Visual Plane
every picture states the point it is correct from
Drawn confidently

The rule that draws another room

The taught rule for spacing receding boards — each gap a fixed fraction of the last — is not a projection of anything, and it produces a pavement that is a correct perspective to within a fifth of a pixel. Of a room whose horizon is a hundred and seventy pixels from the one the panel drew. The error is not incoherence; it is a disagreement between two halves of one drawing.
18 min read 8 figures Taught, and never measuredNot a projection at allDetermined up to something

Worth reading first: Three procedures, one panel.

The rule is taught everywhere and it is one sentence: each gap a fixed fraction of the one before it. Draw the near edge of the first board, the near edge of the second, and let every subsequent gap be two-thirds — or three-quarters, or whatever looks right — of the last.

It is not a projection of an evenly boarded floor — the rule is exact for a floor that lengthens finds the floor it is an exact projection of — and parallel projection is not primitive perspective makes the general point about mistaking one drawing system for a degraded version of another. A perspective pavement’s gaps do not fall in a geometric series; they fall as the difference of successive reciprocals, which is a different function. So the wrong field’s usual expectation applies: the rule should fail a reader’s projective test outright.

It does, arithmetically. What that failure is worth in pixels is the surprise.

Calibrating the rule as favourably as possible

The comparison has to be fair, and fairness here means giving the rule its best case.

A draughtsman using it does not pick a ratio at random. They fix the near edge of the pavement and the far one — the first is on the ground line and the last is where the floor meets the wall — and choose the ratio that lands the series on both. So the rule is set here the same way: the first and last transversals go exactly where the correct construction puts them, and only the ones between are the rule’s own.

That removes every objection about a badly chosen ratio, and it means the departure measured below is the smallest the rule can produce.

The rule's pavement is a correct perspective to 0.17 px — of a room whose horizon is 171 px awayThe taught rule for spacing receding boards — each gap a fixed fraction of the last — with its ratio chosen so that the first and last transversals land exactly where the correct ones do. The drawn pavement is 9.2 pixels from the correct one at its worst, which is plainly visible; and it is itself a correct perspective, to 0.174 pixels, of a room whose horizon sits 171 pixels above the one the panel's orthogonals meet at — off the top of this figure. The rule's error is not incoherence. It is a disagreement between two parts of one drawing.the pavement implies a horizon 171 px off the topthe horizon the panel drewcorrect from 12 cm at 160 mm wide9.2 px from the truth, 0.17 px from a perspective
Fig. 1 The correct pavement dashed and the rule’s solid, with the ratio chosen so that the ends coincide exactly.

What the rule’s pavement is worth as a picture

At its worst transversal the rule’s pavement is nine and a half pixels from the correct one on this panel. That is a large error — three or four line widths, and obvious in a side-by-side comparison.

Now run the reader’s test. Fit the best correct perspective to the rule’s transversals, with the horizon free, and ask how far the worst mark is from where that perspective would have put it.

Under a fifth of a pixel.

The rule’s pavement is not merely close to a correct perspective. To within a reader’s own ability to place a mark on a reproduction, it is one.

Of what

Of a room whose horizon is a hundred and seventy pixels from the one the panel’s own orthogonals meet at — off the top of the drawing on this panel, and off the top of most panels.

That is the whole finding. The rule’s transversals are internally consistent to a degree nobody would have predicted. They are consistent with a different picture from the one the rest of the panel is drawing, and the disagreement is between the two halves rather than inside either.

The rule's pavement is a correct perspective to 0.18 px — of a room whose horizon is 172 px awayThe taught rule for spacing receding boards — each gap a fixed fraction of the last — with its ratio chosen so that the first and last transversals land exactly where the correct ones do. The drawn pavement is 9.2 pixels from the correct one at its worst, which is plainly visible; and it is itself a correct perspective, to 0.178 pixels, of a room whose horizon sits 172 pixels above the one the panel's orthogonals meet at — off the top of this figure. The rule's error is not incoherence. It is a disagreement between two parts of one drawing.the pavement implies a horizon 172 px off the topthe horizon the panel drewcorrect from 12 cm at 160 mm wide9.2 px from the truth, 0.18 px from a perspective
Fig. 2 A longer pavement by the same rule, whose implied horizon has moved closer and whose internal departure has grown.

