A point and a line are one object

The word duality has appeared in eight essays here and has never once been an object. Four lines have a cross-ratio says a pencil is the dual of a range and then measures the range. The polar with a straightedge builds a line out of a point and calls the relation a polarity in passing. Two triangles and the line nobody drew proves Desargues and mentions that its converse is its dual.

In every case the word is doing the work of an explanation and none of the work of a measurement. This essay makes it a map with a matrix in it, so that the sentences above become claims that can be run and can fail.

What the exchange is #

A point of the projective plane is three numbers up to scale. A line is three numbers up to scale. The condition that the point lies on the line is ax + by + cw = 0, which is symmetric in the two triples: it does not know which of them is the point.

That symmetry is the whole content. Choose any invertible three-by-three matrix D and send the point p to the line Dp. If D is symmetric the map is its own inverse — a polarity — and then

$$p \text{ lies on } l \quad\Longleftrightarrow\quad \mathbf{D}^{-1}l \text{ lies on } \mathbf{D}p$$

because (D⁻¹l)ᵀ(Dp) = lᵀp. One line of algebra, and it is why every incidence theorem in the subject arrives in pairs.

The second half of that figure is the half worth having. A correlation that sent every point to one fixed line would preserve every incidence perfectly and would preserve the non-incidences too, which is the failure mode this collection has recorded twice under other names — a cross-ratio test that measured nothing and a conformality test evaluated at the one input where it cannot fail. So the measurement is a pair: the incidences that held hold to 1.2 × 10⁻¹⁶, and the ones that did not fail by 0.39.

The choice, and the one place there is none #

D is a choice. Any invertible matrix gives a duality, different matrices give different dual figures, and every one of them is equally true. A treatment that hard-coded one would be reporting the matrix rather than the theorem, so every function behind these figures takes it as an argument and defaults to the identity — the polarity with respect to x² + y² + w² = 0, a conic with no real points at all, which is exactly why it treats every real point alike and singles out no line.

There is one duality on this collection that is not a choice, and it is the horizon has a pole: a calibrated camera fixes a conic nobody picked, and under the polarity that conic defines, the horizon of a plane and the vanishing point of its normal are pole and polar. That is a fact about an instrument. Everything in this essay is a fact about the plane.

Joining and meeting are one operation #

The most useful consequence is the smallest. Two points determine a line; two lines determine a point. In coordinates both are the cross product of two triples, and a straightedge does both with the same gesture.

This collection has been leaning on that since its first phase without saying so. Everything in the polar with a straightedge, everything in the bay-repeating and depth-dividing constructions, and every recovery of a vanishing point is a sequence of joins and meets — and nothing else. No length, no angle, no midpoint, no compass. That is what makes those constructions survive a projection and therefore run on a photograph.

Duality is why the list of such operations is closed under swapping the two words. A construction with seven joins and four meets has a partner with seven meets and four joins, exactly as long, using exactly the same instrument.

The configuration, and its dual drawn beside it #

An incidence theorem is a statement about a configuration: a set of points, a set of lines, and a list of which lie on which. Desargues’ configuration is ten points and ten lines with every point on exactly three lines and every line through exactly three points, thirty incidences in all.

Its dual is another configuration with the same two counts, which is forced: exchanging points and lines exchanges the two counts and they were equal. What is not forced, and what is measured here, is that the dual is another Desargues configuration rather than some other object with the same census — checked by finding a centre in the dual and requiring the three lines that came from the axis’s points to pass through it. They do, to 4.3 × 10⁻¹⁶.

The dual is as short and not as drawable #

Duality preserves the form of a construction exactly and its conditioning not at all, and the distinction is the reason this collection keeps meeting constructions that are exact and undrawable.

A join of two points $a$ apart, each located to $\varepsilon$, gives a line whose direction is uncertain by about $\varepsilon/a$: the conditioning parameter is a separation.

A meet of two lines crossing at $\alpha$, each located to $\varepsilon$, gives a point uncertain by about $\varepsilon/\sin\alpha$: the conditioning parameter is an angle.

