Solid angle of a star

Posted on 20 March 2023 by John

The apparent size of a distant object can be measured by projecting the object onto a unit sphere around the observer and calculating the area of the projected image.

A unit sphere has area 4π. If you’re in a ship far from land, the solid angle of the sky is 2π steradians because it takes up half a sphere.

If the object you’re looking at is a sphere of radius r whose center is a distance d away, then its apparent size is

$\Omega = 2\pi\left(1 - \frac{\sqrt{d^2 - r^2}}{d}\right)$

steradians. This formula assumes d > r. Otherwise you’re not looking out at the sphere; you’re inside the sphere.

If you’re looking at a star, then d is much larger than r, and we can simplify the equation above. The math is very similar to the math in an earlier post on measuring tapes. If you want to measure the size of a room, and something is blocking you from measuring straight from wall to wall, it doesn’t make much difference if the object is small relative to the room. It all has to do with Taylor series and the Pythagorean theorem.

Think of the expression above as a function of r and expand it in a Taylor series around r = 0.

$\Omega = 2\pi\left(1 - \frac{\sqrt{d^2 - r^2}}{d}\right) = 2\pi\left(\frac{\sqrt{r^2}}{2d^2} + \frac{r^4}{8d^4} + \cdots \right)$

and so

$\Omega \approx \frac{\pi r^2}{d^2}$

with an error on the order of (r/d)⁴. To put it another way, the error in our approximation for Ω is on the order of Ω². The largest object in the sky is the sun, and it has apparent size less than 10^-4, so Ω is always small when looking at astronomical objects, and Ω² is negligible.

So for practical purposes, the apparent size of a celestial object is π times the square of the ratio of its radius to its distance. This works fine for star gazing. The approximation wouldn’t be as accurate for watching a hot air balloon launch up close.

Square degrees

Sometimes solid angles are measured in square degrees, given by π/4 times the square of the apparent diameter in degrees. This implicitly uses the approximation above since the apparent radius is r/d.

(The area of a square is diameter squared, and a circle takes up π/4 of a square.)

Examples

When I typed

    3.1416 (radius of sun / distance to sun)^2

into Wolfram Alpha I got 6.85 × 10^-5. (When I used “pi” rather than 3.1416 it interpreted this as the radius of a pion particle.)

When I typed

    3.1416 (radius of moon / distance to moon)^2

I got 7.184 × 10^-5, confirming that the sun and moon are approximately the same apparent size, which makes a solar eclipse possible.

The brightest star in the night sky is Sirius. Asking Wolfram Alpha

    3.1416 (radius of Sirius / distance to Sirius)^2

we get 6.73 × 10^-16.

Fixed points of the Fourier transform

Posted on 17 March 2023 by John

This previous post looked at the hyperbolic secant distribution. This distribution has density

$f_H(x) = \frac{1}{2} \text{sech} \left(\frac{\pi x}{2} \right)$

and characteristic function sech(t). It’s curious that the density and characteristic function are so similar.

The characteristic function is essentially the Fourier transform of the density function, so this says that the hyperbolic secant function, properly scaled, is a fixed point of the Fourier transform. I’ve long known that the normal density is its own Fourier transform, but only recently learned that the same is true of the hyperbolic secant.

Hermite functions

The Hermite functions are also fixed points of the Fourier transform, or rather eigenfuctions of the Fourier transform. The eigenvalues are 1, i, -1, and i. When the eigenvalues are 1, we have fixed points.

There are two conventions for defining the Hermite functions, and multiple conventions for defining the Fourier transform, so the truth of the preceding paragraph depends on the conventions used.

For this post, we will define the Fourier transform of a function f to be

$\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \exp(-i \omega x)\, f(x)\, dx$

Then the Fourier transform of exp(-x²/2) is the same function. Since the Fourier transform is linear, this means the same holds for the density of the standard normal distribution.

