The theorem that is obvious one dimension up

Two triangles and the line nobody drew draws the plane Desargues configuration and measures it: two triangles perspective from a point have their three pairs of corresponding sides meeting at three points that lie on one line. Drawn, checked, exact.

What that essay does not say is why. In the plane there is no reason visible in the figure — the collinearity is a genuine theorem, and the classical proofs are either an exercise in coordinates or a page of case analysis about which lines meet where.

In space it needs no proof at all. Take the same configuration with the two triangles in two different planes, and the collinearity becomes something a reader can see in one sentence.

That is the whole argument. Two corresponding sides, say the side of the first triangle through A and B and the side of the second through A′ and B′, lie in the plane spanned by the centre and the two rays — because A, A′ and the centre are collinear, and so are B, B′ and the centre. Two lines in one plane meet. Their meeting point lies on the first triangle’s plane and on the second’s, so it lies on the intersection of those two planes. Do that three times and all three meets are on that one line.

No coordinates, no case analysis, and nothing that depends on the tilt: at 16° the meets are 1.0 × 10⁻¹⁴ metres off the line and at 38° they are 8.5 × 10⁻¹⁶, both at the arithmetic floor.

It is worth noticing what the spatial argument does not use. It never asks where the centre is, only that the three rays pass through it; it never asks anything about the shape of either triangle; and it never uses the tilt, which is why sweeping the tilt changes no residual. The three steps are a join, a meet, and the observation that a point in two planes is on their intersection — and every one of those is an incidence rather than a measurement. A proof built only from incidences is a proof that survives projection, which is precisely why the plane case follows.

The plane figure is the shadow, and this collection has the projector #

The step from the spatial statement to the plane one is a projection, which is the operation this whole collection is about.

Photograph the spatial configuration. A projection carries lines to lines and preserves meets, so every incidence in the space figure appears in the photograph: the three joining lines still concur at the image of the centre, corresponding sides still meet, and the three meets still lie on a line — the image of the line where the two planes crossed. The plane Desargues configuration in the photograph is not analogous to the spatial one. It is a picture of it.

That is the sense in which the theorem is obvious one dimension up. The plane statement is true because it is the shadow of a spatial statement that needs no proof, and this collection can say so with a camera rather than with an argument.

Two claims that look like one #

The figure above appears to make a single statement. It makes two, and they come apart under a perturbation chosen to separate them — which is the only way to find out that a measurement is measuring two things at once.

The first claim is that the three meets are collinear. The second is that the line they lie on is the image of the planes’ intersection. In the drawing they are the same line, so nothing distinguishes them.

The perturbation is exact rather than approximate, and that is what makes it a good one. Sliding a vertex along its own ray is precisely the operation that preserves the hypothesis of plane Desargues and destroys the spatial configuration, because being perspective from a point is a condition on rays and lying in a plane is not. Any other perturbation would have broken both at once and settled nothing.

So the honest reading of the exact figure is: the collinearity is a plane fact, true whether or not the drawing came from a spatial configuration; and the axis being the image of a real line in the room is an additional fact, true only when it did.

And the theorem is a test, not a description #

A reader might reasonably ask whether the collinearity is worth measuring at all — three lines produce three intersection points, and three points always lie on a line unless they happen not to.

Three points in general position are collinear only by accident, and the accident does not happen here: break the hypothesis and the collinearity goes with it, by fifty-three pixels rather than by a rounding. That contrast is what makes Desargues usable as a test — a reader who finds three such meets collinear on a photograph has learned that the two triangles are perspective from a point, which is a fact about the scene.

The 68.6 pixels is worth quoting beside the 53.46, because the two are the input and the output of the same perturbation and their ratio is the sensitivity: a construction that lost the hypothesis by sixty-eight pixels and kept the conclusion to within one would be telling a reader that the conclusion did not depend on the hypothesis.

Which solid cast this picture, and the answer is a family #

Saying the plane figure is a shadow invites the obvious question. A shadow of what, exactly? The photograph came from one spatial configuration, but a reader holding only the photograph cannot know which.

