A grid on the wall is a scale without a projection

The one thing a single view cannot give is scale: a photograph fixes every ratio in a scene and not a single length, so a reader has to bring one length in from outside the picture before anything in it can be measured. That is a fact about perspective, and it is easy to mistake for a fact about pictures in general. An Egyptian canon of proportion shows it is not.

A canon rules a wall, or a panel, into a grid of equal squares and sets a standing figure at a fixed count of them — eighteen squares for a king, eighteen for most other men, twelve for a lesser figure, by the convention this drawing follows. That single rule supplies a comparison between any two figures on the wall with no horizon, no station point and no centre of projection anywhere in the construction, because the comparison is never between two lengths in the world — it is between two counts on the same ruling.

The picture: counted, not triangulated #

A canon needs a wall ruled fronto-parallel and a camera looking straight at it, which is the plainest photograph this collection draws.

A canon of this kind is what this essay calls a canon of proportion: a fixed unit, ruled once, against which every figure on the same wall is counted rather than measured. It is a drawing convention in the same sense every other subject in this field is — a decision about what to record and what to let go — and it is a drawing system distinct from any of the ones built around a camera, because nothing about it answers to a station point at all. The rest of this essay puts it through the same battery of questions this collection asks of a camera-based system, and the answers turn out to be as sharp as any this site has measured, in a different currency each time.

Every other convention on this site measures a picture that is a projection from a point onto a plane, and every measurement in that convention is a statement about the angle that point subtends. This one is not. The wall is photographed, so the recording camera is a pinhole exactly like every other camera on this site — but the reading of the picture never uses the camera at all. A ruler laid across the ruling counts squares, and a figure drawn at eighteen of them reads as eighteen wherever on the fronto-parallel wall it stands, because a uniform scaling multiplies every square by the same factor and a ratio of two equal things survives any uniform scaling exactly. The counting error above, 1.8e-15 squares, is the floating-point residue of a machine doing that division — it is not a measurement of anything optical, and that is the whole point being made by quoting it.

The count survives being ruled differently #

If the convention’s honesty depends on some particular density of ruling, it is a much weaker claim than “no horizon and no centre.” The obvious test is to rule the same wall coarser or finer and see whether the reading changes.

Nothing about the reading moved. The pitch of one square on the page — 14.7 px in both figures — is set by the physical size of one square in the world and the camera photographing the wall, not by how many of them the wall happens to be divided into; ruling the same wall into more or fewer squares changes how tall the grid looks on the page and changes nothing about what any one square, or any one figure counted in squares, means. That is the ruling-independence a horizon-and-centre construction does not have: move the horizon in a perspective picture and every depth reading on the page moves with it, because the horizon is doing the work of defining the scale. A canon’s ruling does the equivalent job here, and moving it changes nothing, because the comparison it supports was never between the ruling and anything outside the picture — only between two counts on it.

A different kind of measuring point #

The construction field already has a device called a measuring point, and it is worth being precise about why the canon’s ruling is not a second name for it, because both are, in a loose sense, “a way of getting a length off a picture that a horizon alone will not give.”

The measuring point sits on the horizon of a genuine perspective construction, at the distance a station point would be found by rotating it into the picture plane, and it lets a draughtsman step off equal receding intervals along a line that is already converging toward a vanishing point. It is a real construction, and it works — but it works inside a perspective drawing, using that drawing’s own horizon and its own vanishing point, so everything the earlier essay on the distance point is the viewing distance establishes about a station point applies to it as well: move the assumed viewing distance and the measuring point moves with it. A canon’s ruling needs none of that apparatus. It is drawn on a wall with no vanishing point at all, its squares do not converge anywhere, and a reader does not need to know, or assume, or recover a station point before the ruling means anything. Both devices let a reader step off a length that a bare outline would not supply; only one of them depends on first getting a horizon and a distance right.

No centre to find #

This collection runs one battery of five questions against every drawing system it studies, and the canon is worth putting through it, because the answer to one of those five questions is the reason a canon needs no horizon at all.