We will define the Hermite polynomials by

$H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$

using the so-called physics convention. H_n is an nth degree polynomial.

The Hermite functions ψ_n(x) are the Hermite polynomials multiplied by exp(-x²/2). That is,

$\psi_n(x) = H_n(x) \exp(-x^2/2)$

With these definitions, the Fourier transform of ψ_n(x) equals (-i)ⁿ ψ_n(x). So when n is a multiple of 4, the Fourier transform of ψ_n(x) is ψ_n(x).

[The definition Hermite functions above omits a complicated constant term that depends on n but not on x. So our Hermite functions are proportional to the standard Hermite functions. But proportionality constants don’t matter when you’re looking for eigenfunctions or fixed points.]

Hyperbolic secant

Using the definition of Fourier transform above, the function sech(√(π/2) x) is its own Fourier transform.

This is surprising because the Hermite functions form a basis for L²(ℝ), and all have tails on the order of exp(-x²), but the hyperbolic secant has tails like exp(-x). Each Hermite function eventually decays like exp(-x²), but this happens later as n increases, so an infinite sum of Hermite functions can have thicker tails than any particular Hermite function.

Interpolating rotations with SLERP

Posted on 15 March 2023 by John

Naive interpolation of rotation matrices does not produce a rotation matrix. That is, if R₁ and R₂ are rotation (orthogonal) matrices and 0 < t < 1, then

is not in general a rotation matrix.

You can represent rotations with unit quaternions rather than orthogonal matrices (see details here), so a reasonable approach might be to interpolate between the rotations represented by unit quaternions q₁ and q₂ using

but this has a similar problem: the quaternion above is not a unit quaternion.

One way to patch this up would be to normalize the expression above, dividing by its norm. That would indeed produce unit quaternions, and hence correspond to rotations. However, uniformly varying t from 0 to 1 does not produce a uniform rotation.

The solution, first developed by Ken Shoemake [1], is to use spherical linear interpolation or SLERP.

Let θ be the angle between q₁ and q₂. Then the spherical linear interpolation between q₁ and q₂ is given by

$q(t) = \frac{\sin((1-t)\theta)}{\sin\theta}q_1 + \frac{\sin(t\theta)}{\sin\theta}q_2$

Now q(t) is a unit quaternion, and uniformly increasing t from 0 to 1 creates a uniform rotation.

[1] Ken Shoemake. “Animating Rotation with Quaternion Curves.” SIGGRAPH 1985.

Shuffle product

Posted on 13 March 2023 by John

riffle shuffle

The shuffle product of two words, w₁ and w₂, written

w₁ Ш w₂,

is the set of all words formed by the letters in w₁ and w₂, preserving the order of each word’s letters. The name comes from the analogy with doing a riffle shuffle of two decks of cards.

For example, bcd Ш ae, the shuffle product of bcd and ae, would be all permutations of abcde in which the consonants appear in alphabetical order and the vowels are also in alphabetical order. So abecd and baecd would be included, but badec would not be because the d and c are in the wrong order.

Side note on Ш

Incidentally, the symbol for shuffle product is the Cyrillic letter sha (Ш, U+0428), the only Cyrillic letter commonly used in mathematics, at least internationally. Presumably Russian mathematicians use other Cyrillic letters, but the only Cyrillic letter an American mathematician, for example, is likely to use is Ш.

The uses of Ш that I’m aware of are the Dirac comb distribution, the Tate–Shafarevich group, and the shuffle product.

Duplicate letters

What is the shuffle product of words containing duplicate letters? For example, what about the shuffle product of bread and crumb? Each word contains an r. The shuffle product, defined above as a set, doesn’t distinguish between the two rs. But another way to define the shuffle product is as a formal sum, with coefficients that count duplicates.

Imagine coloring the letters in abc blue and the letters in cde red. Then abccde and abccde would count as two different possibilities, one with blue c followed by red c, and one the other way around. This term in the formal sum would be 2abccde, the two capturing that there are two ways to arrive at this word.

