Desargues read the other way

Two triangles and the line nobody drew proves Desargues’ theorem and closes with a remark: its converse is its dual, so proving one proves both. That remark was doing a lot of work and nothing was measuring it.

This essay measures it. The configuration is dualised as an object — ten points, ten lines and thirty incidences exchanged — and the converse is then read off the resulting drawing rather than argued for.

The theorem, as a configuration #

Two triangles in perspective from a centre: three lines join corresponding vertices and meet at that centre, and the three points where corresponding sides cross lie on a line.

Written as a configuration, that is ten points — the centre, six vertices, three points of the axis — and ten lines — three joins, six sides, the axis. Every point lies on exactly three of the lines, and every line carries exactly three of the points. Thirty incidences, and this collection’s own implementation closes them to 4.4 × 10⁻¹⁵ before anything is dualised.

The counting is necessary and not sufficient #

Exchanging points and lines exchanges the two counts — points-per-line and lines-per-point — and here they are both three. So the dual of a 10₃ configuration is a 10₃ configuration, by arithmetic and with no geometry in it.

That is the argument usually given for self-duality, and on its own it proves nothing about this configuration. There are ten distinct 10₃ configurations; the counts do not tell them apart, and a dual that landed on a different one would satisfy the arithmetic perfectly.

The same shape of mistake has been recorded twice on this site under other names: a cross-ratio test evaluated at four consecutive divisions, which gives the wrong method a perfect score, and a conformality test differenced along one tangent basis, which gives the cylinder one. A necessary condition evaluated where it cannot fail is not evidence, and a count is a necessary condition.

What is measured instead #

Three things, and together they are the theorem rather than its census.

Every incidence survives. All thirty of them, at 5.1 × 10⁻¹⁶ measured scale-free. That is not automatic in the way the counts are; it is the statement that this particular correlation carries this particular figure.

The counts come out right on the dual as well — three lines at every point, three points on every line — which is the arithmetic, checked rather than assumed because an implementation error would break it before it broke anything else.

And the converse is legible on the dual. The object that came from the axis is now a point; the three objects that came from the axis’s points are now lines; and those three lines pass through that point to 4.3 × 10⁻¹⁶. Read in words: if corresponding sides meet on a line, corresponding vertices join through a point. That is the converse, and nothing proved it — it was drawn.

Why the converse is the dual and not something else #

A theorem is an implication, so its dual is an implication between the dual statements. Dualise Desargues’ hypothesis — the three joins are concurrent — and it becomes the three meets are collinear, which is the original conclusion. Dualise the conclusion and it becomes the hypothesis. The implication comes out reversed, which is exactly what a converse is.

That is why the pair is unusually strong. Most theorems have duals that are genuinely different statements needing their own figures: Pascal’s dual is Brianchon’s, which is about tangents rather than points and looks nothing like it. Desargues’ dual lands back on Desargues, and the direction of the arrow is the only thing that moved.

Ten theorems in one figure #

The configuration’s counts — ten points, ten lines, three of each on each of the other — hide a symmetry that is stronger than self-duality and explains it.

Label the ten points with the pairs from a five-element set and the ten lines with the triples, and let a point lie on a line when the pair is contained in the triple. Each triple contains three pairs, so three points to a line; each pair sits in three triples, so three lines through a point; and there are ten of each. That is the Desargues configuration exactly, and the model makes both of its famous properties immediate.

Self-duality is complementation. A pair’s complement is a triple and a triple’s is a pair, and containment reverses — so exchanging each object for its complement swaps points with lines and reverses incidence, which is precisely what the dual figure does.

And every point is a centre. The relabelling group is the symmetric group on the five elements, which acts transitively on the pairs, so no point of the configuration is distinguished from any other. Take any of the ten as the centre of perspective and the remaining nine sort themselves into two triangles and an axis, giving a perfectly good instance of the theorem.

So the figure a reader draws as one Desargues theorem is ten of them at once, and which pair of triangles they see is a fact about where their eye started rather than about the drawing. The two triangles and the axis are not the figure’s structure; they are one of ten equally valid decompositions of it.

That reframes the converse as well. It is not a second theorem the figure happens also to satisfy — it is the same ten instances read with points and lines exchanged, which the complementation map performs. A figure with one distinguished centre would have a dual with one distinguished axis and the two would be different objects. This one has neither, so the theorem and its converse are one statement about one object, and the “direction of the arrow” the section above says is the only thing that moved is itself an artefact of choosing a starting point.

It also explains why the closure is so robust under the sliders. The concurrences and collinearities are not ten separate coincidences maintained by ten separate mechanisms; they are one incidence structure, and a projectivity carries the whole structure or none of it.

The three points that cannot be drawn #

The dual of a line through the origin of the picture is a point at infinity, and a Desargues configuration centred on the origin has three lines through it — the three joins.

So the dual of the most natural way to draw the figure has three of its ten points nowhere on the page. The theorem is untouched; the incidences still hold, measured scale-free; and the drawing is impossible. Every figure here therefore places the centre off the origin, and says so.

This is worth more than a footnote because it is the clearest small example of a distinction the whole field rests on. The projective plane has no origin and no scale. Writing a point as a pair of numbers with an implied 1 chooses both, and the choice is invisible until a construction lands on the part of the plane the choice cannot reach.

What the theorem is for here #

Desargues is not decoration in this collection. It is the licence for a large class of straightedge constructions — the ones a point and a line are one object characterises as joins and meets and nothing else — and dualising it doubles that class.

The constructions in the bay repeated by a straightedge, carrying a height across the room and seven is not a power of two are sequences of joins and meets whose correctness rests on incidence theorems holding in the picture. Every one of them has a dual sequence of the same length using the same instrument, and the dual is available for free — which is what the quadrilateral that finds the middle exploits to build a harmonic conjugate from four lines instead of four points, and what the diagonals find the middle established in its original form.

