Two rules for one pavement

Vignola’s treatise on perspective gives two rules for laying a tiled floor and says that they agree. The first draws the pavement in plan below the panel, puts an observer in front of the plan, and reads each tile corner off where the ray to it crosses the picture plane’s trace. The second draws nothing outside the panel: orthogonals to the centric point, one further point on the horizon, and the diagonal from that point cuts each orthogonal at a corner.

Executed from the same ground line, the same horizon and the same one number, they agree to two parts in ten trillion of a pixel. That claim is true, it is worth checking, and it is not the finding.

Each rule contains one free parameter that its own wording never names, both parameters are the same quantity, and they are parked in places that invite different answers. The first calls it the observer’s distance and puts it below the sheet, in a plan drawn at whatever scale fits. The second calls it the distance point’s place and puts it along the horizon, usually off the edge. An executor who follows both recipes and chooses what looks reasonable in each makes two correct perspectives of two different rooms, and the difference between them is measured below in centimetres of where the reader has to stand.

Where each rule hides its parameter #

The two hiding places are worth setting out side by side, because the whole essay is a consequence of them being different.

The first rule’s parameter is a distance in a plan. The plan is a separate drawing below the panel, and the observer stands in front of the picture plane’s trace in it at a distance the draughtsman chooses. Nothing about the panel says what the distance should be; the plan is not even drawn at the panel’s scale — in the hero above it is at four tenths of it, which is free, and the crossings map back to the full-size corners to three parts in ten trillion of a pixel because only the proportions of the rays matter. What is not free is the distance itself, and at the true focal length it lands 208 pixels below the bottom of the sheet.

The second rule’s parameter is a point on the horizon. There is no plan and no elevation. The distance point goes somewhere along the horizon line, and the further out it goes the shallower the pavement. At the same true value it lands 175 pixels off the right-hand edge.

So both rules have exactly one thing to decide, both decisions are the same decision, and a draughtsman looking at either recipe alone has no way to know that. The plan’s observer looks like a matter of how much room the drawing needs below the panel. The distance point looks like a matter of how far the horizon can be extended. Neither looks like a statement about a reader’s eye.

Given the same number, they are one map #

Before the disagreement is worth anything, the agreement has to be established properly rather than asserted, and it has to be established at more than one value.

They are not two constructions that happen to agree at one setting. Both reduce to the same expression for a transversal’s height — the horizon plus a rise divided by one plus the tile’s depth over the parameter — so the agreement is an identity rather than a coincidence, and it holds at every value.

That is worth stating carefully, because an agreement that is an identity is exactly the kind of measurement that can be made vacuous. Two implementations of one formula agreeing to the last bit of a double is a check on the arithmetic and not on the geometry, and this collection has shipped that mistake. What makes the number above informative is what happens when the identity’s one input is not shared, which is the next figure and is where the two rules stop being one map.

What the wordings invite #

An executor does not choose 520 twice. An executor reads each recipe, looks at the sheet, and picks the number that makes that recipe convenient.

Twenty-four pixels is not an error in either construction. Both are exact; both are pictures of a real room; both would pass any check that asks whether a pavement is a projection of a flat floor of square tiles. They are pictures of different rooms, and the quantity that separates the rooms appears nowhere in either recipe.

That is a sharper failure than the ones this collection usually records. Dividing depth by eye produces a pavement no camera could photograph; the constant-ratio rule produces one whose depths are not a perspective of anything. Here nothing is wrong with the drawing at all. What is wrong is that the procedure does not determine the picture, and every gate that reads a finished pavement reports success on both.

The disagreement, in centimetres of the reader’s room #

Twenty-four pixels is one pair of choices. The honest measurement is over the choices the two wordings actually make available.

Eleven point eight centimetres is the finding. It is not a residual, not a tolerance and not an accuracy: it is the range of rooms the pair of recipes will produce from one brief, and it is nearly a factor of four between the nearest and the furthest.

The eight rows are not extremes chosen to make the spread look bad. Every one of them is a reason a draughtsman actually gives. A plan across thirty degrees, or across sixty. A distance point at the edge of the pavement, or half a pavement beyond it, or at the edge of the sheet, or a sheet’s width out. Each is a statement about the drawing’s convenience, and each is silently a statement about where a reader must put an eye — which is the collection’s founding identity arriving where nobody looked for it.