You could also have duplicate letters within a single word. So in banana, for example, you could imagine coloring each a a different color and coloring the two ns different colors.

Mathematica code

This page gives an implementation of the shuffle product in Mathematica.

shuffleW[s1_, s2_] := 
     Module[{p, tp, ord}, 
          p = Permutations@Join[1 & /@ s1, 0 & /@ s2]\[Transpose];
          tp = BitXor[p, 1];
          ord = Accumulate[p] p + (Accumulate[tp] + Length[s1]) tp;
          Outer[Part, {Join[s1, s2]}, ord, 1][[1]]\[Transpose]]

This code takes two lists of characters and returns a list of lists of characters. You can use this to compute both senses of the shuffle product. For example, let’s compute abc Ш ac.

The Mathematica command

    shuffleW[{a, b, c}, {a, c}]

returns a list of 10 lists:

 
    {{a, b, c, a, c}, {a, b, a, c, c}, {a, b, a, c, c}, 
     {a, a, b, c, c}, {a, a, b, c, c}, {a, a, c, b, c}, 
     {a, a, b, c, c}, {a, a, b, c, c}, {a, a, c, b, c}, 
     {a, c, a, b, c}}

If we ask for the union of the set above with Union[%] we get

    {{a, a, b, c, c}, {a, a, c, b, c}, {a, b, a, c, c}, 
     {a, b, c, a, c}, {a, c, a, b, c}}

So using the set definition, we could say

abc Ш ac = {aabcc, aacbc, abacc, abcac, acabc}.

Using the formal sum definition we could say

abc Ш ac = 4aabcc + 2aacbc + 2abacc + abcac + acabc.

Photo by Amol Tyagi on Unsplash

Prime numbers and Taylor’s law

Posted on 13 March 2023 by John

The previous post commented that although the digits in the decimal representation of π are not random, it is sometimes useful to think of them as random. Similarly, it is often useful to think of prime numbers as being randomly distributed.

If prime numbers were samples from a random variable, it would be natural to look into the mean and variance of that random variable. We can’t just compute the mean of all primes, but we can compute the mean and variance of all primes less than an upper bound x.

Let M(x) be the mean of all primes less than x and let V(x) be the corresponding variance. Then we have the following asymptotic results:

M(x) ~ x / 2

and

V(x) ~ x²/12.

We can investigate how well these limiting results fit for finite x with the following Python code.

    from sympy import sieve
    
    def stats(x):
        s = 0
        ss = 0
        count = 0
        for p in sieve.primerange(x):
            s += p
            ss += p**2
            count += 1
        mean = s / count
        variance = ss/count - mean**2
        return (mean, variance)

So, for example, when x = 1,000 we get a mean of 453.14, a little less than the predicted value of 500. We get a variance of 88389.44, a bit more than the predicted value of 83333.33.

When x = 1,000,000 we get closer to values predicted by the limiting formula. We get a mean of 478,361, still less than the prediction of 500,000, but closer. And we get a variance of 85,742,831,604, still larger than the prediction 83,333,333,333, but again closer. (Closer here means the ratios are getting closer to 1; the absolute difference is actually getting larger.)

Taylor’s law

Taylor’s law is named after ecologist Lionel Taylor (1924–2007) who proposed the law in 1961. Taylor observed that variance and mean are often approximately related by a power law independent of sample size, that is

V(x) ≈ a M(x)^b

independent of x.

Taylor’s law is an empirical observation in ecology, but it is a theorem when applied to the distribution of primes. According to the asymptotic results above, we have a = 1/3 and b = 2 in the limit as x goes to infinity. Let’s use the code above to look at the ratio

V(x) / a M(x)^b

for increasing values of x.

If we let x = 10^k for k = 1, 2, 3, …, 8 we get ratios

0.612, 1.392, 1.291, 1.207, 1.156, 1.124, 1.102, 1.087

which are slowly converging to 1.

