2026-11-09 07:23:49
The idea of finite geometries sounds like an oxymoron the first time you hear of it, something like dry water or cold fire, but the axioms of some kinds of geometry can indeed be satisfied by finite models.
The axioms of projective plane geometry are as follows.
The most famous example of a finite projective plane is the Fano plane.
It helps to switch back and forth between thinking of finite geometries geometrically and in terms of set theory. The fact that the Fano plane has an easy-to-understand illustration is fortunate, but not to be taken for granted. Also, it could be misleading to lean to heavily on the fact that the lines and points of finite geometry can, in this case, be represented by points and line segments (and a circle) in the Euclidean plane.
The set-theoretic perspective is that the points are the one-element sets {1}, {2}, …, {7}, and the lines are the sets listed in the legend of the graph: {1, 2, 5}, {1, 3, 7}, …, {5, 6, 7}. It’s simple to confirm that any pair of points lines in exactly one of the three-element sets, that the intersection of any pair of the three-element sets is a one-element set, and that there are three points—1, 2, 3, and 4 will do—such that no three line on the same line, i.e. no three belong to the same three-element set.
The Fano plane is the unique projective plane with 7 points, but there are four non-equivalent projective planes with 10 points.
There are many other examples of non-isomorphic finite projective planes with the same number of points, but there are also many unsettled conjectures, at least as of an hour ago, regarding the existence of finite projective planes of various orders.
Finite geometry might seem like a technicality, spaces that satisfy the axioms but are not what anyone had in mind when the axioms were codified. So why care about finite geometries? For one thing, unexpected examples are quite valuable since they help separate the consequences of a set of axioms from common expectations.
For another, finite geometries come up in applications, such as combinatorics and cryptography. These discrete applications take advantage of intuition from traditional geometry. This intuition is an imperfect guide, so conclusions need to be verified by concrete calculations, but geometric intuition suggests what one ought to verify.
2026-11-03 22:41:25
A few years ago I wrote about a curious result: a/b is usually greater than c/d when a + d is greater than b + c. And “usually” means with probability 11/12.
As stated in the post, you can make this statement rigorous by choosing integers up to size N at random and taking the limit as N goes to infinity.
We can also choose a, b, c, and d uniformly from [0, 1]. This is similar to choosing four integers up to N and dividing all the integers by N, which doesn’t change the probabilities.
Now a/b > c/d if and only if ad > bc, so another way to state the theorem is that
Prob(ad > bc | a + d > b + c) = 11/12.
Or to state it yet another way, pick two pairs of numbers at random. The probability that the pair with the larger sum also has the larger product is 11/12. That seems more intuitive to me than the original statement.
If a + d is greater than b + c, it’s likely that a/b is greater than c/d. It seems plausible that if a + d is much greater than b + c, it’s more likely that a/b is greater than c/d. Or in terms of cross products, the more the sum of one pair is bigger than the sum of the other, the more likely the product of the former is to be greater than the latter. That is
f(r) = Prob(ad > bc | (a + d)/(b + c) ≥ r)
should be an increasing function of r. We know the probability above equals 11/12 when r = 1. We expect that the probability should approach 1 as r goes to infinity.
This is indeed the case, as shown in the plot below.
In particular, f(2) ≈ 0.99 and f(6) ≈ 0.999. So if a + d is two (or six) times bigger than b + c, there is a 0.99 (or 0.999) probability that ad is larger than bc.
You can calculate f(r) analytically or estimate it by simulation. I had Claude do both and they produce comparable results. There is a closed-form expression for f(r) but it’s really complicated.
The post Fraction comparison trick revisited first appeared on John D. Cook.2026-11-03 20:33:10
Spherical geometry is very tangible because we live on a sphere. Projective geometry seems much more exotic, and yet they’re closely related. To show how they’re connected, let’s go back to the origins of non-Euclidean geometry.
Here are Euclid’s five postulates, statements about what geometric objects can be constructred.
