John D. Cook

The last chapter of George Box’s book Improving Almost Anything contains the lyrics to “I Am the Very Model of a Professor Statistical,” to be sung to the tune of “I Am the Very Model of a Modern Major General” by Gilbert & Sullivan. Here’s the original: The original song has a few funny math-related […]The post The Very Model of a Professor Statistical first appeared on John D. Cook.

Let s be an integer equal to at least 5 and let n be a positive integer. Look at all tuples of s integers, each integer being between 0 and n-1 inclusive, and look at the sum of their squares mod n. About 1/n will fall into each possible value. So, for example, if you look […]The post Sum of squares mod n uniformly distributed first appeared on John D. Cook.

If you ask Mathematica for a list of Mexican states via CountryData["Mexico", "RegionNames"] you will get a list of strings: "Aguascalientes", "Baja California", ..., "Zacatecas"} However, when you try to turn this into a list of objects representing these states via states = Entity["AdministrativeDivision", {#, "Mexico"}] & /@ CountryData["Mexico", "RegionNames"] something strange happens. Some items […]The post Quirks in Mathematica’s administrative division data for Mexico first appeared on John D. Cook.

A few days ago I wrote a post comparing English and Japanese vowel sounds in a 2D chart. In this post I’d like to do something similar for English and Swedish. As before the data come from [1]. A friend of mine who learned Swedish would joke about how terribly he had to contort his […]The post Visualizing Swedish vowels first appeared on John D. Cook.

Here is a Traveling Salesman tour of Canada’s provinces and territories created by Mathematica. This is the shortest path connecting the geographic centers of the regions. Here is a much larger (4.5 MB) PDF file of the same map with higher resolution. Starting in the northwest, the tour is Yukon Northwest Territories Nunavut Quebec Newfoundland […]The post A traveling salesman tour of Canada first appeared on John D. Cook.

A ligature in typesetting is a way of presenting two (or more) consecutive characters differently the individual characters would be displayed. For example, “fi” is often rendered with the top of the ‘f’ dotting the ‘i’. Here’s an example from Computer Modern, the default font in LaTeX. Usually the difference is subtle—ordinarily readers are not […]The post Ligatures for Logic first appeared on John D. Cook.

Vowel sounds can be visualized in a two-dimensional space according to tongue position. The vertical axis is runs from open down to closed, and the horizontal runs from front to back. See a linguistics textbook for far more detail. English has five vowel letters, but a lot more than five vowel sounds. Scholars argue about […]The post Visualizing English and Japanese vowels first appeared on John D. Cook.

I rarely intentionally use emoji, and yet I often run into them unbidden. This is because some Unicode characters double as emoji. For example, the zodiac symbol for Aries is used both in celestial navigation and in astrology. The latter is much more common, and so when some software sees U+2648 it interprets the character […]The post Preventing characters from displaying as emoji first appeared on John D. Cook.

It’s necessary now and then to find the day of the year for a date. For example, today is 272nd day of 2022. How hard would it be to calculate the day of the year in your head? Each month has about 30 days, so the dth day of the mth month is approximately day […]The post Day of the year first appeared on John D. Cook.

A few days ago I wrote about Room squares, squares named after Thomas Room. This post will be about Room’s original square. You could think of a Room square as a tournament design in which the rows represent locations and the columns represent rounds (or vice versa). Every team plays every other team exactly once, […]The post The original Room square first appeared on John D. Cook.

Yesterday I found a statistics textbook for geologists [1] for $1 at a library book sale. When I thumbed through the book an image similar to the one below caught my eye. This image approximates Figure 15.2 in [1], The nodes represent six factors of the thickness of rock formations and the edges are labeled […]The post Visualizing correlations with graphs first appeared on John D. Cook.

I figured out something that I wasn’t able to find by searching, so I’m posting it here in case other people have the same question and the same difficulty finding an answer. I’m sure other people have written about this, but I couldn’t find it. Maybe lots of people have written about this in Japanese […]The post Katakana, Hiragana, and Unicode first appeared on John D. Cook.