Reference: Joel E. Cohen. Statistics of Primes (and Probably Twin Primes) Satisfy Taylor’s Law from Ecology. The American Statistician, Vol. 70, No. 4 (November 2016), pp. 399–404

The coupon collector problem and π

Posted on 12 March 2023 by John

How far do you have to go down the decimal digits of π until you’ve seen all the digits 0 through 9?

We can print out the first few digits of π and see that there’s no 0 until the 32nd decimal place.

3.14159265358979323846264338327950

It’s easy to verify that the remaining digits occur before the 0, so the answer is 32.

Now suppose we want to look at pairs of digits. How far out do we have to go until we’ve seen all pairs of digits (or base 100 digits if you want to think of it that way)? And what about triples of digits?

We know we’ll need at least 100 pairs, and at least 1000 triples, so this has gotten bigger than we want to do by hand. So here’s a little Python script that will do the work for us.

    from mpmath import mp
    
    mp.dps = 30_000
    s = str(mp.pi)[2:] 
    
    for k in [1, 2, 3]:
        tuples = [s[i:i+k] for i in range(0, len(s), k)]
        d = dict()
        i = 0
        while len(d) < 10**k:
            d[tuples[i]] = 1
            i += 1
        print(i)

The output:

    32
    396
    6076

This confirms that we at the 32nd decimal place we will have seen all 10 possible digits. It says we need 396 pairs of digits before we see all 100 possible digit pairs, and we’ll need 6076 triples before we’ve seen all possible triples.

We could have used the asymptotic solution to the “coupon collector problem” to approximately predict the results above.

Suppose you have an urn with n uniquely labeled balls. You randomly select one ball at a time, return the ball to the run, and select randomly again. The coupon collector problem ask how many draws you’ll have to make before you’ve selected each ball at least once.

The expected value for the number of draws is

n H_n

where H_n is the nth harmonic number. For large n this is approximately equal to

n(log n + γ)

where γ is the Euler-Mascheroni constant. (More on the gamma constant here.)

Now assume the digits of π are random. Of course they’re not random, but random is as random does. We can get useful estimates by making the modeling assumption that the digits behave like a random sequence.

The solution to the coupon collector problem says we’d expect, on average, to sample 28 digits before we see each digit, 518 pairs before we see each pair, and 7485 triples before we see each triple. “On average” doesn’t mean much since there’s only one π, but you could interpret this as saying what you’d expect if you repeatedly chose real numbers at random and looked at their digits, assuming the normal number conjecture.

The variance on the number of draws needed is asymptotically π² n²/6, so the number of draws with usually be an interval of the expected value ± 2n.

If you want the details of the coupon collector problem, not just the expected value but the probabilities for different number of draws, see Sampling with replacement until you’ve seen everything.

The NBA and MLB trees are isomorphic

Posted on 7 March 2023 by John

NBA MLB games

An isomorphism is a structure-preserving function from one object to another. In the context of graphs, an isomorphism is a function that maps the vertices of one graph onto the vertices of another, preserving all the edges.

So if G and H are graphs, and f is an isomorphism between G and H, nodes x and y are connected in G if and only if nodes f(x) and f(y) are connected in H.

There are 30 basketball teams in the National Basketball Association (NBA) and 30 baseball teams in Major League Baseball (MLB). That means the NBA and MLB are isomorphic as sets, but it doesn’t necessarily mean that the hierarchical structure of the two organizations are the same. But in fact the hierarchies are the same.

Both the NBA and MLB have two top-level divisions, each divided into three subdivisions, each containing five teams.

Basketball has an Eastern Conference and a Western Conference, whereas baseball has an American League and a National League. Each basketball conference is divided into three divisions, just like baseball leagues, and each division has five teams, just as in baseball. So the tree structures of the two organizations are the same.