And the direction of the mistake is consistent. Both wordings push toward a small parameter, because a small one keeps the observer near the plan and the distance point on the sheet, and a small parameter is a wide-angle picture and a short viewing distance. The convenient choice is the one that makes the depicted room deeper than the room drawn, which is exactly what standing in the wrong place prices for a reader who holds the page at arm’s length.

The parameter is on the sheet exactly when the picture is unusable #

The trade is worth drawing rather than describing, because it inverts the instinct a draughtsman brings to it.

The convenience and the correctness are in direct opposition, and the exchange rate is one for one: the distance point’s offset from the centric point is the focal length, and the focal length scaled to the displayed width is the viewing distance. Moving the point onto the paper does not approximate a shorter lens. It specifies one.

That is the same measurement both vanishing points on the paper makes for a two-point layout, where keeping both points on the sheet commits the reader to eighty millimetres and a room five times too deep. Two different taught composition rules, one underlying quantity, and in both cases the rule is stated as advice about the drawing surface.

The first rule has the identical trade in a different coordinate. Its observer sits below the sheet at the plan’s own scale, so shrinking the plan brings the observer up onto the paper — and the pavement it draws is unchanged, because the plan’s scale really is free. What is not free is how far in front of the trace the observer stands, measured in plan units, and that is what a draughtsman adjusts to fit.

The control, which is a parameter that really is free #

A rule with one unnamed choice in it is only interesting if the reader can be shown what an innocuous choice looks like, and the first rule happens to contain one — which makes it the control this essay needs rather than a separate observation.

Vignola’s plan rule takes two numbers, not one. The plan below the panel is drawn at some scale, and the observer stands some distance in front of the picture plane’s trace within it. The recipe names neither, and to an executor they look like the same kind of decision: how much room the drawing gets underneath, and where in that room the observer goes.

They are not the same kind of decision at all. The plan’s scale is genuinely free. The hero above draws it at four tenths of the panel’s, and the crossings map back to the full-size corners to three parts in ten trillion of a pixel, because a ray from the observer to a corner crosses the trace at a place decided by the ray’s proportions and not by how large the triangle is drawn. Halve the plan and halve the observer’s distance with it and the panel is byte-for-byte identical. Nothing about the finished picture knows what scale the plan was at.

The observer’s distance in plan units is not free, and it is the whole picture. Move it and every transversal moves.

So one rule carries two unnamed numbers, one of which may be chosen carelessly and one of which is the reader’s position, and the recipe distinguishes them nowhere. That is the sharpest form of the complaint, because it removes the obvious defence. It is not that a fifteenth-century treatise declines to compute a focal length. It is that the same sentence structure — draw it where it is convenient — is used for a quantity that has no effect and for a quantity that decides everything, and an executor following the text has no way to tell which sentence is which.

The test that separates them is the one this collection applies everywhere: change the input and see whether the output moves. It costs one extra drawing and it is not in any manual.

The third rule, and what a section does differently #

Alberti’s construction predates both of Vignola’s and this collection has already measured it in Alberti draws a pavement, which makes its figure the right thing to borrow and put the two rules against.

The difference is not accuracy and it is not the number of lines. It is that Alberti’s section draws the eye. The distance from the eye to the panel is a length on the paper, in the same drawing, measured with the same ruler as everything else, and a draughtsman who changes it sees the section change.

Vignola’s two rules each remove that drawing, in different directions. The plan rule keeps a second view and loses the elevation of the eye; the distance-point rule keeps neither and compresses the whole of the eye’s position into one point’s offset along a line. Both are genuine improvements in labour — the second rule needs no auxiliary drawing at all — and both convert a measured length into an unnamed choice.

Which is a pattern rather than a criticism of one treatise. What survives being copied tracks what a workshop procedure loses as it is transmitted, and the thing lost first is nearly always the step whose purpose is not visible in the finished drawing. A section through the eye leaves no mark on the panel. A plan below it leaves none either. The distance point leaves one dot on the horizon, off the edge, and the shortest of the three recipes is the one in which the reader’s position has become a dot nobody can see.

Why the algebra makes it inevitable #

The reduction is short enough to write down, and having it is what turns an observed agreement into a reason.