A Room square is a variation on a Latin square. Room squares are named after Thomas Room, though there is an application to rooms as in compartments of a building that we’ll discuss below. In a Latin square of size n you have to assign one of n symbols to each cell so that each […]The post Room squares and Tournaments first appeared on John D. Cook.

HCPCS stands for Healthcare Common Procedure Coding System. HCPCS codes are commonly pronounced like “hick pics.” I was curious/afraid to see what DALL-E would create with the prompt “HCPCS hick pics” and was pleasantly surprised that it produced the image above. More on that latter. Searching for medical codes I occasionally need to search for […]The post HCPCS (“hick pics”) codes first appeared on John D. Cook.

I was spelunking around in Unicode and found that there are assigned characters for representing domino tiles and that the characters are enumerated in a convenient order. Here is the code chart. There are codes for representing tiles horizontally or vertically. And even though, for example, the 5-3 is the same domino as the 3-5, […]The post Dominoes in Unicode first appeared on John D. Cook.

Ran across a great quote from Liv Boeree recently: The problem with raging against the machine is that the machine has learned to feed off rage. Someone appropriately replied with a screenshot from an episode of Star Trek TOS, Day of the Dove, about a being that feeds off anger, like contemporary media.The post Raging against the machine first appeared on John D. Cook.

This post will give a detailed example of working in a field with nine elements. This is important because finite fields are not often treated concretely except for the case of prime order. In my first post on Costas arrays I mentioned in a footnote that Lempel’s algorithm works more generally over any finite field, […]The post Field of order 9 first appeared on John D. Cook.

A couple days ago I wrote about Costas arrays. In a nutshell, a Costas array of size n is a solution to the n rooks problem, with the added constraint that if you added wires between the rooks, no two wires would have the same length and slope. See the earlier post for more details. […]The post Costas arrays in Mathematica first appeared on John D. Cook.

The ISO 3166-1 standard defines three codes for each country: a 2-letter abbreviation, a 3-letter abbreviation, and a 3-digit code. The 2-letter abbreviations may be familiar because it is very often (but not always [1]) also the country code top-level domain (ccTLD). For example, AU is the ISO abbreviation for Australia, and .au is the […]The post Two-letter vs Three-letter Country Abbreviations first appeared on John D. Cook.

A week ago I posted some pairs of similar flags on Twitter, and later I found that Mathematica’s CountryData database contains flag descriptions. So I thought I’d use the flag descriptions to see which flags Mathematica things are similar. For example, the FlagDescription attribute for Chad in Mathematica is Three equal vertical bands of blue […]The post Finding similar world flags with Mathematica first appeared on John D. Cook.

The famous n queens problem is to find a way to position n queens on a n×n chessboard so that no queen attacks any other. That is, no two queens can be in the same row, the same column, or on the same diagonal. Here’s an example solution: Costas arrays In this post we’re going […]The post Costas arrays first appeared on John D. Cook.

Suppose you have an even number of teams that you’d like to schedule in a Round Robin tournament. This means each team plays every other team exactly once. Denote the number of teams as 2n. You’d like each team to play in each round, so you need n locations for the games to be played. […]The post Balanced tournament designs first appeared on John D. Cook.

The two previous posts looked at adjacency networks. The first used examples of US states and Texas counties. The second post made suggestions for using these networks in a classroom. This post is a continuation of the previous post using examples from Japan. Japan is divided into 8 regions and 47 prefectures. Here is a […]The post Graphing Japanese Prefectures first appeared on John D. Cook.

In the previous post I looked at graphs created from representing geographic regions with nodes and connecting nodes with edges if the corresponding regions share a border. It’s an interesting exercise to recover the geographic regions from the network. For example, take a look at the graph for the continental United States. It’s easy to […]The post Classroom exercise with networks first appeared on John D. Cook.

Suppose you want to color a map with no two bordering regions having the same color. If this is a map on a plane, you can do this using only four colors, but maybe you’d like to use more. You can reduce the problem to coloring the nodes in a graph. Each node corresponds to […]The post Adjacency networks first appeared on John D. Cook.

