2025-12-29 19:00:48
If the earth were a perfect sphere, “down” would be the direction to the center of the earth, wherever you stand. But because our planet is a bit flattened at the poles, a line perpendicular to the surface and a line to the center of the earth are not the same. They’re nearly the same because the earth is nearly a sphere, but not exactly, unless you’re at the equator or at one of the poles. Sometimes the difference matters and sometimes it does not.
From a given point on the earth’s surface, draw two lines: one straight down (i.e. perpendicular to the surface) and one straight to the center of the earth. The angle φ that the former makes with the equatorial plane is geographic latitude. The angle θ that the latter makes with the equatorial plane is geocentric latitude.
For illustration we will draw an ellipse that is far more eccentric than a polar cross-section of the earth.
At first it may not be clear why geographic latitude is defined the way it is; geocentric latitude is conceptually simpler. But geographic latitude is easier to measure: a plumb bob will show you which direction is straight down.
There may be some slight variation between the direction of a plumb bob and a perpendicular to the earth’s surface due to variations in surface gravity. However, the deviations due to gravity are a couple orders of magnitude smaller than the differences between geographic and geocentric latitude.
The conversion between the two latitudes is as follows.
Here e is eccentricity. The equations above work for any elliipsoid, but for earth in particular e² = 0.00669438.
The function atan2(y, x) returns an angle in the same quadrant as the point (x, y) whose tangent is y/x. [1]
As a quick sanity check on the equations, note that when eccentricity e is zero, i.e. in the case of a circle, φ = θ. Also, if φ = 0 then θ = φ for all eccentricity values.
Next we give a proof of the equations above.
We can parameterize an ellipse with semi-major axis a and semi-minor axis b by
The slope at a point (x(t), y(t)) is the ratio
and so the slope of a line perpendicular to the tangent, i.e tan φ, is
Now
and so
where e² = 1 − b²/a² is the eccentricity of the ellipse. Therefore
and the equations at the top of the post follow.
For the earth’s shape, e² = 0.006694 per WGS84. For small eccentricities, the difference between geographic and geocentric latitude is approximately symmetric around 45°.
But for larger values of eccentricity the asymmetry becomes more pronounced.
[1] There are a couple complications with programming language implementations of atan2. Some call the function arctan2 and some reverse the order of the arguments. More on that here.
The post The center of the earth is not straight down first appeared on John D. Cook.2025-12-29 09:23:19
One of the things I found off-putting about category theory when I was first exposed to it was its reliance on the notion of “collections” that are not sets. That seemed to place the entire theory on dubious foundations with paradoxes looming around every corner.
It turns out you can mostly ignore such issues in application. You can, for example, talk about the forgetful functor that maps a group to the set of its elements, ignoring the group structure, without having to think deeply about the collection of all sets, which Russell’s paradox tells us cannot itself be a set.
And yet issues of cardinality are not entirely avoidable. There is a theorem [1] that says in effect that category theory would be uninteresting without collections too large to be sets.
Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions.
If the collection of arrows (morphisms) between objects in C is so small as to be a set, then C is a sub-category of the category of sets. As Awodey explains, “the only special properties such categories can possess are ones that are categorically irrelevant, such as features of the objects that do not affect the arrows in any way.”
Most categories of interest have too many objects to be a set, and even more morphisms than objects.
[1] Category Theory by Steve Awodey. Theorem 1.6.
The post Interesting categories are big first appeared on John D. Cook.2025-12-27 19:14:04
One of my daughters gave me a Klein bottle for Christmas.
Imagine starting with a cylinder and joining the two ends together. This makes a torus (doughnut). But if you twist the ends before joining them, much like you twist the ends of a rectangular strip to make a Möbius strip, you get a Klein bottle. This isn’t possible to do in 3D without making the cylinder pass through itself, so you’re supposed to imagine that the part where the bottle intersects itself isn’t there.
But is a Klein bottle real? My Christmas present is a real physical object, so it’s real in that sense. Is a Klein bottle real as a mathematical object? Can it be defined without any appeal to imagining things that aren’t true? Yes it can.
Start with a unit square, the set of points (x, y) with 0 ≤ x, y ≤ 1. If you identify the top and bottom of the square, the points with y coordinate equal to 0 or 1, you get a cylinder. You can imagine curling the square in 3D and taping the top and bottom together.
Similarly, if you start with the unit square and identify the vertical sides together with a twist, you get a Möbius strip. You won’t be able to physically do this with a square, but you could with a rectangle. Or you could imagine the square to be made out of rubber, and you stretch it before you twist it and join the edges together.
If you start with the unit square and do both things described above—join the top and bottom as-is and join the sides with a twist—you get a Klein bottle. You can’t quite physically do both at the same time in 3D; you’d have to cut a little hole in the square to let part of the square pass through, as in the glass bottle at the top of the post.