Four of the five questions this battery asks — is there a centre, does it keep true measure, does it diminish with depth, does it draw a straight line straight — are questions a perspective construction answers by having a station point: a centre exists because every ray meets at the eye, diminution exists because a further ray subtends a smaller angle. A canon has none of that machinery, which is why it fails to have a centre in the specific, arithmetical sense the battery tests for — not approximately, not to within some tolerance, but as a refusal: the equations a centre-fitting routine solves for a pinhole picture have no solution at all when every ray runs parallel, because parallel rays are rays to a centre infinitely far away, and “infinitely far away” is not a point the arithmetic can return.

That refusal is a genuinely different event from a bad fit. Recovering the camera takes a real perspective picture and works backward to the station point that drew it, and the whole exercise depends on the rays it is handed converging somewhere, however poorly conditioned that convergence might be. Handed a canon’s rays instead, the same recovery has nothing to converge on even in principle — not a centre recovered with large uncertainty, but a system with no solution, because “no solution” and “a solution far away with large error bars” are different findings and the battery is built to report which one it has met. A canon of proportion is the drawing this collection’s own recovery machinery has nothing to do with, by construction, and that absence is itself a measurement worth having on the record: most conventions in this field bend the recovery’s answer; this one removes the question.

What the canon has and a pinhole does not #

The battery’s sixth column — scale — is the one no other system in this collection’s comparisons has needed, because it asks a question that only makes sense for a system with no centre and no diminution to begin with: given the picture alone, how accurately can a length be read off it.

That is the sharp form of this essay’s argument, and it deserves to be stated as a trade rather than a verdict. The canon supplies exactly what a pinhole picture cannot — a scale, read from the picture with nothing brought in from outside it — and a pinhole picture supplies exactly what the canon cannot, which is depth. Each system answers its own question already makes this the organising idea of the whole field: no system on this site is a worse version of any other, because each is built to answer a different question, and the 58% figure above is not a failure of perspective. It is the size of the question perspective was never trying to answer with a single reading — a pinhole’s scale genuinely is not one number, and the 58% is what happens when a reader insists on treating it as though it were.

The gap this closes, that a single perspective picture cannot #

The one thing a single view cannot give states the general problem a canon happens to solve completely: a photograph fixes every ratio between lengths in the scene it shows, and fixes none of their absolute sizes, because scaling the whole world and the camera together by the same factor leaves every pixel exactly where it was. Two views give shape and no size shows that a second photograph, taken from a different point, still does not close that gap on its own — a second camera adds a genuine three-dimensional shape and still leaves the whole reconstruction free to be scaled, because the ambiguity is in the relationship between the cameras and the scene rather than in how many cameras there are. What closes it, in every camera-based construction this collection has measured, is a length brought in from outside the picture: a known height standing somewhere in the frame, a stated distance between two points, some fact that did not come from the photograph itself. Five facts that close the same gap catalogues the various forms that outside fact can take.

A canon of proportion is the one construction in this field that needs no outside fact at all, because the ruling supplies its own unit and every figure on the same wall is drawn against that unit rather than against the world. It does not close the single-view scale gap by being a cleverer camera; it closes it by declining to be a camera in the first place, which is the same move a carpet and the people on it describes from the opposite direction — a construction that gives up depth outright, rather than compromise on it, in exchange for keeping something else exactly.

The control: reading a pinhole picture the canon’s way #

The trade above is worth demonstrating rather than asserting, and the demonstration is to take the canon’s own rule — rule the picture into equal squares from the nearest figure, count every other figure against that ruling — and apply it to a picture the rule was never built for.

