NASA’s Kepler mission has discovered a planet orbiting two stars, something like Tatooine in Star Wars. However, unlike Tatooine, this planet is a gas giant about the size and mass of Saturn. But if you had a place to stand near the surface of this planet, you might see a sunset something like the one Luke Skywalker saw.
Source: Science Daily.
Don Fredkin left a comment on my previous blog post that surprised me. I found out that latitude doesn’t exactly mean what I thought.
Imagine a line connecting your location with the center of the Earth. I thought that your latitude would be the angle that that line makes with the plane of the equator. And that’s almost true, but not quite.
Instead, you should imagine a line perpendicular to the Earth’s surface at your location and take the angle that that line makes with the plane of the equator.
If the Earth were perfectly spherical, the two lines would be identical and so the two angles would be identical. But since the Earth is an oblate spheroid (i.e. its cross-section is an ellipse) the two are not quite the same.
The angle I had in mind is the geocentric latitude ψ. The angle made by a perpendicular line and the plane of the equator is the geographic latitude φ, also known as the astronomical latitude. The following drawing from Wikipedia illustrates the difference, exaggerating the eccentricity of the ellipse.
How do these two ideas of latitude compare? I’ll sketch a derivation for equations relating geographic latitude φ and geocentric latitude ψ.
Let f(x, y) = (x/a)2 + (y/b)2 where a = 6378.1 km is the equatorial radius and b = 6356.8 km is the polar radius.. The gradient of f is perpendicular to the ellipse given by the level set f(x, y) = 1. At geocentric latitude ψ, y = tan(ψ) x and so the gradient is proportional to (1/a2, tan(ψ) / b2). From taking the dot product with (1, 0) it follows that the cosine of φ is given by
(1 + (a/b)4 tan2 ψ)-1/2.
It follows that
φ = tan-1( (a/b)2 tan ψ )
and
ψ = tan-1( (b/a)2 tan φ ).
The geocentric and geographic latitude are equal at the poles and equator. Between these extremes, geographic latitude is larger than geocentric latitude, but never by more than 0.2 degrees. The maximum difference, as you might guess, is near 45 degrees.
Here’s a graph of φ – ψ in degrees. The difference between these two angles is called the angle of vertical.
The maximum occurs at 44.9 degrees and equals 0.1917.
The curve looks very much like a parabola, and indeed it is. The approximation
φ = ψ + 0.186 – 0.0000946667 (ψ – 45)2
is very accurate, within about 0.005 degrees.
Related post: Journey away from the center of the Earth
What point on Earth is farthest from its center? Mt. Everest comes to mind, but its summit is the point highest above sea level, not the point farthest from the center. These are not the same because the Earth is not perfectly spherical.
Our planet bulges slightly at the equator due to its rotation. The equatorial diameter is about 0.3% larger than the polar diameter. Sea level at the equator is about 20 kilometers farther from the center of the Earth than sea level at the poles.
Mt. Everest is about nine kilometers above sea level and located about 28 degrees north of the equator. Chimborazo, the highest point in Ecuador, is seven kilometers above sea level and 1.5 degrees south of the equator.
So how far are Mt. Everest and Chimborazo from the center of the Earth? To answer that, we first need to how far sea level at latitude θ is from the center of the Earth.
Imagine slicing the Earth with a plane containing its polar diameter. To a good approximation (within 100 meters) the resulting shape would be an ellipse. The equation of this ellipse is
(x / a)2 + (y / b)2 = 1
where a = 6378.1 km is the equatorial radius and b = 6356.8 km is the polar radius. A line from the center of the ellipse to a point at latitude θ has equation y = tan(θ) x. Solving the pair of equations for x shows that the distance from the center to the point at latitude θ is
d = sqrt( a2b2 sec2 θ / (a2 tan2 θ + b2 ) )
For Mt. Everest, θ = 27.99 degrees and so d = 6373.4. For Chimborazo, θ = -1.486 degrees and so d = 6378.1. So sea level is 4.7 km higher at Chimborazo. Mt. Everest is 2.6 km taller, but the summit of Chimborazo is about 2.1 km farther away from the center of the Earth.
Update: See my next post for a slight correction. A more accurate calculation would compute sea level is about 4.65 km higher at Chimborazo than Mt. Everest.