This is the final post in a series of three posts about shortest tours, solutions to the so-called traveling salesmen problem. The first was a tour of Africa. Actually two tours, one for the continent and one for islands. See this post for the Mathematica code used to create the tours. The second was about […]The post Shortest tours of Eurasia and Oceania first appeared on John D. Cook.

The previous post looked at an optimal tour of continental Africa. This post will give analogous tours of continental North America, North American Islands, and South America. The next post looks at Eurasia and Oceania. North American Continent Here’s the North American continental tour. The order of the tour is as follows. Canada United States […]The post Three tours of the Americas first appeared on John D. Cook.

Suppose you’d like to tour Africa, visiting each country once, then returning to your starting point, minimizing the distance traveled. Here’s my first attempt at a solution using Mathematica, based on an example in the documentation for FindShortestTour. africa = CountryData["Africa"] FindShortestTour[africa] GeoGraphics[{Thick, Red, GeoPath[africa[[%[[2]]]]]}] This produced the following map: Hmm. Maybe I should have […]The post A traveling salesman tour of Africa first appeared on John D. Cook.

I recently wrote posts about spherical analogs of the Pythagorean theorem, the law of cosines, and the law of sines. The corresponding formulas for hyperbolic space mostly just replace circular functions with hyperbolic functions, i.e. replace sine with hyperbolic sine and cosine with hyperbolic cosine. Triangles on a sphere or on a hyperbolic space like […]The post Trig in hyperbolic geometry first appeared on John D. Cook.

In an earlier post I looked in detail at a series for inverse cosine centered at 1. The function arccos(z) is multivalued in a neighborhood of 1, but the function arccos(z) / √(2 – 2z) is analytic in a neighborhood of 1. We cancel out the bad behavior of inverse cosine at 1 by dividing […]The post Eliminating a Bessel function branch cut first appeared on John D. Cook.

As far as I know, all contemporary math libraries use the same branch cuts when extending elementary functions to the complex plane. It seems that the current conventions date back to Kahan’s paper [1]. I imagine to some extent he codified existing practice, but he also settled some issues, particularly regarding floating point implementation. I’ve […]The post Branch cuts for elementary functions first appeared on John D. Cook.

Suppose you need to estimate the inverse cosine of an argument near 1. There’s a series for that: You can find this series, for example, here. This comes in handy, for example, when working with the analog of the Pythagorean theorem on a sphere. You could just use the series and be on your way. […]The post Series for inverse cosine at 1 first appeared on John D. Cook.

I had some errors in a recent blog post that might have been eliminated if I had programmatically generated the content of the post rather than writing it by hand. I rewrote the example in this post in using org-mode. My org file looked like this: #+begin_src python :session :exports none lax_lat = 33.94 lax_lon […]The post Literate programming to reduce errors first appeared on John D. Cook.

I’ve had to work a little with VIN numbers lately, and so I looked back at a post I wrote on the subject three years ago. That post goes into the details of Vehicle Identification Numbers and the quirky algorithm used to compute the check sum. This post captures the algorithm in Python code. See […]The post Computing VIN checksums first appeared on John D. Cook.

Here’s a cute little poem: I wish I knew The root of two. O charmed was he To know root three. So we now strive To find root five. The beginning of each stanza is a mnemonic for the number mentioned in the following line. √ 2 = 1.414 √ 3 = 1.732 √ 5 […]The post Square root mnemonics first appeared on John D. Cook.

If a password of length n is made up of lower case letters and digits, with each symbol appearing with equal probability, how many times would you expect to shift between letters and digits when typing the password? If you’re typing the password on a mobile device, how many times would you change between alphabetic […]The post Runs of letters and digits in passwords first appeared on John D. Cook.

Blog posts about modal logic Typesetting modal logic Modal and temporal logic for security Word problems, modal logic, and regular expressions Axioms for S1 through S5 Naming and numbering modal logic systems Modal logic and science fiction Dual axioms Temporal and polymodal logics I created the image above by asking DALL-E 2 for an image […]The post Modal logic posts first appeared on John D. Cook.

