Math

Fraction comparison trick revisited

Posted on by John

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.

Euclidean, Spherical, and Projective Geometry

Posted on by John

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.

Euclidean geometry

Here are Euclid’s five postulates, statements about what geometric objects can be constructred.

  1. To draw a straight line from any point to any point.
  2. To produce (extend) a finite straight line continuously in a straight line.
  3. To describe a circle with any centre and distance (radius).
  4. That all right angles are equal to one another.
  5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

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.

Non-Euclidean geometry

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.

Spherical geometry

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.

Projective geometry

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.

Posts in this series

How a programming might reinvent the projective plane

Posted on by John

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.

Making peace with the projective plane

Posted on by John

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.

A unified view of poles and polars

Posted on by John

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.

Projective space

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.

Poles and polars

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−1 u.

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 Unbalanced Group Theorem

Posted on by John

The classification of finite simple groups (CFSG) has been called the biggest theorem in mathematics. It is big, not necessarily in terms of its importance, but in the sense that its proof developed over the course of decades and thousands of pages of journal articles.

Two days ago Axiom announced it had formalized the proof of CFSG in 43.1 million lines of Lean. If you click on the link for the accompanying paper, you’ll find that the title of the paper is The Unbalanced Group Theorem, not The Classification of Finite Simple Groups. Why is that?

The unbalanced group theorem (UGT) is a key component of the CFSG. In the course of their effort to formalize CFSG Axiom found counterexamples to an assertion that the 1986 proof of UGT depends on. The UGT, and the CFSG that depends on it, is correct, but the proof was flawed. Axiom formalized the entire CFSG but focused their paper on their new proof of the UGT.

***

What exactly does the UGT say? For that matter, what is a finite simple group?

A short post like this can only go into so much detail, but I’ll see what I can do.

A group represents some kind of symmetry. Groups can be built out of simple groups sorta like the way integers can be built out of primes. See more on the comparison between simple groups and prime numbers here.

The CFSG says that every finite simple group is cyclic, alternating, of Lie type, or sporadic. The UGT is a technical result used in proving that nothing is missing from this list of possibilities.

What does it mean for a group to be “unbalanced”? This depends on a few preliminary definitions.

  • Define O(G) to be the largest normal subgroup of G with an odd number of elements.
  • An element t in G is called an involution if it has order 2: t is not the identity but t² is.
  • The centralizer of an element t of a group G, written CG(t), is the set of elements of G that commute with t.

A group G is unbalanced if O(CG(t)) is a not a subgroup of O(G) for some involution t.

The UGT is a classification theorem for unbalanced groups.

Related posts

Inversion in an ellipse

Posted on by John

The previous three post have been part of a series on geometric duality. The traditional but somewhat mysterious terminology is the pole of a line and the polar of a point. As explained here, the terminology was first coined in the context of spherical geometry where the term “pole” makes sense. It was then extended to the plane where the meaning is less obvious.

The use of poles and polars is that they preserve certain geometric information, and so they can transform one problem into a dual problem. This was illustrated in the first post in the series. Equations in spherical trigonometry have a buy-one-get-one-free property: applying an identity to the dual of a triangle gives another identity.

This post discussed how to define the pole of a line and the polar of a point with respect to a circle in the plane. The post stated that more generally you can do this in the projective plane. Since there’s no difference between conic sections in the projective plane, anything you can do for a circle you can do for an ellipse or hyperbola.

Given a point P, we can define its polar line geometrically or algebraically.

Algebraic approach

Given an ellipse with equation

x²/a² + y²/b² = 1

where a > b, the foci are at ± c where c = √(a² − b²).

For a point P = (x0, y0), its polar with respect to the ellipse is the line

(x0/a²) x + (y0/b²) y = 1.

Geometric approach

To find the polar of a point P, we start by inverting P in the ellipse. Draw any line through P that meets the ellipse at two points A and B. On that line there is exactly one point Q that divides AB harmonically with P:

QA/QB = PA/PB

with Q on the opposite side of AB from P if P is inside the ellipse, or between A and B if P is outside. The polar is the line through Q parallel to the tangent at R, where R is the point where the ray OP meets the ellipse.

