon, and the whole apparatus of parallelism comes back. That is the content of the affine rung in what one picture determines: a picture on its own is projective and has no midpoints; the vanishing line of a plane upgrades it to affine, and midpoints, parallelograms and equal steps come back with it.

Once equal steps along any one ground direction are available, equal steps along every direction are, and the cutting parallels in step 3 turn n equal steps on one line into n equal steps on another. Which is the schoolbook construction for dividing a segment into n parts, transported into a picture without change.

The field a straightedge generates #

The sharp statement is worth making, because it says exactly what is and is not available.

Take the near end, the far end and the vanishing point as 0, 1 and infinity. Everything the straightedge can construct on that line, using those three and any auxiliary points off it, is a rational fraction. Joins and meets are rational operations on coordinates, so the set they generate is closed under addition, subtraction, multiplication and division and contains 0 and 1 — it is a field, and the smallest such field is the rationals.

So:

• every whole fraction is constructible, and the construction above builds it in a fixed number of lines;
• no irrational fraction is, at any length.

The second half is not a statement about how hard the construction is. There is no construction. A routine that accepted a request for the point 0.7071 of the way along, and returned something, would be returning a nearby rational and calling it the answer.

The machinery here therefore refuses. Asked for p/q it wants two whole numbers; handed √2⁄2 it declines, rather than approximating and reporting success.

That refusal earns its keep two essays later. The circle in the square wants a number the straightedge cannot make is the case where the wanted fraction is exactly √2⁄2, and where the taught rule quietly substitutes a ratio measured on the paper for one that lives in the room.

The auxiliary direction, and where to put it #

Step 1 says “pick any third point C not on the line”. Any is true, and it is not good advice.

The rungs of the ladder crowd toward the vanishing point of the direction A→C, and how fast they crowd depends on how steeply that direction recedes. A direction running straight into the picture crowds hard: the fifth rung and the sixth are close together on the paper, and the construction that produces the seventh is a crossing of two lines that are nearly the same line.

A direction lying nearly across the picture — nearly parallel to the picture plane — barely crowds at all. Its vanishing point is far away, the rungs are almost evenly spaced on the paper, and every crossing is clean.

This is why a draughtsman lays the divisions off along the ground line, the horizontal at the bottom of the picture in the plane of the paper, where equal steps in the room really are equal steps on the page. It is the same instinct as the measuring point in the measuring point, which transfers a scale from that line into the picture.

But it is worth being clear about what kind of decision it is. The construction is exact for every auxiliary direction. What varies is whether it can be drawn — the same distinction as the vanishing point that runs off the paper in the height transfer, met again in a different place. Exactness is a fact about projective geometry; drawability is a fact about paper and about how squarely two lines cross.

Two routes, and the reason for keeping both #

There are two ways to build the ladder in step 1 and the machinery here does both.

The parallelogram. To add the vector B − A to a point P, take an auxiliary point off the line, draw four lines and take two crossings. It is the schoolbook construction of a translation and it is exactly what a translation is.

The harmonic step. The next rung is the harmonic conjugate of the last but one with respect to the last and the vanishing point. In the affine coordinate that conjugate is 2b − a, which is the next step; it is the same complete quadrangle that finds a midpoint, run outwards instead of inwards.

Both are joins and meets, both are exact, and they do not fail the same way. The parallelogram route holds its auxiliary point fixed, and as the ladder crowds toward its own vanishing point the two lines through that auxiliary converge on each other — so the meet that defines the step becomes a meet of two nearly identical lines, and it gives out after about six steps on a receding direction. The harmonic route holds up further.

That is a fact about conditioning rather than about correctness, and the site keeps having to make the distinction. Keeping both routes means the distinction can be measured rather than asserted: the two agree exactly where both work, and where one stops the other says how far past it the other goes.

What the ladder costs, counted #

It is worth putting a number on the comparison, because “three lines against a stack of quadrangles” is a slogan and the accounting is better than the slogan.

Dividing a depth into n parts by the ladder costs n − 1 steps to build the rungs, one join to the far end, and n − 1 cutting parallels. For sevenths that is thirteen lines and twelve crossings, and every one of them is drawn on points that were themselves drawn — but only one layer deep, because all thirteen use the original A, B and C rather than the previous answer.

Halving toward a seventh costs one complete quadrangle per halving. A quadrangle is four lines and three crossings, and each one is built on the previous quadrangle’s output. Getting inside a tenth of a pixel of a seventh takes about ten of them: forty lines, thirty crossings, and a chain of dependence ten deep.

The difference that matters is not the count. It is the depth of the chain. Errors in a drawing accumulate along a chain of constructions roughly as the square root of its length if they are independent, and rather worse than that when each construction is built on a point the previous one placed slightly wrong in a consistent direction. Thirteen lines drawn off three fixed points is a shallow tree; forty lines drawn in a chain of ten is not.

This is why the exactness claims in this collection have to be read carefully. Both constructions are exact in the sense the machinery reports, which is that a computer carrying them out in double precision lands on the projected truth. What separates them for a person is the shape of the dependence graph, and no assertion about arithmetic sees it.

Why the books stop at halves #

It is worth asking why the halving construction is universal in the literature and the ladder is not, because the answer is not that the ladder is harder.

Halving is self-contained. It needs the rectangle already drawn and nothing else — no auxiliary direction, no choice, no extra vanishing point. It is one crossing, and the result is visibly in the middle, so it can be checked by eye. Everything about it is reassuring.

The ladder needs a decision — where to put C — and its result cannot be checked by eye at all, because a seventh of a receding depth does not look like anything. The construction has to be trusted, and the only grounds for trusting it are the argument about incidence.

That is the same asymmetry that runs through the whole of this collection’s account of constructions that are taught and never measured. A method that produces a result a reader can sanity-check gets used; a method that produces a result nobody can sanity-check gets used only by people who believe the theorem. And the methods that produce checkable-looking results are, more often than not, the ones that are wrong — as dividing depth by eye measures, where the best of three taught spacings misplaces a post by three and a half metres and looks perfectly convincing.

The practical summary #

For a draughtsman, three sentences.

Any whole fraction of a receding line is constructible with a straightedge alone, given the horizon, in n + 2 lines. Repeated halving reaches only the fractions with a power of two underneath, and reaching any other one that way is an approximation whose error compounds faster than a pencil can be trusted. And no irrational fraction is constructible at all, which matters exactly once and is the subject of the next essay.