Why the rule is nearly right

It is worth knowing why, because “it happens to be close” is not an explanation and the actual reason is short.

A correct pavement’s transversal positions, measured as reciprocals of the distance below the horizon, are exactly linear in the braccio number. The rule’s are exactly geometric in it. Over a short range a geometric series and a linear one can be matched at both ends and they part company only in the middle, by an amount that goes as the square of the range — which is small when the range is small.

So the rule is a two-term approximation to a one-term truth, calibrated at the ends, and the error is a bow. The bow’s size grows with the pavement and its shape does not: it is one hump, always the same way up, and that is why the reading calls it systematic.

The instrument that does convict it

If the pavement alone cannot, something else must, and it is the cheapest reading available: the horizon.

A panel shows where its orthogonals meet. That is a mark on the page, found with a straightedge, and it needs no arithmetic. The transversals separately imply a horizon, because their ratios only fall on a line for one value of it. On a correctly constructed panel the two are the same point; on a panel spaced by the rule they are a hundred and seventy pixels apart.

The threshold between “a slipped hand moved the implied horizon” and “the pavement was not drawn to this panel” had to be measured rather than chosen, and it is the part of this reading most easily got wrong. At a reader’s own ruler — a fifth of a pixel — the horizon test convicts nine drawings in ten, including the ones drawn correctly by a slipping hand, because a hand slip moves the implied horizon too. What separates them is not whether it moved but how far: a hand at a pixel’s precision moves it by one to twelve pixels on this panel, and the rule moves it by a hundred and seventy. An eighth of the panel’s rise sits between the two at every pavement length measured, and that is where the threshold is.

The pavement alone: the distance point vanishes and the two hands blur60 drawings by each procedure at a hand precision of 1.2 pixels, classified from the transversals alone. Every distance-point drawing reads as correct however badly the point was placed; the rule reads as correct too, because its pavement is internally consistent; and the two hands are told apart on about half the panels. The threshold between a rule and a hand is √6, which is arithmetic; the threshold between the two hands is the midpoint of two overlapping distributions, and is a guess.correctno shapefrom a zerosteppeda rulethe distance pointAlberti's sectionthe measuring pointa photographthe constant ratio60144157203036060read from the transversals alone60 drawings each
Fig. 3 What the pavement alone says about sixty drawings by each procedure. The rule’s row reads as correct.

The size at which the pavement gives it away

The rule’s internal inconsistency is not zero, it is small, and it grows. So there is a length at which a reader with nothing but the transversals catches it, and the length is a number.

At a fifth of a pixel it is nine braccia. Below nine the rule’s pavement is a correct perspective as far as any reading of it goes; above nine it betrays itself without help.

Nine braccia is larger than most painted pavements. That is not a rhetorical flourish — a painted tiled floor typically shows three to six rows before something interrupts it, and four marks before anything is said counts what three to six rows can carry, which is between zero and three statements. So the rule has been surviving for five centuries in exactly the conditions it survives, and the drawings on which it would be caught are the demonstration diagrams in the treatises rather than the paintings.

What a reader can conclude from a small pavement

Nothing about the rule, from the pavement. Something decisive, from the panel.

The transversals alone. Below nine braccia they are consistent with a correct perspective at a reader’s precision, so no reading of them is evidence against the rule. Reporting “the spacing is projectively consistent” about a five-tile floor is reporting the count rather than the painter.