In the earlier post about the MLB tree structure, I showed how you could number baseball teams so that the team number n could tell you the league, division, and order within a division by taking the remainders when n is divided by 2, 3, and 5. Because the NBA tree structure is isomorphic, the same applies to the NBA.

Here’s a portion of the graph with numbering. The full version is available here as a PDF.

Here’s the ordering.

Los Angeles Clippers
Miami Heat
Portland Trail Blazers
Milwaukee Bucks
Dallas Mavericks
Brooklyn Nets
Los Angeles Lakers
Orlando Magic
Utah Jazz
Chicago Bulls
Houston Rockets
New York Knicks
Phoenix Suns
Washington Wizards
Denver Nuggets
Cleveland Cavaliers
Memphis Grizzlies
Philadelphia 76ers
Sacramento Kings
Atlanta Hawks
Minnesota Timberwolves
Detroit Pistons
New Orleans Pelicans
Toronto Raptors
Golden State Warriors
Charlotte Hornets
Oklahoma City Thunder
Indiana Pacers
San Antonio Spurs
Boston Celtics

Incidentally, the images at the top of the post were created with DALL-E. They look nice overall, but you’ll see bizarre details if you look too closely.

Numbering minor league baseball teams

Posted on 6 March 2023 by John

El Paso Chihuahuas team logo
Last week I wrote about how to number MLB teams so that the number n told you where they are in the league hierarchy:

n % 2 tells you the league, American or National
n % 3 tells you the division: East, Central, or West
n % 5 is unique within a league/division combination.

Here n % m denotes n mod m, the remainder when n is divided by m.

This post will do something similar for minor league teams.

There are four minor league teams associated with each major league team. If we wanted to number them analogously, we’d need to do something a little different because we cannot specify n % 2 and n % 4 independently. We’d need an approach that is a hybrid of what we did for the NFL and MLB.

We could specify the league and the rank within the minor leagues by three bits: one bit for National or American league, and two bits for the rank:

00 for A
01 for High A
10 for AA
11 for AAA

It will be convenient later on if we make the ranks the most significant bits and the league the least significant bit.

So to place a minor league team on a list, we could write down the numbers 1 through 120, and for each n, calculate r = n % 8, d = n % 3, and k = n % 5.

The latest episode of 99% Invisible is called RoboUmp, a show about automating umpire calls. As part of the story, the show discusses the whimsical names of minor league teams and how the names allude to their location. For example, the El Paso Chihuahuas are located across the border from the Mexican state of Chihuahua and their mascot is a chihuahua dog. (The dog was named after the state.)

The El Paso Chihuahuas are the AAA team associated with the San Diego Padres, a team in the National League West, team #3 in the order listed in the MLB post. The number n for the Chihuahuas must equal 7 mod 8, 111_two, the first bit for National League and the last two bits for AAA. We also require n to be 2 mod 3 because it’s in the West, and n = 3 mod 5 because the Padres are #3 in the list of National League West teams in our numbering. It works out that n = 23.

How do minor league and major league numbers relate? They have to be congruent mod 30. They have to have the same parity since they represent the same league, and must be congruent mod 3 because they have in the same division. And they must be congruent mod 5 to be in the same place in the list of associated major league teams.

So to calculate a minor league team’s number, start with the corresponding major league number, and add multiples of 30 until you get the right value mod 8.

For example, the Houston Astros are number 20 in the list from the earlier post. The Triple-A team associated with the Astros is the Sugar Land Space Cowboys. The number n for the Space Cowboys must be 6 mod 8 because 6 = 110_two, and they’re a Triple-A team (11) in the American League (0). So n = 110.

The Astros’ Double-A team, the Corpus Christi Hooks, needs to have a number equal to 100_two = 4 mod 8, so n = 20. The High-A team, the Asheville Tourists, are 50 and the Single-A team, the Fayetteville Woodpeckers, is 80.