Reading a pinhole picture as though it were a canon does not fail gently. It is exact at the one figure the ruling happens to have been taken from, because at that single depth the two conventions agree by construction, and it degrades by exactly the ratio of depths thereafter — a figure at 32 m against a ruling taken from 6 m reads at 19%, which is 6/32 read the other way round, because a pinhole’s image height falls as one over depth and a fixed ruling has no way to know that. This is the failure that says the counting itself is sound and the picture is the wrong kind of picture for it, which is the distinction the whole essay turns on: the canon was never wrong about anything it was built to read, and a pinhole picture was never wrong about anything either, and the 19% belongs to neither convention — it belongs to the mismatch between them.

Another convention that needs no horizon #

The canon is not the only drawing system in this field that manages without a horizon, and the comparison is worth making because the other one gets there by a completely different route.

A register avoids a horizon by equal-height convention: every figure is drawn the same size regardless of depth, so no perspective convergence is ever set up to find a vanishing point in. A canon avoids a horizon differently — it does not equalise drawn height, it counts height in a fixed unit, so two figures of different declared rank are visibly different sizes on the page and are still comparable, because both sizes are read against the same ruling rather than against each other’s apparent depth. Both constructions land on the same absence — no horizon, no vanishing point — from opposite premises: one throws the size variable away, the other keeps it and supplies an independent unit to read it in. A register can state an order and nothing about magnitude; a canon states an exact magnitude and nothing about depth. Neither is the more complete convention — each keeps a different one of the two things a drawing might want to say about a figure standing further along a wall.

The honest limit #

Every number above is a statement about a fronto-parallel wall, ruled and photographed square-on. Tilt the wall, or move the figures off it onto a receding floor, and the uniform scaling this whole argument rests on stops holding — a wall at a slant images with a foreshortening that a fixed square grid does not correct for, which is exactly why a canon of proportion was drawn on flat walls and flat panels rather than on receding floors, and exactly why the several drawing systems in this field that do handle a receding floor need a horizon or an equivalent construction to do it. The canon’s exactness is bought by refusing the third dimension its own convention altogether: it is not a system for drawing depth badly, it is a system that was never asked to draw depth at all, and asking what it says about a figure standing further back is asking a question it has no vocabulary for — which is exactly what the control above demonstrated by applying the wrong rule to the wrong picture and watching the answer degrade by the ratio of the depths involved.

And the ruling-independence demonstrated above rests on counting itself being a legitimate measurement, which is not something to take for granted just because the arithmetic is simple. Counting is a measurement makes the general case: a count carries an uncertainty exactly as a length or an angle does, it is simply an uncertainty of a different and usually much smaller kind, bounded by whether a whole square was correctly identified as whole rather than by any question of calibration. A canon’s exactness is not exemption from measurement error — it is a convention that has arranged for its own measurement error to be the error of counting discrete, equal-sized marks rather than the error of judging a continuously varying angle, which is why 1.8e-15 squares is the floor this collection’s own arithmetic sits at rather than a floor the wall itself achieves.

Nor does anything here say which of the two conventions a given wall or panel is using without independent evidence. A grid of squares behind a figure is consistent with a canon; the absence of a visible grid is consistent with either a lost ruling or a picture that was never using one, and this essay’s own battery presumes the ruling is known and the figures’ declared counts are known, which a real surviving wall does not always hand over intact.

What this is an instance of #

A canon of proportion is this field’s cleanest instance of a fact stated more abstractly by each system answers its own question and by a centre and a measure are exclusive: no drawing system on this site keeps a centre of projection and a true measure of length at once, and the canon is the limiting case of that trade rather than an exception to it — it keeps the measure completely, at 1.4e-14% of error, by having no centre of projection whatsoever. A carpet and the people on it makes the equivalent point about a different pair of conventions colliding on one wall: a plan view keeps a carpet’s shape true and destroys a standing figure’s height, an elevation keeps the figure and destroys the carpet, and no single camera supplies both. A canon and a pinhole picture are the same collision restated with scale and depth in the two seats the carpet and the standing figures occupied there, and the moral carries over unchanged: a length read off the wrong convention is not a small error waiting to be corrected. It is a question answered by a system that was never asked it.