The transversals against the orthogonals. Decisive at any size for a floor meant to be square, and the reading nobody takes — though it needs a fitted horizon rather than only a straightedge, and the straightedge test for square tiles is the diagonal. Find the orthogonals’ meet, fit the horizon the transversals imply, and compare. A hundred and seventy pixels is not a subtle discrepancy.

And the shape, if there is enough of it. The rule’s residual is smooth and a hand’s is not, which one hand step each measures with a statistic whose threshold is arithmetic. On a pavement long enough for the roughness to mean anything, the two are separated on nearly every drawing.

The horizon, out of the repetition4 uprights of one height, 6 pairs, 6 meeting points — every one of them on the horizon to 1e-13 px. The horizon is drawn afterwards, and it is not used to find them.the camera's horizoncorrect from 19 cm, at 160 mm wide46° across
Fig. 4 The horizon recovered from the picture’s own contents, which is the step that turns a pavement into a panel and the step this reading depends on.

The rule against the eye

There is a natural comparison and it is not the one the rule usually gets.

Dividing depth by eye measures what a good draughtsman produces when asked for equal receding depths with no construction at all. The answer is a residual that is smooth in the middle of the pavement and rough at its ends — a mixture, because an eye interpolates smoothly and terminates badly.

The rule is smooth everywhere. So a rule-spaced pavement is, by the reading here, more internally consistent than an eye-spaced one — and that is a fair description of what the rule is for. It is a device for making a drawing look regular, and it works.

What it does not do is make the drawing a projection, and the difference between those two only becomes visible when the drawing is asked what solid it depicts. That is the wrong field’s whole method, and this is one of the cleaner instances: a construction that is not a projection, passing every test of its own output, and failing the one test that compares it against the rest of the picture.

The room the rule actually draws

It is worth working out what room that is, because “a different room” is vague and the answer is specific.

The implied horizon sits a hundred and seventy pixels above the drawn one on this panel, which means the pavement is consistent with a camera whose eye is a good deal higher above the floor than the panel’s own orthogonals say. The implied slope of the transversals then gives an implied viewing distance, and it is longer than the panel’s.

So the rule draws a floor seen from further away and from higher up than the rest of the picture is drawn from. That is a recognisable look. A pavement spaced by the rule under a figure group drawn to the panel’s own horizon produces a picture in which the floor recedes too gently for the people standing on it — which is a complaint made about a great many quattrocento floors, usually attributed to inexperience, and which is here a consequence of a specific rule with a specific sign.

The sign is worth stating because it is testable. The rule’s implied horizon is always on the far side, and the implied viewing distance is always longer. A pavement that errs the other way was not spaced by this rule.

Why the sign is fixed

The claim that the rule’s implied horizon is always on the far side is stated as testable and it is also provable, in one line, from the shape of the departure.

Two sequences matched at both ends and differing in between depart with a single hump — the essay’s own description, and it follows from one being strictly convex against the other over the whole range, which a geometric series is against a linear one for any ratio other than one. A single-signed departure cannot be absorbed by a perspective that tilts either way: the best-fitting horizon has to move in the one direction that flattens the hump, and which direction that is depends only on the sign of the convexity.

Every taught ratio is below one — two-thirds, three-quarters, seven-tenths — so the convexity has one sign for all of them, and therefore so does the implied horizon’s displacement. The rule cannot draw a room seen from lower down than the panel’s, at any ratio a manual quotes, on a pavement of any length.

That makes the horizon test one-sided rather than two-sided, which is worth having because a one-sided test is stronger evidence. A pavement whose implied horizon sits a hundred and seventy pixels below the drawn one was not spaced by this rule and was not spaced by a hand slip either, since a hand’s displacement is small in both directions. It was spaced by something else, and the reader knows that from the sign alone before measuring the size.

Why the horizon test is not circular

An objection worth answering, because the test compares a drawing with itself.

A reader might say that the orthogonals and the transversals are both the draughtsman’s work, so a disagreement between them shows only that the draughtsman was inconsistent — which everybody knew — rather than that the spacing rule is not a projection.