Clearly there’s something different about the fifth postulate, the parallel postulate. It’s much longer and more complicated than the other postulates. There was speculation from ancient times that the parallel postulate might be unnecessary, that it could be derived as a theorem rather than stated as an axiom.
Giovanni Girolamo Saccheri (1667–1733) tried to prove the fifth postulate by contradiction, and in the process began the study of non-Euclidean geometry. He wasn’t the first to try to prove the fifth postulate. Ptolemy, for example, tried to prove the fifth postulate as a theorem in the second century AD. But Saccheri was the first to try a proof by contradiction, or at least the first to extensively explore contradictions to the parallel postulate.
A more modern formulation of the parallel postulate says
Given a line ℓ and a point P not on ℓ, there is exactly one line through P parallel to ℓ.
Non-Euclidean geometry replaces this axiom with an alternative, such as saying there are multiple lines parallel to ℓ or that there are no lines parallel to ℓ.
Note that non-Euclidean geometry is not an arbitrary tinkering with the axioms of geometry just to see what happens. It tinkers with one specific axiom for motivated reasons.
You can create models of non-Euclidean geometry using Euclidean geometry. For example, spherical geometry is constructed by defining a (spherical) point as a (Euclidean) point on a sphere, and defining a (spherical) line to be a great circle, the intersection of a (Euclidean) plane with the sphere. In spherical geometry there are no parallel lines: all lines intersect.
This example shows that non-Euclidean geometry need not be non-physical. If you leave general relativity out of the picture and think of physical space as undeniably Euclidean, it can still be useful to think of different geometries to describe objects in Euclidean space.
In projective geometry, all distinct lines intersect at one point.
In spherical geometry, all distinct lines intersect at two points, on opposite sides of a great circle. If you consider antipodal pairs of points on a sphere to be a single point in the projective plane, now all lines intersect at a single point. Projective geometry is just one step away from spherical geometry.
The recent thread of posts started with a post a couple weeks ago about two versions of the law of cosines on a sphere, one for sides and one for angles. You can go from one to the other by going from a spherical triangle to its polar triangle. That started me going down the rabbit hole of poles and polars, first in spherical geometry, then in plane geometry, then in projective geometry, culminating here.
Now we have two ways of producing poles and polars on a sphere: the original construction in terms of spherical geometry, and the projective construction. They are consistent: you can do spherical geometry, then identify antipodal points, or you can identify antipodal points, then do projective geometry. You end up at the same place by either path.
One theme in the last few posts has been that projective geometry is not an arbitrary exercise in abstraction but a framework that can simplify how we think about things. For example, when I defined the polar triangle of a spherical triangle here, I said that the vertices of the polar triangle are the poles of the sides of the original triangle. Imagine the side as the equator of the sphere and take one of the poles; that’s where the abstract concept of “pole” comes from. But which pole do you take? There is a convention for answering this question, but in projective geometry it doesn’t matter: both (spherical) poles constitute one (projective) point.
2026-11-03 18:45:19
This post will show how a graphic programmer’s hack could lead to a rigorous definition of the projective plane.
As mentioned in the previous post, we start by adding some “points at infinity” to Euclidean plane to take care of exceptional situations. For example, instead of saying most pairs of lines intersect at a single point but some don’t intersect at all, you could say all pairs of lines intersect at a point, but for some pairs of lines, namely parallel lines, that intersection point is “at infinity.”
Suppose you need to write a program to find the intersection of two lines. The usual way of doing this would be to implement a partial function that takes two lines as input, and sometimes returns a point of intersection and sometimes does something else: it crashes, or throws an exception, or hangs. But if we add in points at infinity, our program could always return a point.
But how are you going to represent points at infinity? It’s not enough to use a floating point infinity, because different lines intersect at different points at infinity. For example, the lines y = 1 and y = 2 intersect at the point at infinity in the horizontal direction, but the lines x = 1 and x = 2 intersect at the point at infinity in the vertical direction. You could say the former lines intersect at (o, inf) and the latter at (inf, 0), but that won’t work in general.