I’ve started to read through Michael Fogus’ list of recommended technical papers and ran across this gem from Out of the Tar Pit:
Complexity breeds complexity. There are a whole set of secondary causes of complexity. This covers all complexity introduced as a result of not being able to clearly understand a system. Duplication is a prime example of this — if (due to state, control or code volume) it is not clear that functionality already exists, or it is too complex to understand whether what exists does exactly what is required, there will be a strong tendency to duplicate. This is particularly true in the presence of time pressures.
Within a few hours of reading that quote I had a conversation illustrating it. I talked with someone who needed to make a small change to a complex section of code. He said the code had three minor variations on the same chunk of functionality. He could get his job done much faster in the short term if he simply added a forth mutation to the code base. He refused to do that, but many developers would not refuse.
Suppose the rate of growth in complexity of a project is proportional to how complex the project is. And suppose, as the quote above suggests, that the proportionality constant is the time pressure. Then the complexity over time is given by
y‘(t) = a y(t)
where y(t) is complexity at time t and a is the time pressure. Then complexity grows exponentially. The solution to the equation is
y(t) = y0 eat
where y0 is the initial complexity. This isn’t meant to be an exact model, just a back-of-the-envelope illustration. On the other hand, I’ve seen situations where it gives a fairly good description of a project for a while. Complexity can grow exponentially like compound interest, and the greater the pressure, the greater the rate of compounding.
Now suppose there’s a different kind of time pressure, a pressure to simplify a project. This would correspond to a negative value of the proportionality constant a. If there were such pressure, this would mean that complexity would decrease exponentially.
I don’t think this kind of negative pressure on complexity is as realistic as positive pressure, but it’s not entirely unrealistic either. In the rare case of pressure to simplify, removing one source of complexity could lead to cascading effects. Because we don’t need this one thing any more, we don’t need these other things that were only there to prop it up, etc. There could be a sustained decrease in complexity, though it probably would not be exponential.
Related post: A little simplification goes a long way
Sometimes it’s clear what’s meant by one topic being more advanced than another. For example, algebra is more advanced than arithmetic because you need to know arithmetic before you can do algebra. If you can’t learn A until you’ve learned B, then A is more advanced. But often advanced is used in a looser sense.
When I became a software developer, I was surprised how loosely developers use the word advanced. For example, one function might be called more “advanced” than other, even though there was no connection between the two. The supposedly more advanced function might be more specialized or harder to use. In other words, advanced was being used as a synonym for obscure. This is curious since advanced has a positive connotation but obscure has a negative connotation.
I resisted this terminology at first, but eventually I gave in. I’ll say advanced when I’m sure people will understand my meaning, even if I cringe a little inside. For example, I have had a Twitter account SansMouse that posts one keyboard shortcut a day [1]. These are in a cycle, starting with the most well-known and generally useful shortcuts. When I say the shortcuts progress from basic to advanced, people know what I mean and they’re happy with that. But it might be more accurate to say the shortcuts regress from most useful to least useful!
I’m not writing this just to pick at how people use words. My point is that the classification of some things as more advanced than others, particularly in technology, is largely arbitrary. The application of this: don’t assume that ‘advanced’ necessarily comes after ‘basic’.
Maybe A is called more advanced than B because most people find B more accessible. That doesn’t necessarily mean that you will find B more accessible. For example, I’ve often found supposedly advanced books easier to read than introductory books. Whether the author’s style resonates with you may be more important than the level of the book.
Maybe A is called more advanced than B because most people learn B first. That could be a historical accident. Maybe A is actually easier to learn from scratch, but B came first. Teachers and authors tend to present material in the order in which they learned it. They may think of newer material as being more difficult, but a new generation may disagree.
Finally, whether one thing is more advanced than another may depend on how far you intend to pursue it. It may be harder to master A than B, but that doesn’t mean it’s harder to dabble in A than B.
In short, you need to decide for yourself what order to learn things in. Of course if you’re learning something really new, you’re in no position to say what that order should be. The best thing is to start with the conventional order. But experiment with variations. Try getting ahead of the usual path now and then. You may find a different sequence that better fits your ways of thinking and your experience.
* * *
[1] Sometime after this post was written I renamed SansMouse to ShortcutKeyTip. I stopped posting to that account in September 2013, but the account is still online.
How many k × k magic squares are possible? If you start from a liberal definition of magic square, there’s an elegant result. For the purposes of this post, a magic square is a square arrangement of non-negative numbers such that the rows and columns all sum to the same non-negative number m called the magic constant. Note that this allows the possibility that numbers will be repeated, and this places no restriction on the diagonals.