The answer is that the orthogonals are not a free choice. They run from the braccia marks on the ground line to the centric point, and once the ground line is divided and the centric point is chosen, every orthogonal is determined. There is no rule to apply and no ratio to pick; a draughtsman who can lay a ruler between two points draws them correctly. So the orthogonals carry the panel’s actual geometry and the transversals carry whatever the draughtsman did about depth, and the comparison is between something determined and something chosen.

That asymmetry is what makes the test informative rather than circular, and it is also why the test says nothing about a drawing whose orthogonals are wrong. A panel whose orthogonals do not meet at a point at all is a different problem, and both vanishing points on the paper is where this collection measures what that costs.

What the rule shares with a lens

The rule and a photograph leave residuals of the same shape, and it is worth naming the shared structure because it is the reason the row needs a further essay.

Both are smooth departures from a projective law. The rule’s comes from replacing a linear function of the braccio number with a geometric one; a lens’s comes from displacing every mark radially about the principal point. Neither has anything to do with a hand, both bow the transversals the same way, and the roughness statistic reports the same value for both — 0.885 against 0.841 on this panel, which is the same number to the precision that matters.

What separates them is that a lens has to explain the whole page. Its coefficient, fitted from the pavement, predicts the bow of any straight edge anywhere in the picture; the spacing rule predicts nothing about an edge because it says nothing about edges. The lens a pavement can hide is that test, and it is a prediction made on one part of a picture and checked on another — which is the only kind of test that separates two explanations fitting the same data equally well.

The lens bows the edge by 7.1 px and the rule leaves it exactly straightTwo explanations for the same pavement. A photograph taken through a lens of k₁ = -0.5 and a pavement spaced by the constant-ratio rule leave residuals of the same shape — roughness 0.885 against 0.841 — so no reading of the transversals separates them. What separates them is a second family of marks: a straight edge drawn obliquely across the page, which the lens bows by 7.09 pixels by an amount the pavement already fixed, and which the spacing rule leaves straight to 2.5e-14 pixels because it says nothing about edges at all. The test is a prediction, made on one part of the picture and checked on another.the edge, as drawn and as the lens delivers itcorrect from 12 cm at 160 mm widesagitta 7.09 px against 2e-14
Fig. 5 The separation: the lens bows an oblique edge by an amount the pavement already fixed, and the spacing rule leaves it exactly straight.

Two things this does not say

Worth setting out, because the finding is easy to over-read in either direction.

It does not say the rule is fine. A pavement nine and a half pixels from where it should be is nine and a half pixels wrong, and every measurement taken from it — a height, a plan, a viewing distance — inherits that. What it says is that the error does not show up as incoherence, so a reader checking for incoherence will not find it.

And it does not say the rule is a good approximation to perspective. It is a good approximation to a perspective, of a room the draughtsman did not intend, seen from a distance the draughtsman did not choose. Calling that an approximation to the intended picture is exactly the confusion the slip that leaves no trace is about, arriving here from the other side: there the wrong drawing was a correct picture of another room and the reading could not see it; here the wrong drawing is a nearly correct picture of another room and only the panel can see it.

The rule's pavement is a correct perspective to 0.17 px — of a room whose horizon is 173 px awayThe taught rule for spacing receding boards — each gap a fixed fraction of the last — with its ratio chosen so that the first and last transversals land exactly where the correct ones do. The drawn pavement is 9.2 pixels from the correct one at its worst, which is plainly visible; and it is itself a correct perspective, to 0.174 pixels, of a room whose horizon sits 173 pixels above the one the panel's orthogonals meet at — off the top of this figure. The rule's error is not incoherence. It is a disagreement between two parts of one drawing.the pavement implies a horizon 173 px off the topthe horizon the panel drewcorrect from 12 cm at 160 mm wide9.2 px from the truth, 0.17 px from a perspective
Fig. 6 The rule at a length where it does give itself away, with its implied horizon back inside the drawing.