The lines y = 3x + 1 and y = 3x + 1 intersect at the point at infinity with slope 3, but the lines y = 5x + 1 and y = 5x + 1 intersect at the point at infinity with slope 5. If you say both pairs of lines meet at (inf, inf) you’ve lost information.
So you come up with a hack. You represent points not as (x, y) pairs but as (x, y, f) where x and y are real numbers but f is a Boolean flag. Now (x, y, true) represents the ordinary point (x, y) and (x, y, false) represents the point at infinity in the direction of the line from the origin to (x, y). Problem solved!
After a while, you realize that this hack can be useful for more than just finding the intersection of lines. You discover that if you use a real number instead of a Boolean for the last coordinate, you can implement translations and rotations using matrix multiplication. So you replace true and false with 1 and 0, do your matrix calculation, then divide by the last coordinate so the result ends in 1 or 0 again. And this works really well.
This last coordinate is starting to act more like the other coordinates. You treat it as a peer when you multiply triples by a matrix, but only temporarily. You immediately normalize at the end to get back to more familiar territory.
Maybe it occurs to you after a while that you’re doing something analogous the introduction of imaginary numbers. At first the square root of −1 was seen as a hack that produced correct answers. It wasn’t really a number but something imagined to be a number as an aid to calculation. Over time mathematicians got more comfortable with these imaginary numbers and explored what would happen if they were treated as first class citizens. Maybe you should do the same with this third coordinate that started out as a hack.
This leads you to homogenous coordinates. A point in the extended plane you’ve created is represented by a triple (x, y, z) with the understanding that given a number λ ≠ 0, (x, y, z) and (λx, λy, λz) represent the same point. This is the natural extension of always dividing by the last coordinate, if it’s not zero, so that the triple always ends in 1. And you discover after a while that you need to add in the requirement that not all the coordinates can be zero at the same time.
You show this to a mathematician friend and he puts your construction in fancier language. He says you’re extended plane is ℝ³ minus the origin, modulo non-zero scalar multiplication. In other words, the points in your extended plane are equivalence classes of points in ℝ³.
Now if you think about this geometrically, your equivalence classes are lines through the origin because that’s what you get when you look at all scalar multiples of a given vector. What started out as the plane with extra points added is now the set of all lines though the origin. That’s one way to define the projective plane: a point in the projective plane corresponds to a line through the origin in ℝ³. Furthermore, a line in the projective plane corresponds to a plane through the origin in ℝ³.
Lines through the origin are determined by where they cross the unit sphere, so you could think of the projective plane as a sphere with one modification: because a line through the origin crosses the sphere in two antipodal points, we consider those two points in ℝ³ to be a single point in the projective plane.
The post How a programming might reinvent the projective plane first appeared on John D. Cook.2026-11-02 22:44:02
In my newsletter I give an introduction to the latest few posts and occasionally give some additional commentary. In my latest newsletter I wrote
Projective space and homogeneous coordinates seemed like arbitrary abstractions when I was a student because they were presented without any motivation. Much later I came to realize that they’re natural, almost inevitable. Exceptional cases and hand-wavy references to objects “at infinity” go away when moving to projective space.
My mathematics education was very top-down: abstractions first, specifics later (if ever). I was OK with that for a while, until I hit my abstraction tolerance sometime in grad school. Now I would advocate more of a bottom-up approach, motivating abstractions first rather than presenting them as a fait accompli.
The first time I saw the projective plane was probably in a topology class where someone said you could construct the plane by starting with a square and gluing opposite edges in a physically impossible way. This seemed like a gimmick, a way to construct a non-orientable manifold but not anything practical. Another construction is to say the projective plane is an ordinary sphere with antipodal points identified. I found that a little more satisfying, but it still unmotivated.