Focus and directrix

Plane and projective geometry are very mature and so some of the terminology sounds quaint. One such term is the directrix of a conic section, explained here. The polar of a focus is a directrix, and the pole of a directrix is a focus.

Related posts

Poles and polars

Posted on by John

The previous two posts have been about polar triangles. The first used a polar triangle to derive a theorem, and the second showed how to compute polars using linear algebra. But what exactly is a polar triangle and why is it called that?

Poles and polars on a sphere

Let’s start with a sphere S and a great circle G. Then G is the intersection of S with a plane through the center of S. The circle G has two poles, two points on opposite sides of S that are as far as possible (i.e. π/2 radians) from G. If you rotate the sphere so that G is the equator, the poles are the north and south pole.

Conversely, if you start with a point P on the sphere S, the polar of P is the great circle for which P is a pole. So if you think of P as the north pole, the polar of P is the equator. The term equator would have been more intuitive than polar, but the terminology was established a couple centuries ago.

In my opinion it would be clearer to call poles and polars both duals. The dual of a great circle is a point, and the dual of a point is a great circle.

Now let’s look at a spherical triangle ABC sides a, b, and c. The vertices of the polar triangle of ABC are the poles of the sides a, b, and c. This definition is essentially correct, but it needs a little refinement.

The pole A′ of the side a is one of the two poles of the great circle containing a, the one that that lies in the same hemisphere as A. Define B′ and C′ similarly, and the polar triangle is the spherical triangle with vertices A′B′C′.

Poles and polars in the plane

The terms pole and polar were coined for spherical geometry, then generalized to plane (projective) geometry. If you think of great circles as the lines of spherical geometry, then you can carry the idea of poles and polars over to plane geometry where the pole (dual) of a line is a point, and the polar (dual) of a point is a line.

However, in the plane you can’t define the pole of a line without more context. You have to start with a circle (or more generally a conic section) and define the pole of a line with respect to the circle and the define the polar of a point with respect to the same circle.

Given a circle S with center O, to find the polar of a point P ≠ O, let Q be the inversion of P in S, and let ℓ be the line through Q perpendicular to the line OP.

Conversely, the pole of a line ℓ with respect to a circle S is the inversion of Q in S where Q is the intersection of ℓ with OQ. Here we assume the line ℓ does not pass through the center of the circle O because otherwise O = Q and it doesn’t make sense to speak of the line OQ.

Projective geometry

The definitions above have a couple holes that can be filled in by extending the Euclidean plane to the projective plane.

We defined the pole of a line ℓ except when ℓ passes through the center of the circle. If ℓ passes through O then the pole of ℓ is the point at infinity in the direction of ℓ. Similarly, we defined the polar of all points except O. In projective geometry, we can define the polar of O to the be the line at infinity, the line containing all the points at infinity.

Computing polar triangles with linear algebra

Posted on by John

The previous post defined the polar triangle of a spherical triangle in terms of its sides and angles, but it doesn’t construct the triangle. It only specifies one of a family of triangles located anywhere on the sphere.

How do we know that polar triangles exist? There is a geometric construction that starts with a spherical triangle and constructs a unique new spherical triangle, and one can show that this new triangle has the angles and sides described in the previous post.

There is also a linear algebraic construction of polar triangles, and that’s the topic of this post. In a nutshell, take your sphere to be the unit sphere in 3D. Write the coordinates of each vertex as the columns of a matrix. Invert that matrix and normalize each row. Then the rows are the coordinates of the polar vertices.

This process is an involution: the polar of the polar of a triangle is the original triangle.