You can determine what major league team is associated with a minor league team by taking the remainder by 30. For example, the Rocket City Trash Pandas has number 77, so they’re associated with the major league team with number 17, which is the Los Angeles Angels. The remainder when 77 is divided by 8 is 5 = 101_two, which tells you they’re a Double-A team since the high order bits are 1 and 0.

John Conway’s mental factoring method and friends

Posted on 3 March 2023 by John

There are tricks for determining whether a number is divisible by various primes, but many of these tricks have to be applied one at a time. You can make a procedure for testing divisibility by any prime p that is easier than having to carry out long division, but these rules are of little use if every one of them is different.

Say I make a rule for testing whether a number is divisible by 59. That’s great, if you routinely need to test divisibility by 59. Maybe you work for a company that, for some bizarre reason, ships widgets in boxes of 59 and you frequently have to test whether numbers are multiples of 59.

When you want to factor numbers, you’d like to test divisibility by a set of primes at once, using fewer separate algorithms, and taking advantage of work you’ve already done.

John Conway came up with his 150 Method to test for divisibility by a sequence of small primes. This article explains how Conway’s 150 method and a couple variations work. The core idea behind Conway’s 150 Method, his 2000 Method, and analogous methods developed by others is this:

Find a range of integers, near a round number, that contains a lot of distinct prime factors.
Reduce your number modulo the round number, then test for divisibility sequentially, reusing work.

Conway’s 150 Method starts by taking the quotient and remainder by 150. And you’ll never guess what his 2000 Method does. :)

This post will focus on the pattern behind Conway’s method, and similar methods. For examples and practical tips on carrying out the methods, see the paper linked above and a paper I’ll link to below.

The 150 Method

Conway exploited the fact that the numbers 152 through 156 are divisible by a lot of primes: 2, 3, 5, 7, 11, 13, 17, 19, and 31.

He starts his method with 150 rather than 152 because 150 is a round number and easier to work with. We start by taking the quotient and remainder by 150.

Say n = 150q + r. Then n – 152q = r – 2q. If n has three or four digits, q only has one or two digits, and so subtracting q is relatively easy.

Since 19 divides 152, we can test whether n is divisible by 19 by testing whether r – 2q is divisible by 19.

The next step is where sequential testing saves effort. Next we want to subtract off a multiple of 153 to test for divisibility by 17, because 17 divides 153. But we don’t have to start over. We can reuse our work from the previous step.

We want n – 153q = (n – 152q) – q, and we’ve already calculated n – 152q in the previous step, so we only need to subtract q.

The next step is to find n – 154q, and that equals (n – 153q) – q, so again we subtract q from the result of the previous step. We repeat this process, subtracting q each time, and testing for divisibility by a new set of primes each time.

The 2000 method

Conway’s more extensive method exploited the fact that the numbers 1998 through 2021 are divisible by all primes up to 67. So he would start by taking the quotient and remainder by 2000, which is really easy to do.

Say n = 2000q + r. Then we would add (or subtract) q each time.

You could start with r, then test r for divisibility by the factors of 2000, then test r – q for divisibility by the factors of 2001, then test r – 2q for divisibility by the factors of 2002, and so on up to testing r – 21q for divisibility by the factors of 2021. Then you’d need to go back and test r + q for divisibility by the factors of 1999 and test r + 2q for divisibility by the factors of 1998.

In principle that’s how Conways 2000 Method works. In practice, he did something more clever.

Most of the prime factors of the numbers 1998 through 2021 are prime factors of 1998 through 2002, so it makes sense to test this smaller range first hoping for early wins. Also, there’s no need to test divisibility by the factors of 1999 because 1999 is prime.

Conway tested r – kq for k = -2 through 21, but not sequentially. He would try out the values of k in an order most likely to terminate the factoring process early.

The 10,000 method

This paper gives a much more extensive approach to mental factoring than Conway’s 150 method. The authors, Hilarie Orman and Richard Schroeppel, outline a strategy for factoring any six-digit number. Conway’s rule is more modest, intended for three and four digit numbers.