What I would have appreciated was someone saying that the projective plane was invented to make geometry more elegant by eliminating exceptional cases and making rigorous the idea of things “at infinity.” For example, geometric calculations often have to treat vertical lines specially, but vertical lines are not special in projective geometry.
You could start by saying we’re going to represent points in the plane not simply as (x, y) but as (x, y, 1) and use (x, y, 0) for the point at infinity in the direction (x, y). That may seem like a hack, but it’s certainly motivated. It’s exactly the sort of thing a graphics programmer would do. In fact, I imagine many programmers have done just that, independently bumping into projective geometry.
After having introduced this extra coordinate, it’s natural to ask what would it mean if the last coordinate were something other than 0 or 1. You’ll quickly run into this when you start doing calculations on these extended coordinates. You could have a rule that you always normalize by dividing by the last coordinate, if it’s not zero, and then you have essentially discovered homogeneous coordinates. (Update; I develop this idea in more detail in the next post.)
You’ll see something like the approach above in elliptic curve cryptography. Programmers wanting to implement encryption algorithms are thinking concretely in terms of (x, y) points, until they have to worry about a mysterious “point at infinity” and handle it correctly. Then homogenous coordinates appear as a tool for getting out of a bind.
***
I thought about titling this post something like “Dr. Strangelove, or: How I Learned to Stop Worrying and and Love the Projective Plane” but I wasn’t sure how well that would go over. I doubt many readers have seen the 1964 movie this is alluding to, but the trope “How I Learned to Stop Worrying and Love …” is more widely known than the movie itself.
The post Making peace with the projective plane first appeared on John D. Cook.2026-11-02 04:29:54
I wrote several posts on poles and polars a few days ago. First, I wrote about the dual of a spherical triangle, called the polar triangle. Then I wrote about poles and polars in plane geometry and briefly mentioned the extension to projective geometry.
Poles and polars are a little awkward in the plane because you have to pay attention to special cases. Poles and polars are relative to a conic section, and are defined/computed differently depending on whether you’re taking the dual with respect to an ellipse, parabola, or hyperbola. Also, the polar of the center of a conic section isn’t defined.
If you move from the Euclidean plane to the projective plane, the exceptions go away, though it may not seem so if you’re talking about points at infinity or lines at infinity. If you fully commit to working in projective space, there aren’t any hand-wavy points or lines “at infinity.” There’s not even a distinction between ellipses, parabolas, and hyperbolas.
There’s no such thing as a free lunch, and the cost of this elegant unification is moving to a bigger geometric space with a different coordinate system. Instead of working in the Euclidean spaces R² or Rn, we work in Pn over a field F which could be R but doesn’t need to be [1].
The coordinates on Pn are homogeneous coordinates, (n + 1)-tuples of field elements, not all equal to zero, with two (n + 1)-tuples representing the same point if they are proportional, i.e. one is a non-zero multiple of the other.
We can embed Rn in Pn by adding an extra coordinate on the end, set equal to 1. You could think of this extra coordinate as a switch, set to 1 for finite points and set to 0 for infinite points. But this distinction between finite and infinite points is an artifact of the embedding, not a property of Pn: there are no special points in Pn. A point that is “at infinity” in one embedding of Rn in Pn need not be in another embedding.
Once you accept the need to work in homogeneous coordinates, everything becomes simple and consistent.
A quadric (a conic section when n = 2) is the set of points
{x : xᵀAx = 0}
where A is a non-singular (n + 1) by (n + 1) symmetric matrix over F.
The polar of a point p is the hyperplane
{x : pᵀAx = 0}
and the pole of a hyperplane
H = {x : uᵀx = 0}
is the point
p = A−1u.
Notice there’s no mention of “infinity” in this section. All the calculations take place in the field F with no points at infinity added.
[1] The only requirement is that the field F cannot have characteristic 2. So F could be the reals, the rationals, the complex numbers, etc. F could even be a finite field as long as the number of elements in F is not a power of 2.
The post A unified view of poles and polars first appeared on John D. Cook.