Although you can’t construct a physical Klein bottle without a bit of cheating, there’s nothing wrong with the mathematical definition. There are some details that have been left out, but there’s nothing illegal about the construction.
To fill in the missing details, we have to say just what we mean by identifying points. When we identify the top edge and bottom edge of the square to make a cylinder, we mean that we imagine that for every x, (x, 0) and (x, 1) are the same point. Similarly, when we identify the sides with a twist, we imagine that for all y, (0, y) and (1, 1 − y) are the same point.
But this is unsatisfying. What does all this imagining mean? How is this any better than imagining that the hole in the glass bottle isn’t there? We can define what it means to “identify” or “glue” edges together in a way that’s perfectly rigorous.
We can say that as a set of points, the Klein bottle is
K = [0, 1) × [0, 1),
removing the top and right edge. But what makes this set of points a Klein bottle is the topology we put on it, the way we define which points are close together.
We define an ε neighborhood of a point (x, 0) to be the union of two half disks, the intersection with K of an open disk of radius ε centered at (x, 0) and the intersection with K of an open disk centered at (x, 1). This is a way to make rigorous the idea of gluing (x, 0) and (x, 1) together.
Along the same lines, we define an ε neighborhood of a point (0, y) to be the intersection with K of an open disk of radius ε centered at (0, y) and an open disk of radius ε centered at (1, 1 − y).
The discussion with coordinates is more complicated than the talk about imagining this and that, but it’s more rigorous. You can’t have simplicity and rigor at the same time, so you alternate back and forth. You think in terms of the simple visualization, but when you’re concerned that you may be saying something untrue, you go down to the detail of coordinates and prove things carefully.
Topology can seem all hand-wavy because that’s how topologist communicate. They speak in terms of twisting this and glueing that. But they have in the back of their mind that all these manipulations can be justified. The formalism may be left implicit, even in a scholarly publication, when it’s assumed that the reader could fill in the details. But when things are more subtle, the formalism is written out.
In the construction above, we define the Klein bottle as a set of points in the 2D plane with a new topology. That works, but there’s another approach. I said above that you can’t join the edges to make a Klein bottle in three dimensions. I added this disclaimer because you can join the edges without cheating if you work in higher dimensions.
If you’d like a parameterization of the Klein bottle, say because you want to calculate something, you can do that, but you’ll need to work in four dimensions. There’s more room to move around in higher dimensions, letting you do things you can’t do in three dimensions.
The post Klein bottle first appeared on John D. Cook.
2025-12-24 22:06:41
It’s been said whatever you can validate, you can automate. An AI that produces correct work 90% of the time could be very valuable, provided you have a way to identify the 10% of the cases where it is wrong. Often verifying a solution takes far less computation than finding a solution. Examples here.
Validating AI output can be tricky since the results are plausible by construction, though not always correct.
One way to validate output is to apply consistency checks. Such checks are necessary, but not sufficient, and often easy to implement. An simple consistency check might be that inputs to a transaction equal outputs. A more sophisticated consistency check might be conservation of energy or something analogous to it.
Some problems have certificates, ways of verifying that a calculation is correct that can be evaluated with far less effort than finding the solution that they verify. I’ve written about certificates in the context of optimization, solving equations, and finding prime numbers.
Correctness is more important in some contexts than others. If a recommendation engine makes a bad recommendation once in a while, the cost is a lower probability of conversion in a few instances. If an aircraft collision avoidance system makes an occasional error, the consequences could be catastrophic.
When the cost of errors is extremely high, formal verification may be worthwhile. Formal correctness proofs using something like Lean or Rocq are extremely tedious and expensive to create, and hence not economical. But if an AI can generate a result and a formal proof of correctness, hurrah!
But if an AI result can be wrong, why couldn’t a formal proof generated to defend the result also be wrong? As the Roman poet Juvenal asked, Quis custodiet ipsos custodes? Who will watch the watchmen?
An AI could indeed generate an incorrect proof, but if it does, the proof assistant will reject it. So the answer to who will watch Claude, Gemini, and ChatGPT is Lean, Rocq, and Isabelle.
Isn’t it possible that a theorem prover like Rocq could have a bug? Of course it’s possible; there is no absolute certainty under the sun. But hundreds of PhD-years of work have gone into Rocq (formerly Coq) and so bugs in the kernel of that system are very unlikely. The rest of the system is bootstrapped, verified by the kernel.
Even so, an error in the theorem prover does not mean an error in the original result. For an incorrect result to slip through, the AI-generated proof would have to be wrong in a way that happens to exploit an unknown error in the theorem prover. It is far more likely that you’re trying to prove the wrong thing than that the theorem prover let you down.