With this admittedly non-standard definition, the number of k × k magic squares with magic constant m is always a polynomial in m of degree no more than (k – 1)2. For k = 3, the result is
(m + 1)(m + 2)(m2 + 3m + 4)/8
There is no general formula for all k, but there is an algorithm for finding a formula for each value of k.
Source: The Concrete Tetrahedron
Update: I had reported the polynomial degree as k + 1, but looking back at Concrete Tetrahedron, that book lists the order as (k + 1)2. However, the paper cited in the comments lists the exponent as (k – 1)2, which I believe is correct.
The Economist posted an article online this weekend about the scandal over irreproducible cancer research by Anil Potti. My colleagues Keith Baggerly and Kevin Coombes have been crying foul about this since 2007. I first blogged about it in January 2008.
The story started getting widespread attention last summer when the Cancer Letter reported that Dr. Potti had lied on grant applications. Since then there have been articles in the popular press, and people are staring to file lawsuits.
Apparently the tipping point in the story was finding a fib on Potti’s resume. According to The Economist
He falsely claimed to have been a Rhodes Scholar in Australia (a curious claim in any case, since Rhodes scholars only attend Oxford University).
So what finally got people to pay attention was not accusations of incompetent or fraudulent science, but résumé padding. As Keith Baggerly commented,
I find it ironic that we have been yelling for three years about the science, which has the potential to be very damaging to patients, but that was not what has started things rolling.
A Mersenne prime is a prime number that is one less than a power of 2. These primes are indexed by the corresponding power of two, i.e. Mp = 2p – 1. It turns out p must be prime before 2p – 1 can be prime.
Here are five things I find interesting about Mersenne primes.
1. Record size primes
The largest known prime number is a Mersenne prime, M82,589,993, proved prime in 2018. And ever since M521 was proven prime in 1952, the largest known prime has always been a Mersenne prime (with one exception in 1989). See a history of prime number records.
One reason for the prevalence of Mersenne primes in the record books is that there is a special algorithm for testing whether a number of the form 2p – 1 is prime, the Lucas-Lehmer test.
2. Finiteness
There may only be a finite number of Mersenne primes. Only 51 are known so far. There are conjectures that predict there are an infinite number of Mersenne primes, but these have not been settled.
3. Connection with perfect numbers
Euclid proved that if M is a Mersenne prime, M(M+1)/2 is a perfect number. Two millennia later, Euler proved that if N is an even perfect number, N must be of the form M(M+1)/2 where M is a Mersenne prime. (More details here.)
Since we only know of 51 Mersenne primes at the moment, and we don’t know of any odd perfect numbers, there are only 51 known perfect numbers.
4. Connection with random number generation
The Mersenne twister is a popular, high-quality random number generator. The generator is so named because its period is a Mersenne prime, M19,937.
5. History
Mersenne primes are named after the French monk Marin Mersenne (1588–1648) who compiled a list of Mersenne primes. Mersenne wasn’t the first to be aware of such primes. As mentioned above, Euclid connected these primes with perfect numbers.
Marin Mersenne is one of my academic ancestors. I studied under Ralph Showalter, who studied under Tsuan Ting, and so forth back to Frans van Shooten Jr., who studied under Marin Mersenne.
What I find fascinating about this is not my particular genealogy, but that adequate records exist to construct such genealogies. The Mathematics Genealogy Project has over 150,000 records, some reaching back to the Middle Ages.
The property “is a normal subgroup of” is not transitive.
If A is a subgroup of B, and B is a subgroup of C, then A is a subgroup of C. But the corresponding statement about normal subgroups is false. And there’s a simple example that shows it is false.
We need to find a group C with subgroups A and B such that A is normal in B, B is normal in C, but A is not normal in C.
The subgroup A must have at least two elements, otherwise A would just be the group identity and would then be a normal subgroup of C. The order of a subgroup divides the order of the group, so B must have at least twice as many elements as A, and C must have twice as many elements as B. So the smallest possible example would be a group with 8 elements and subgroups of order 2 and 4.
We’re in luck, because there’s a group of order 8 that will work, D8. This is the group of symmetries of a square under flips and rotations. Let A be the subgroup of flips about the vertical axis of symmetry. Let B the symmetries you can find by combinations of such flips and 180 degree rotations. You can show that A is normal in B, and B is normal in C.
Now let c be a 90 degree clockwise turn and let a be a flip. You can show that cac-1 is not a flip or the identity, so A is not a normal subgroup of C.
Related post: A 3,000 page proof (classification of finite simple groups)