Where the ratio comes from

One last question, because the rule is always taught with a number attached and the number varies.

Two-thirds, three-quarters, and “about seven-tenths” all appear in drawing manuals, and none of them is derived from anything. What the calibrated version here shows is that the ratio is not a free choice at all once the pavement’s ends are fixed: the first and last transversals determine it, and it comes out at whatever the geometric series needs to land on both.

So a manual quoting a ratio is quoting the answer for one particular depth of room, and applying it to a deeper or shallower one produces a pavement whose ends are wrong as well as whose middle is. The version measured here — ratio chosen from the ends — is strictly the best case, and the taught version with a fixed number is worse by however much the room differs from the one the number came from.

Four transversals give 1.333333, and the answer is always 4/3The transversals of an evenly divided pavement are the images of evenly spaced collinear points, so every four consecutive ones have the cross-ratio four equally spaced points have: 4/3. Braccia 1 to 4 are marked; their cross-ratio is 1.333333333, which is 6.7e-16 from 4/3. Nothing about the panel enters — no horizon, no focal length, no assumption about the scale — so this is a test a reader can run on a photograph of a drawing.1234cross-ratio 1.333333four evenly spaced points give 1.333333correct from 12 cm at 160 mm wide6 quadruples, worst 2e-15 off
Fig. 7 The reading the rule survives, run on the quadruple nearest the reader.

The short version

The taught spacing rule, calibrated at both ends, produces a pavement nine and a half pixels from the correct one — and that pavement is itself a correct perspective, to under a fifth of a pixel, of a room whose horizon is a hundred and seventy pixels from the one the panel drew.

So the rule is not caught by the reader’s projective test until the pavement reaches nine braccia, which is longer than most painted floors. It is caught immediately by comparing the horizon the pavement implies with the one the panel’s own orthogonals meet at, which nobody does — and more cheaply still by a straightedge along a diagonal of its tiles. The horizon, and the fraction is where this collection establishes how much a horizon is worth once it is found.

The rule's pavement is a correct perspective to 0.11 px — of a room whose horizon is 180 px awayThe taught rule for spacing receding boards — each gap a fixed fraction of the last — with its ratio chosen so that the first and last transversals land exactly where the correct ones do. The drawn pavement is 6.5 pixels from the correct one at its worst, which is plainly visible; and it is itself a correct perspective, to 0.113 pixels, of a room whose horizon sits 180 pixels above the one the panel's orthogonals meet at — off the top of this figure. The rule's error is not incoherence. It is a disagreement between two parts of one drawing.the pavement implies a horizon 180 px off the topthe horizon the panel drewcorrect from 12 cm at 160 mm wide6.5 px from the truth, 0.11 px from a perspective
Fig. 8 And the size a reader is most likely to meet, where the pavement says nothing and the panel says everything.

Where this goes next

horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon11.72 mhalve the remaining gap225.13 mtaper by eye347.80 mcorrect from 26 cm, at 160 mm wide34° across Dividing depth by eye Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score. the distance pointAlberti's sectionthe measuring pointcorrect from 12 cm at 160 mm widethree routes, 0e+0 px apart Three procedures, one panel Alberti's lateral section, the distance-point construction and a pinhole camera put every transversal at the same pixel — and every reading of the finished drawing therefore returns the same number for all three. The methods are distinguishable on the desk and indistinguishable on the panel, which is the fact any attribution has to start from. 2 transversalsnoneunfalsifiable3 transversalsnoneunfalsifiable4 transversals15 transversals26 transversals38 transversals510 transversals713 transversals1017 transversals14independent statements a pavement makesthree unknowns, one equation per mark Four marks before anything is said A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.

What links here

Computed from the collection, not written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AttributionConstant ratioCross-ratioFalsifiabilityHorizonProjective invariantResidualToleranceTransversalViewing distance