The following Python code illustrates the process described above. It begins with a triangle on the globe with vertices A, B, and C at Chicago, Quito, and Paris respectively. It computes the polar triangle, then computes the polar of that, and shows that the result is the original triangle.

import numpy as np

def latlon_to_cartesian(lat_deg, lon_deg):
    lat = np.radians(lat_deg)
    lon = np.radians(lon_deg)
    x = np.cos(lat) * np.cos(lon)
    y = np.cos(lat) * np.sin(lon)
    z = np.sin(lat)
    return np.array([x, y, z])

def cartesian_to_latlon(v):
    v = v / np.linalg.norm(v)
    lat = np.degrees(np.arcsin(np.clip(v[2], -1, 1)))
    lon = np.degrees(np.arctan2(v[1], v[0]))
    return lat, lon

def polar_triangle(M):
    """
    Polar triangle via matrix inversion.
    M: 3×3 matrix with columns = unit vectors of the original vertices.
    Returns 3×3 matrix whose columns are the polar unit vectors.
    """
    Minv = np.linalg.inv(M)
    # Normalize each row of the inverse (rows are parallel to the polar directions)
    polar_rows = Minv / np.linalg.norm(Minv, axis=1, keepdims=True)
    return polar_rows.T          # columns = polar vertices

# Original vertices (Chicago, Quito, Paris)
A = latlon_to_cartesian(41.85003, -87.65005)
B = latlon_to_cartesian(-0.22985, -78.52495)
C = latlon_to_cartesian(48.85341,  2.3488)

M = np.column_stack([A, B, C])
print("M (columns = original vertices):\n", M)
print("det(M) =", np.linalg.det(M))

# Polar triangle
M_prime = polar_triangle(M)
print("\nM' (columns = polar vertices):\n", M_prime)

print("\nPolar geographic coordinates (lat, lon):")
for name, col in zip(["A'", "B'", "C'"], range(3)):
    print(name, cartesian_to_latlon(M_prime[:, col]))

# Polar of the polar
M_double = polar_triangle(M_prime)
print("\nPolar-of-polar matrix:\n", M_double)
print("max |difference from original M| =",
      np.max(np.abs(M_double - M)))

Duality in spherical trigonometry

Posted on by John

I’ve written before about the law of cosines on a sphere:

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C).

But that’s just one of two laws of cosines on a sphere. The equation above is the law of cosines for sides. The law of cosines for angles is typographically very similar, just inverting the case of the letters used as arguments.

cos(C) = cos(A) cos(B) + sin(A) sin(B) cos(c).

Incidentally, the law of cosines for sides shows how the sides of a spherical triangle determine the angles. The law of cosines for angles shows how, unlike in the case of plane geometry, angles determine the sides.

Two kinds of angles

Let’s back up and say what the letters refer to. As is conventional in plane geometry, the sides of a triangle are a, b, and c, and the angles are A, B, and C, with a being the side opposite the angle A, and similarly for the other letters.

However, unlike plane geometry, the sides are also angles, but of a different sort. The sides a, b, and c are central angles, and the “angles” A, B, and C are dihedral angles.

The sides are arcs of a great circle. The side a, for example, is a portion of a great circle, running between the vertices at B and C [1]. Think of a plane that contains the center of the sphere O and the vertices B and C. The length of a is the angle at O.

The angles A, B, and C are the angles at which great circles meet. In the formalism of differential geometry, these angles exist in the tangent plane at the vertices. These angles are intrinsic, measured along the (tangent plane of) the surface of the sphere. The central angles are extrinsic, measured in a plane that cuts through the sphere to its center.

Polar triangles

Now because sides are angles and vertices are angles, we can interchange the two kinds of angles to form the polar triangle of the original triangle. This is a new triangle whose central angles are the complements of the original dihedral angles, and whose dihedral angles are the complements of the original central angles. In symbols,

a′ = π − A
b′ = π − B
c′ = π − C
A′ = π − a
B′ = π − b
C′ = π − c

Now we return to the laws of cosines at the top of the post: the law of cosines for angles is the law of cosines for sides applied to the polar triangle. Every equation in spherical trigonometry has a polar counterpart.

 

[1] There’s some overloaded terminology here, just as in plane geometry. The symbol a is used to denote a side of a triangle and also the length of that side. The symbol A is used to denote both a vertex and the dihedral angle at that vertex.