The lift is a one-parameter family and every member of it is a genuine spatial Desargues configuration casting exactly this photograph. Their second planes lean anywhere from 53.6° down to 29.6°, which is not a small spread, and nothing in the picture chooses among them.

There is a trap in reading that figure and it is worth naming. The re-drawing residual — 2.0 × 10⁻¹² pixels — is not the measurement. The lifted points are constructed on the camera’s own rays, so of course they re-draw the picture; that number is checking the arithmetic and nothing else. The measurement is the gap between two lines that have to meet: the ray from the lifted centre through a drawn vertex, and the ray from the camera through the same vertex. Those are two skew lines in general, and they meet to 6.4 × 10⁻¹⁴ metres here only because the drawn triangles are perspective from a point. Slide a drawn vertex off the ray and the gap opens to 0.13 metres.

So the lift is a reconstruction, and like every reconstruction in this collection it recovers a family rather than an answer. It is the same shape as what one picture determines: the picture fixes the projective structure exactly and leaves a metric parameter free, and one extra fact from outside — here, the height of the centre — collapses the family to a point.

Ten points and ten lines, each doing the same job #

It is worth counting the configuration, because the count is what makes the duality below inevitable rather than surprising.

There are ten points: the centre, three vertices of the first triangle, three of the second, and the three meets. There are ten lines: the three joining lines through the centre, three sides of the first triangle, three of the second, and the axis. Every point lies on exactly three of the lines, and every line passes through exactly three of the points.

That regularity is not decoration. It means no point of the figure is distinguishable from the others by its incidences alone — the centre looks, to the incidence structure, exactly like one of the meets. A reader handed the ten points and the thirty incidences with the labels removed could not say which point was the centre, and the configuration would look the same if a different point were nominated for the role.

Which gives the theorem a stronger reading than the one it is usually stated with. The configuration does not contain one pair of triangles perspective from a point; it contains ten such pairs, one for each choice of which point plays the centre, and all of them hold at once. The version drawn above is one reading of a figure that supports ten.

The spatial picture makes that plausible in a way the plane one does not. Lift the configuration and the ten points are five planes’ worth of intersections; there is nothing in the space that singles out the centre either, because the centre was only ever the point two triangles were projected from, and projection is symmetric in a way the drawing hides.

Why the plane statement is still a theorem #

It would be tidy to conclude that plane Desargues is merely spatial Desargues seen flat, and that the plane proof is therefore unnecessary. That is nearly right and the exception is the interesting part.

The lift above shows that this plane configuration comes from a spatial one. It does not show that every plane Desargues configuration does, and in the real projective plane the general argument requires the plane to be embedded in a space — which is available for the plane this collection works in and is not a consequence of the plane’s own axioms. There exist projective planes, perfectly consistent as incidence geometries, in which Desargues’ theorem is false; they cannot be embedded in any projective space, and that is exactly why.

So the honest statement is conditional. In the plane a camera photographs, Desargues holds, and it holds because that plane sits in a space and the configuration lifts. The theorem is a statement about the ambient geometry as much as about the figure, and the lift figure is the evidence for the premise rather than a proof of the conclusion.

That is not an abstraction with no consequence here. It is the reason a projection of a projection composes to a projection at all, and the reason the straightedge constructions in this collection are reliable: they are executing incidences in a plane that is a genuine slice of space, and every one of them would be available to a draughtsman working on a photograph.

The axis is a line in the room, and that is worth something #

The perturbation section separated two claims and called the second one additional. It is worth saying what that additional claim buys, because it is the only part of this essay that a reader could use on a photograph of something real.

When the configuration does lift — when the two triangles really are in two planes — the axis a straightedge draws is the image of the line where those planes cross. That line is an object in the room, and its image has been obtained without measuring anything, without knowing either plane, and without knowing where the camera was.