Put the horizon at height $h$ and the ground line at $h + r$, so $r$ is the drop from the horizon to the near edge of the pavement. A tile boundary $k$ units of depth into the room images at

$$y_k \;=\; h + \frac{r}{1 + k\,u/d},$$

where $u$ is one tile’s depth in the room’s units and $d$ is the eye’s distance from the panel in the panel’s own units. That is the perspective divide and nothing else.

Vignola’s plan rule produces it by similar triangles in the plan and the elevation, with $d$ appearing as the observer’s standoff from the trace. His distance-point rule produces it by cutting the orthogonals with a diagonal, and the diagonal’s slope is fixed by where the distance point sits — at offset $d$ from the centric point. Both rules take exactly one number and it is the same $d$, so there is nothing for them to disagree about except its value, and the agreement to two parts in ten trillion of a pixel is the identity being confirmed rather than two methods converging.

Which is why the essay’s measurement had to be the spread rather than the residual. An identity checked against itself is a check on the arithmetic, and this site has recorded three cases of a necessary condition evaluated at the one input where it cannot fail. The number that could have come out otherwise is the one produced when each rule is given the value its own wording invites, and it comes out at twenty-four pixels for one plausible pair and at ninety-six for the worst.

There is a second reading of the same formula worth having. As $d$ grows the transversals crowd toward $h + r$ more slowly and the pavement flattens; as $d$ shrinks they pile onto the horizon and the room deepens. So the parameter is not a scale factor on the picture — it changes the shape of the depth sequence, which is why two pavements built at different $d$ cannot be brought into agreement by any adjustment of the ground line or the horizon. They are different projections, and the lens a pavement can hide is what that looks like when somebody tries to read the focal length back off a finished floor.

What this does not settle #

Three limits.

The first is about attribution. The two rules measured here are executed as they are stated, on one panel, at one set of divisions, and the reading is about the recipes rather than about what any particular workshop did with them. A draughtsman who used both rules on one panel would notice the disagreement immediately, because the transversals would not coincide; the failure mode this essay describes belongs to somebody who uses one rule per drawing and takes the treatise’s word that it does not matter which. Three procedures on one panel is where the collection asks which was actually used, and it is a different question with different evidence.

The second is that both rules are exact and neither is being accused of anything. The disagreement is a degree of freedom, not an error, and a recipe with a free parameter is not defective — the measuring point has one too, and so does every construction on this site that depends on a focal length. What separates this case is that the parameter is unnamed and appears twice under two different descriptions, so an executor cannot tell that a choice is being made at all.

The third is the honest scope of the centimetres. The eleven point eight is computed for a figure 160 millimetres wide, and the whole quantity scales with the displayed width — the same construction printed at 210 millimetres spreads over more room and at postcard size over less. What does not scale is the ratio between the nearest and furthest choice, which is a property of the two wordings and is the transferable number.

One map, two descriptions, and the number that is in neither #

The general shape is the one this collection keeps finding under taught constructions, and it is worth stating in a form that applies past pavements.

Two procedures that reduce to one formula are the same construction. If each states its inputs differently, the difference is not a difference of method and no amount of executing them carefully will reveal it. What reveals it is writing both down in the same variables and seeing which variable neither of them names — which took, here, one algebraic reduction and four placements of one drawing.

The unnamed variable in this case is the viewing distance, and it has now appeared in this collection under five aliases: the observer’s distance in a plan, the distance point’s offset along a horizon, the separation of two vanishing points, the eye’s distance in a section, and the measuring point’s distance from the direction it measures. Every one of them is the same length, drawn on the horizon of every distance-point construction since the fifteenth century and named as a distance in none of them.

And the alias is what does the damage rather than the omission. A recipe that said choose a distance and it is arbitrary would leave a draughtsman knowing that something had been left to them. A recipe that says put a point on the horizon where convenient leaves them thinking they have made a decision about a drawing. That is the same substitution dividing to a point off the board records on a wall, where a step that supplies a third pair of corresponding points arrives disguised as an instruction about arithmetic, and the same one the bays that are not equal records on a colonnade, where which pairs a recipe may reuse decides what it will ever draw. In all three the procedure is complete, the drawing is correct, and what has gone missing is the reader.