So a construction with seven joins and four meets and its partner with seven meets and four joins are the same length, use the same instrument, and are governed by different quantities — and a figure that is comfortable in one may be hopeless in the other. Two points at opposite corners of the page join beautifully; the dual pair of lines may be nearly parallel and meet a kilometre away.

That has a consequence worth stating in general, because it explains a pattern rather than an incident. A projectively invariant construction has a projectively variable conditioning. Separations and angles are both destroyed by a projection, so the very thing that makes joins and meets survive a photograph — that they use no metric quantity — is what makes their difficulty depend on the photograph. The construction is a fact about the configuration; whether it can be carried out is a fact about the drawing.

Which is why “exact and undrawable” recurs across these fields — a height transferred across a room whose vanishing point lands half a kilometre away, a lamp whose construction reaches for a point off the page — and why the remedy is always the same shape: choose which dual form to execute, since one of the two usually has its conditioning parameter in a comfortable range when the other does not.

And the census does not determine the configuration #

The essay is careful to check that the dual of Desargues’ configuration is another Desargues configuration rather than some other object with the same counts, and the care is warranted: the census genuinely does not determine the object.

There are three distinct combinatorial configurations with nine points and nine lines, three on each, and ten with ten and ten. So “ten points, ten lines, three of each on each” describes ten different objects, of which Desargues’ is one. Its self-duality is a property it has and eight of the others do not — the one other self-dual member of the family is a different figure entirely.

That is what makes the measured $4.3\times10^{-16}$ a result rather than a tautology. Exchanging points and lines is guaranteed to produce a configuration with the same census; that the result is the same configuration is an extra fact, and it is the fact the theorem’s self-converse depends on. A reader who took the counts as proof would have proved something true by an argument that also proves eight false things.

One qualification on the conditioning claim, because it is easy to over-read. The two parameters are not independent when the figure is fixed: a projectivity that opens an angle generally closes a separation somewhere else, and the total difficulty of a construction cannot be driven to zero by choosing a view. What can be chosen is where the difficulty falls, which is enough — a draughtsman needs the hard step to be one they are not taking, not to be absent.

What the dual of Desargues says #

Read the dual figure with the words swapped and the sentence comes out as: if the three lines joining corresponding vertices meet in a point, then the three points where corresponding sides cross lie on a line — with “point” and “line” interchanged throughout, which turns the theorem into its own converse.

That is the strongest form self-duality takes. Most theorems have a dual that is a different theorem needing its own proof; Desargues’ dual is its converse, so proving the theorem proves the converse for free, and the configuration is fixed by the map rather than merely resembling itself.

The one thing that does not carry across: a scale #

A dual figure cannot be measured in pixels, and finding that out is half of what dualising a real drawing teaches.

The dual of a line is a point, and the dual of a line through the origin of the picture is a point at infinity — its third coordinate is the line’s own offset from the origin, which is zero. A Desargues configuration centred on the origin has three such lines, so three of its dual’s ten points cannot be drawn at all.

This is not a defect in the theorem and not an approximation. It is the chart running out: the projective plane has no origin, and writing a point as (x, y) with an implied 1 chooses one. Every residual inside a dual figure here is therefore measured scale-free, as |lᵀp| with both normalised to unit length, and only the incidences in the original are reported as distances on a page.

How the dual figure is drawn at all #

A practical matter that turns out to carry the essay’s one real caution.

A point and a line are each defined only up to a scale, so a dual figure computed from a drawing in pixels lands wherever the arithmetic puts it — typically a thousand times off the canvas, and at a size that has nothing to do with the original’s. There is no natural common scale between a figure and its dual, and there cannot be: the correlation is a linear map on triples and multiplying it by anything gives the same duality.

So the dual here is divided by one number, and the number is a property of the figure rather than of the canvas — the median distance of its finite points from their own centroid. That is reported rather than hidden, because a dual drawn at a scale chosen to make it look right is a picture of the choice.