The practical question is never whether the dual construction is valid. It is whether the marks it needs are the marks the drawing has.

Desargues as a fact about dimension #

There is a reason this particular theorem is the one that dualises so cleanly, and it is not about the plane at all.

Desargues’ configuration is the shadow of a three-dimensional one: two triangles in different planes in space, in perspective from a point, whose corresponding sides meet on the line where the two planes cross. In space the theorem is nearly trivial — the meets are on the intersection of two planes because they are on both planes — and the plane version is that picture flattened.

Duality in space exchanges points and planes, and a line is self-dual. Run the spatial argument dually and it says the same thing with “meet on the line where two planes cross” replaced by “join through the point where two lines cross”, which is the converse again. So the self-duality is inherited from a dimension up, where it is a statement about planes and lines rather than about a ten-point figure.

That also explains a fact this collection uses without remark: Desargues’ theorem holds in every projective plane that is a plane in some three-dimensional projective space, and there are strange planes where it fails. Nothing drawn on paper is one of those, which is why a straightedge construction can be trusted.

The dual figure is not a redrawing #

The point most easily missed about the two panels is that the right-hand one is not the left-hand one seen differently. It is a different set of marks on a different part of the plane, and the only thing they share is a list of thirty facts.

That is worth insisting on because the eye does not cooperate. Two ten-point figures drawn side by side look like two views of one thing, and the temptation is to hunt for the projective map taking one to the other. There is none: a correlation is not a projectivity, and it sends points to lines rather than points to points. The dual of a figure is the same information written in a different alphabet.

The consequence a reader can use is that distances between the two panels mean nothing. A point that looks close to its own dual line is a coincidence of scale; the only quantities with meaning are the incidences, which is why every residual here is scale-free and why the figure states its own normalisation.

What a projection does to the configuration #

Desargues is projectively invariant, which is the reason it is worth anything on a photograph: photograph the two triangles and the theorem holds in the picture, with the picture’s own centre and the picture’s own axis.

That is the property what a projection destroys is a catalogue of the other half of. Lengths, angles, areas and ratios of lengths all go; incidence and the cross-ratio survive; and every theorem in this essay is built out of the surviving half only.

It also means the theorem is available on a picture whose camera is unknown, which is the standing condition of every construction in the construction and metrology fields. A reader with a straightedge and a photograph can carry out both Desargues and its dual, and neither needs the focal length, the station point or a single measured length.

What a reader can do with it #

One thing, and it is small and genuinely useful: a construction that fails can be checked by its dual.

If a straightedge construction on a photograph does not close — three lines that should meet do not, three points that should be collinear are not — the dual construction uses different marks on the same picture and produces an answer that must agree. Where they disagree, the marks are at fault; where they agree and are both wrong, the assumption is.

That is the same discipline as computing a reflection two ways, or checking a measuring-point construction against the depths a camera actually produces. Two independent routes to one number is this collection’s habit, and duality supplies a second route wherever there is a first.

Where the dual construction is the useful one #

An example rather than a principle, because the principle on its own does not tell a reader when to reach for the dual.

A photograph of a building has long straight edges and very few marks that are securely a point: a corner is occluded, a window mullion runs off the frame, the ground line disappears behind a car. Lines are easy and points are hard. A construction whose inputs are lines and whose output is a point is therefore the one to run, and the dual half of every straightedge theorem supplies exactly that.

The reverse arrangement is a photograph of a scatter — a facade with regularly spaced fixings, a floor of studs, a row of posts. There the marks are points and the lines are inferences, and the original half of the theorem is the one to run. The map a row of posts is is the construction field working entirely in that mode.

Neither is more correct. What duality removes is the need to decide in advance which mode a subject’s theorems are going to be written in, because every one of them arrives in both.

The theorem’s two audiences #

Desargues is usually met twice and the two meetings have nothing to say to each other, which is worth naming because this collection needs both.

To a geometer it is the axiom that separates the planes that come from three-dimensional space from the ones that do not, and its content is the classification result mentioned above. To a draughtsman it is a closure test: two triangles that ought to be in perspective, three lines that ought to meet, and a way of telling whether a drawing is a projection of anything at all.

The second reading is the one two triangles and the line nobody drew makes and the one this collection uses, and it is where the converse earns its keep. A drawing supplies whichever half is easier to check. Sometimes three lines visibly meet and the collinearity is the thing to test; sometimes three points are visibly on a line and the concurrence is. The theorem and its converse together mean that either check settles the other, and the dual figure is why they are one theorem rather than two.

The configuration’s own free parameters #

Ten points and ten lines sounds like a great deal of freedom and it is not, which is worth counting once.

A Desargues configuration is fixed by a centre and two triangles in perspective from it: two numbers for the centre, six for the first triangle’s vertices, three for the three scales. Eleven, and a projective map of the plane has eight of its own, so there are three genuine shape parameters — which is why the figure’s slider can move one scale and produce a configuration that is genuinely different rather than a projection of the same one.

Everything else in the figure is determined: the axis, its three points, and the six sides all follow from the eleven. That is what makes the closure a theorem rather than an arrangement, and it is also what makes the dual’s counts inevitable rather than lucky.

What is measured here #

Four numbers, three of them at the arithmetic floor and the fourth a count.

The configuration closes to 4.4 × 10⁻¹⁵ before dualising. Every one of its thirty incidences survives the exchange to 5.1 × 10⁻¹⁶. The converse is legible on the dual figure at 4.3 × 10⁻¹⁶. And the counts — three lines at every point, three points on every line, on both figures — come out as they must, which is the necessary condition that would have been the whole argument if the other three had not been made.