How does the circumference of a circle vary with its radius? What about the area? The answers are simple and familiar in Euclidean geometry, but not as simple or as familiar in non-Euclidean geometry. This post will look at how circumference and area vary as a function of radius in three spaces: a surface with […]The post Radius, circumference, and area in non-Euclidean geometry first appeared on John D. Cook.

A few days ago I wrote that the spherical counterpart of the Pythagorean theorem is cos(c) = cos(a) cos(b) where sides a and b are measured in radians. If we’re on a sphere of radius R and we measure the sides in terms of arc length rather than in radians, the formula becomes cos(c/R) = […]The post Unified Pythagorean Theorem first appeared on John D. Cook.

According to the latest episode of Eclectic Tech, Richard Feynman argued that Brownian motion cannot do work, but researchers at the University of Arkansas have demonstrated that it can by generating an electric current from Brownian motion in a sheet of graphene. You can read more in the physics journal article by the researchers. Unfortunately […]The post Can Brownian motion do work? first appeared on John D. Cook.

This post is the third in a series of posts on spherical trigonometry. We first looked at the analog of the Pythagorean theorem on a sphere, then the analog of the law of cosines on a sphere. Now we look at the analog of the law of sines on a sphere. As before we denote […]The post Direction between two cities first appeared on John D. Cook.

The previous post looked at the analog of the Pythagorean theorem on a sphere. This post looks at the law of cosines on a sphere. Yesterday we looked at triangles on a sphere with sides a and b meeting at a right angle and hypotenuse c. Denote the angle opposite a side with the capital […]The post Law of cosines on a sphere first appeared on John D. Cook.

Suppose you drive a distance a, turn right, then drive a distance b. How far are you from where you started? If a and b are relatively small, then the answer is given by the Pythagorean theorem. Small here means small relative to the radius of the Earth. Any distance you drive is small enough: […]The post Pythagoras on a sphere first appeared on John D. Cook.

I was playing around with DALL-E last night. I pasted the definition of chimera into DALL-E and the results were bizarre. See this Twitter thread for images. I also played around with some more mnemonic images. Students often memorize the sines and cosines of 30°, 45°, and 60°. I thought about making a mnemonic image […]The post Chimera and sine of 60° first appeared on John D. Cook.

I’ve mostly used Windows and Linux for the last several years. When I needed to get a new laptop recently I got a MacBook Pro. I’ve used a Mac before, but it’s been a while, and so I’m starting over. I can move between Windows and Linux and almost forget what OS I’m using because […]The post Cross platform muscle memory first appeared on John D. Cook.

In base 10, four decibel values are approximately integers. Yesterday I explored whether base 10 was unique in this regard. I defined the value of n decibels in base b to be g(n, b) = bn/b. Sticking in 10 for b gives the usual definition of decibel levels. There are two ways to quantify what […]The post More on near-integer decibels first appeared on John D. Cook.

I got a data file from a client recently in “pickle” format. I happen to know that pickle is a binary format for serializing Python objects, but trying to open a pickle file could be a puzzle if you didn’t know this. Be careful There are a couple problems with using pickle files for data […]The post Dump a pickle file to a readable text file first appeared on John D. Cook.

It’s a curious and convenient fact that many decibel values are close to integers [1]: 3 dB ≈ 2 6 dB ≈ 4 7 dB ≈ 5 9 dB ≈ 8 Is base 10 unique in this regard? If we were to look at the analogs of decibels in other bases, would we see a […]The post Duodecibels first appeared on John D. Cook.

With org-mode you can keep data, code, and documentation in one file. Suppose you have an org-mode file containing the following table. #+NAME: mydata | Drug | Patients | |------+----------| | X | 232 | | Y | 351 | | Z | 117 | Note that there cannot be a blank line between the […]The post Keeping data and code together with org-mode first appeared on John D. Cook.

The song YYZ by Rush opens with a theme based on the rhythm of “YYZ” in Morse code: -.-- -.-- --.. YYZ is the designation for the Toronto Pearson International Airport, the main airport serving Toronto. The idea for the song came from hearing the airport identifier in Morse code. However, the song puts no […]The post YYZ and Morse code first appeared on John D. Cook.