The caution is the one this collection keeps meeting: a figure that has been fitted to its frame is not evidence about anything until the fitting is stated. It is the same discipline as the viewing-distance strip that every picture here carries, applied to a diagram that is not a picture at all.

The count that is not a coincidence #

Ten points on three lines each and ten lines through three points each is written 10₃, and its self-duality is arithmetic: exchanging the two roles exchanges the two counts, and the counts are equal.

The arithmetic is necessary and not sufficient, which is the shape of half the findings on this site. There are ten distinct 10₃ configurations, and only some of them are self-dual; Desargues’ is, and it is self-dual in the strong sense that the exchange lands on the same configuration rather than on a different one with the same census. That is what the concurrency measured above tests, and it is the part no counting argument delivers.

The smallest configuration where every pair of points lies on a line and every pair of lines meets in a point is 7₃, the Fano plane, and it is also self-dual — and it cannot be drawn on this page at all, because it does not exist over the real numbers. Duality is a fact about the axioms; drawability is a fact about which field the coordinates come from, and the two come apart at exactly this point.

Why the plane and not space #

Duality in the plane exchanges points and lines because both are given by three numbers. In space it exchanges points and planes, both given by four; a line in space is self-dual, and there are five numbers’ worth of them.

That is worth stating because the obvious generalisation is wrong in a way that matters here. This collection’s central object is a projection from a point onto a plane. Its spatial dual is a projection from a plane onto a point, which is not a picture of anything — so the duality that organises the picture is not a duality of the act of picturing. The symmetry lives inside the image, not between the eye and the page.

What it is worth, in practice #

Three things, and none of them is decorative.

Every construction comes in pairs, so the harder half may be the easier one to run. The quadrilateral that finds the middle builds the harmonic conjugate from four lines rather than from four points, and on a drawing whose points are hard to mark and whose edges are long, the second is the one a hand can execute.

Every fit comes in pairs too. Five points determine a conic; five tangents name the same conic, by the same six-coefficient nullspace problem with the roles exchanged. Whether that is useful is a question about conditioning rather than about duality, and it turns out to have an answer that surprised this round.

And a known theorem gives an unknown one. Pascal’s theorem about six points on a conic dualises into Brianchon’s about six tangents, and Brianchon turns out to be a test a reader can run on a finished drawing with nothing but a straightedge — which is what six tangents and the point nobody drew does to the drawing office’s four-arc ellipse.

A note on what a duality is not #

It is not a symmetry of the world. A point in a room and a line in a room are not interchangeable objects; one has a location and the other has a direction as well. What is interchangeable is their descriptions inside the projective plane, which is a smaller and stranger space than the room and is where every theorem quoted here lives.

Nor is it a symmetry of measurement. Lengths, angles and areas are all destroyed by a projection and none of them appears in any dual pair, which is not a coincidence: duality is a symmetry of the incidence structure, and incidence is exactly what a projection preserves. The catalogue of dual pairs and the catalogue of things a projection destroys are complementary lists, and between them they account for everything.

What is measured here #

Four numbers, and the fourth is the one that makes the other three mean anything.

The polarity is its own inverse to zero — dualising twice returns the point exactly, on every probe. Every incidence in a Desargues configuration survives the exchange to 5.1 × 10⁻¹⁶. The dual is another Desargues configuration, with its converse readable off the drawing at 4.3 × 10⁻¹⁶. And the incidences that were false in the original are false in the dual by 0.39, which is what says the first three numbers are a measurement rather than a description of a map that collapses everything.

The short version #

A point and a line in the projective plane are the same kind of object, and the statement that one lies on the other does not distinguish them. Choosing an invertible symmetric matrix turns that observation into a map, and the map carries every incidence theorem to another true theorem with the words exchanged.

Run on the collection’s own Desargues configuration, the thirty incidences survive to 5 × 10⁻¹⁶ and the non-incidences do not survive at all. What the map does not carry is a scale: the dual of a line through the chosen origin is a point at infinity, and a figure centred there has three of them.