Two planes in a scene cross in a line, and that line is often not visible: a wall meets a ceiling behind a beam, a road meets a verge under a hedge, a tilted board meets the floor where the board’s own edge is not straight. A pair of triangles, one drawn on each plane and perspective from any point at all, hands the reader the image of the invisible line. It is the same species of result as a floor with a referent — a construction on marks that yields something about the room — and it costs six straightedge lines.

The catch is the one the slid vertex exposes. The construction returns a line whether or not the configuration lifts, and the returned line is the image of a real intersection only when it does. Nothing in the drawing announces which case a reader is in, and 11.10 pixels of error looks exactly like 6.5 × 10⁻¹³ pixels of exactness to anybody not measuring. So the construction is only as good as the reader’s grounds for believing the two triangles are each flat on their own plane — which is a fact about the scene, brought from outside, exactly as cheirality is.

That is the recurring shape and it is worth naming once more here: the projective construction is exact and cheap, and the premise it runs on is neither measured nor measurable from the picture.

What the configuration is dual to #

Desargues has the rare property of being its own dual, and the duality is worth stating because it doubles the theorem for free.

Exchange point and line throughout. “Two triangles perspective from a point” becomes “two triangles perspective from a line”, and “their corresponding sides meet on a line” becomes “their corresponding vertices join through a point”. That is the converse of the original statement — and since the dual of a true statement is true, the converse comes free rather than needing its own proof.

The spatial picture makes the symmetry visible: the centre and the axis are the two objects the configuration is built around, one a point and one a line, and the whole figure is symmetric in exchanging their roles. A point and a line are one object measures that exchange on this very configuration — ten points and ten lines, thirty incidences, every one of them surviving the correlation.

What it costs to check, which is one straightedge #

The practical value of Desargues in this collection is that it turns a question about a scene into a question a straightedge can answer on a photograph, and it is worth being explicit about what the straightedge does.

Given two drawn triangles, a reader wanting to know whether they are perspective from a point can draw the three joining lines and see whether they concur. That is three lines and two intersections. Alternatively — and this is the theorem’s contribution — the reader can draw the three pairs of corresponding sides, find the three meets, and see whether those are collinear. That is six lines, three intersections and one straightedge laid across them.

The two tests answer the same question, and having both matters when one of them is unavailable. Joining lines that concur far off the paper are exactly the case dividing to a point off the board is about: the concurrency cannot be checked because the point is not on the sheet. The three meets may be perfectly well placed even then, so the dual test succeeds where the direct one cannot be run.

That is the ordinary reason to want a theorem with two equivalent statements — not elegance, but that one of the two is sometimes off the paper. It is the same argument every projectivity is two perspectivities makes about the intermediate line, and the constructions in this collection lean on it constantly.

What this does not settle #

The spatial configuration here is one arrangement and the framing was harder than the geometry. Hinging the second plane about a convenient ground line puts each meet wherever a triangle’s side happens to cross that line, and a side nearly parallel to it throws its meet metres away and thousands of pixels off the paper. The camera used was found by search against two conditions — all ten points in frame, and no two closer than a label’s width — and the best available separation is 16.6 pixels, which is why the labelling in the spatial figure is sparse. That is a fact about drawing the configuration rather than about the theorem, and no measurement here depends on it.

And the lift family is drawn over a range of centre heights chosen so that every lifted vertex is in front of the camera. Below about 1.6 metres some of them land on the backward half of their own ray — a perfectly good projective configuration, and one this collection has a name for, since it is what is behind the camera arriving inside a reconstruction. Those members are real and no figure here draws them.

One configuration, two dimensions #

The object is ten points and ten lines, and the finding is that its difficulty depends entirely on how many dimensions a reader is willing to use.

Flat, it is a theorem with a proof nobody remembers. Lifted, it is three applications of the observation that two lines in a plane meet. The projection between those two readings is the ordinary one this collection computes everywhere else — and the fact that it carries a hard statement to an easy one is the best argument available for taking the space seriously rather than treating the picture as the primary object.