Orman and Schroeppel suggest a sequence of factoring methods, including more advanced techniques to use after you’ve tried testing for divisibility by small primes. One of the techniques in the paper might be called the 10,000 Method by analogy to Conway’s method, though the authors don’t call it that. They call it “check the m‘s” for reasons that make more sense if you read the paper.

The 10,000 Method is much like the 2000 Method. The numbers 10,001 through 10,019 have a lot of prime factors, and the method tests for divisibility by these factors sequentially, taking advantage of previous work at each step, just as Conway’s methods do. The authors do not backtrack the way Conway did; they test numbers in order. However, they do skip over some numbers, like Conway skipped over 1999.

Major League Baseball and number theory

Posted on 2 March 2023 by John

The previous post took a mathematical look at the National Football League. This post will do the same for Major League Baseball.

Like the NFL, MLB teams are organized into a nice tree structure, though the MLB tree is a little more complicated. There are 32 NFL teams organized into a complete binary tree, with a couple levels collapsed. There are 30 MLB teams, so the tree structure has to be a bit different.

MLB has leagues rather than conferences, but the top-level division is into American and Nation as with the NFL. So the top division is into the American League and the National League.

And as with football, the next level of the hierarchy is divisions. But baseball has three divisions—East, Central, and West—in contrast to four in football.

Each division has five baseball teams, while each football division has four teams.

Here’s the basic tree structure.

Under each division are five teams. Here’s a PDF with the full graph including teams.

Geography

How do the division names correspond to actual geography?

Within each league, the Central teams are to the west of the East teams and to the east of the West teams, with one exception: in the National League, the Pittsburgh Pirates are a Central division team, but they are east of the Atlanta Braves and Miami Marlins in the East division. But essentially the East, Central, and West divisions do correspond to geographic east, center, and west, within a league.

Numbering

We can’t number baseball teams as elegantly as the previous post numbered football teams. We’d need a mixed-base number. The leading digit would be binary, the next digit base 3, and the final digit base 5.

We could number the teams so that you could tell the league and division of the team by looking at the remainders when the number is divided by 2 and 3, and each team is unique mod 5. By the Chinese Remainder Theorem, we can solve the system of congruence equations mod 30 that specify the value of a number mod 2, mod 3, and mod 5.

If we number the teams as follows, then odd numbered teams are in the American League and even numbered teams are in the National League. When the numbers are divided by 3, those with remainder 0 are in an Eastern division, those with remainder 1 are in a Central division, and those with remainder 2 are in a Western division. Teams within the same league and division have unique remainders by 5.

Cincinnati Reds
Oakland Athletics
Philadelphia Phillies
Minnesota Twins
Arizona Diamondbacks
Boston Red Sox
Milwaukee Brewers
Seattle Mariners
Washington Nationals
Chicago Whitesocks
Colorado Rockies
New York Yankees
Pittsburgh Pirates
Texas Rangers
Atlanta Braves
Cleveland Guardians
Los Angeles Dodgers
Tampa Bay Rays
St. Louis Cardinals
Houston Astros
Miami Marlins
Detroit Tigers
San Diego Padres
Toronto Blue Jays
Chicago Cubs
Los Angeles Angels
New York Mets
Kansas City Royals
San Francisco Giants
Baltimore Orioles

Math

Solid angle of a star

Square degrees

Examples

Related posts

Fixed points of the Fourier transform

Hermite functions

Hyperbolic secant

Related posts

Interpolating rotations with SLERP

Shuffle product

Side note on Ш

Duplicate letters

Mathematica code

Related posts

Prime numbers and Taylor’s law

Taylor’s law

Related posts

The coupon collector problem and π

The NBA and MLB trees are isomorphic

Numbering minor league baseball teams

John Conway’s mental factoring method and friends

The 150 Method

The 2000 method

The 10,000 method

More Conway-related posts

Major League Baseball and number theory

Geography

Numbering

Related posts