I mentioned collision avoidance software above. I looked into collision avoidance software when I did some work for Amazon’s drone program. The software that was formally verified was also unrealistic in its assumptions. The software was guaranteed to work correctly, if two objects are flying at precisely constant velocity at precisely the same altitude etc. If everything were operating according to geometrically perfect assumptions, there would be no need for collision avoidance software.
The post Automation and Validation first appeared on John D. Cook.2025-12-24 01:03:58
Newton’s birthday was on Christmas when he was born, but now his birthday is not.
When Newton was born, England was still using the Julian calendar, and would continue to use the Julian calendar until 25 years after his death.
On the day of Newton’s birth, his parents would have said the date was December 25, 1642. We would now describe the date as January 4, 1643.
You’ll sometimes see Newton’s birthday written as December 25, 1642 O.S. The “O.S.” stands for “Old Style,” i.e. Julian calendar. Of course the Newton family would not have written O.S. because there was no old style until the new style (i.e. Gregorian calendar) was adopted, just as nobody living in the years before Christ would have written a date as B.C.
In a nutshell, the Julian year was too long, which made it drift out of sync with the astronomical calendar. The Julian year was 365 1/4 days, whereas the Gregorian calendar has 365 97/400 days, which more closely matches the time it takes Earth to orbit the sun. Removing three Leap Days (in centuries not divisible by 400) put the calendar back in sync. When countries adopted the Gregorian calendar, they had to retroactively remove excess Leap Days. That’s why Newton’s birthday got moved up 10 days.
You can read more on the Julian and Gregorian calendars here.
The winter solstice in the northern hemisphere was two days ago: December 21, 2025. And in 1642, using the Gregorian calendar, the solstice was also on December 21. But in England, in 1642, people would have said the solstice occurred on December 11, because the civil calendar was 10 days ahead of the astronomical calendar.
The post When was Newton born? first appeared on John D. Cook.2025-12-23 23:57:34
A few weeks ago I mentioned that I was reading Stephen Ambrose’s account of the Lewis & Clark expedition and wrote a post about their astronomical measurements. James Campbell left a comment recommending Edwin Danson’s book [1] on the history of the Mason-Dixon line. I ordered the book, and now that work has slowed down for Christmas I have had the time to open it.
In addition to determining their eponymous line, surveyors Charles Mason and Jeremiah Dixon were also the first to measure a degree of latitude in 1767.
What exactly did they measure? We’ll get to that, but first we need some background.
To first approximation a degree of latitude is simply 1/360th of the Earth’s circumference, but Mason and Dixon were more accurate than that. Isaac Newton (1643–1727) deduced that our planet was not a perfect sphere but rather an oblate spheroid. The best measurement in Mason and Dixon’s time was that the Earth’s semi-major axis was 6,397,300 meters with flattening 1/f = 216.8.
It’s a bit of an anachronism to describe the distance in meters since the meter was defined in 1791. The meter was originally defined as one ten-millionth of the distance from the equator to the North Pole along a great circle through Paris.
If the Earth were a perfect sphere, a degree of latitude would be 1/360th of its circumference. Using the original definition of the meter, this would be exactly 10,000,000/360 meters. But because the Earth is not a perfect sphere, each degree of latitude has a slightly different length. To put it another way, the length of a degree of latitude varies by latitude.
Another complication due to the flattening of the Earth is that there are multiple ways to define latitude. The two most common are geocentric and geodetic. The geocentric latitude of a point P on the Earth’s surface is the angle between the equatorial plane and a line between the center of the earth and P. The geodetic latitude (a.k.a. geographic latitude) of P is the angle between the equatorial plane and a line perpendicular to the Earth’s surface at P. More on the difference between geocentric and geodetic latitude here.
Since the length of a degree of latitude varies, we need to say at what latitude they measured the length of a degree. In short, they measured the length of a degree near what we now know as the Mason-Dixon line, the border between Pennsylvania and Maryland.
To be more precise, the starting point was Stargazer’s Stone, a stone placed by Mason and Dixon on John Harland’s farm near Embreeville, Pennsylvania, to a point about a degree and a half due south near what is now Delmar, a town on the Deleware / Maryland border.
I’ve had some difficulty determining how accurate Mason and Dixon were. Some sources I’ve found are obviously wrong. I haven’t verified this, but it seems Mason and Dixon overestimated the length of a degree of latitude at their location by only 465.55 ft or about 0.13%, a remarkable feat given the technology of their day.
[1] Edwin Danson. Drawing The Line: How Mason and Dixon Surveyed the Most Famous Border in America. John Wiley & Sons. 2001.
The post Mason, Dixon, and Latitude first appeared on John D. Cook.
