1. A work of fiction in which a character goes to a math class

2. A math book published at least 50 years ago

3. A biography of a woman in math

4. A math book that makes you laugh

5. A book with a geometric design on the cover

6. A book related to geometry

7. A nonfiction math book written by a person of color

8. A math book that intimidates you

9. A book about connections between math and biology or medicine

10. A math book not originally written in your native language

11. A math book you meant to read in 2020

12. A math history or biography focused on mathematics or a mathematician from Africa, Asia, or a group indigenous to the Americas or Oceania

Here is a pdf you can print out to track your progress: MathReadingChallenge2021. Feel free to connect with me and other math readers via the Goodreads group or on Twitter using the #MathReadingChallenge2021 hashtag. If you’re looking for another place to discuss math-related books, check out the LThMath Book Club, also on Goodreads. For mathematical fiction inspiration, check out Alex Kasman’s page about Math Fiction.

Happy math reading!

]]>The Wild Iris evoked dirt, family, grief, and God for me. Flowers and plants, working close to the ground, the small intimacies of living with someone else, a distant deity, the difficulty of having emotions. The poet wants to be understood but often has no way of reaching the person or entity by which she wants to be understood.

]]>Here are some things for you to find in the cemetery:

- Someone born 100 years before you
- Someone who lived for at least 100 years
- A noun that is not a name, place, date, or family relationship
- A diacritical mark (accent, umlaut, tilde, etc.)
- A name that is also a plant
- A first or last name you’ve never heard before
- A statue
- A (living, not statue) animal that is not a squirrel, dog, or small bird
- A color picture
- A military veteran
- Headstones next to each other with different last names that start with the same letter
- Someone with your first or last name
- The longest-ago birthdate you can find
- Wind chimes
- A name that starts with the letter Q
- A bench
- Two headstones next to each other facing different directions
- An obelisk
- Music notation
- *A Utah pioneer historical plaque
- *A picture of the Salt Lake Temple
- *A President of the Church of Jesus Christ of Latter-day Saints
- Something you did not expect to find in a cemetery

The three asterisked prompts will probably be difficult to find outside of Salt Lake City. Feel free to replace these prompts with your own based on important people or places near you.

Below are 12 prompts to guide your math-related reading in the coming year, along with two or three books you could (but are in no way obligated to) choose for each prompt. This is not a competition and there are no prizes. Feel free to interpret the prompts in any way you wish and count one book for multiple prompts if that’s your style. I’ve set up a public Goodreads group for anyone who is interested in making recommendations or discussing the books they are reading for this challenge. There are threads for each prompt, as well as some general discussion threads. I have not been a moderator of a Goodreads group before, so please bear with me if there are any hiccups as I get this figured out.

**A work of fiction in which a main character is a mathematician**

*The Housekeeper and the Professor*by Yōko Ogawa

*Binti*by Nnedi Okorafor

*The Mathematician’s Shiva*by Stuart Rojstaczer**A math-related book published the year you were born**

For me, that’s 1983. Your mileage may vary.

*Discrete mathematics : A Computational Approach Using BASIC*by Marvin Marcus

*Invitation to Geometry*by Z. A. Melzak**A biography of a mathematician**

*Remembering Sofya Kovalevskaya*by Michele Audin

*John Napier: Life, Logarithms, and Legacy*by Julian Havil

*Julia: A Life in Mathematics*by Constance Reid**A math book that helps you make something**

*Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist*by Ellie Baker and Susan Goldstine

*Making Mathematics with Needlework*, edited by sarah-marie belcastro and Carolyn Yackel

*Crocheting Adventures with Hyperbolic Planes*by Daina Taimina**A book with a number in the title**

*Life of Pi*by Yann Martel

*The 500 Hats of Bartholomew Cubbins*by Dr. Seuss**A book related to number theory**

*An Illustrated Theory of Numbers*by Martin H. Weissman

*Prime Numbers and the Riemann Hypothesis*by Barry Mazur and William Stein**A nonfiction math book written by a woman**

*Beyond Infinity: An Expedition to the Outer Limits of Mathematics*by Eugenia Cheng

*Mathematics in India*by Kim Plofker

*Power in Numbers: The Rebel Women of Mathematics*by Talithia Williams**A graphic novel about math or mathematicians**

*Prime Suspects*by Andrew Granville and Jennifer Granville

*The Thrilling Adventures of Lovelace and Babbage*by Sydney Padua

*Logicomix*by Apostolos Doxiadis and Christos Papadimitriou**A book about connections between math and the arts**

*Music: A Mathematical Offering*by Dave Benson

*Opt Art: From Mathematical Optimization to Visual Design*by Robert Bosch

*Math Art: Truth, Beauty, and Equations*by Stephen Ornes**A book of poetry with mathematical themes**

*Strange Attractors: Poems of Love and Mathematics,*edited by Sarah Glaz and JoAnne Growney

*Proportions of the Heart: Poems that Play with Mathematics*by Emily Grosholz**A children’s or YA book about math or mathematicians**

*Hidden Human Computers: The Black Women of NASA*by Sue Bradford Edwards and Duchess Harris

*The Boy Who Loved Math: The Improbable Life of Paul Erdős*by Deborah Heiligman and LeUyen Pham**A math-related book you want to give to someone who isn’t sure whether they like math**

*How Not to Be Wrong*by Jordan Ellenberg

*Things to Make and Do in the Fourth Dimension*by Matt Parker

*The Joy of X*by Steven Strogatz

Here is a pdf you can print out to track your progress. Feel free to connect with me and other math readers via the Goodreads group or on Twitter using the #MathReadingChallenge2020 hashtag. If you’re looking for another place to discuss math-related books, check out the LThMath Book Club, also on Goodreads. For mathematical fiction inspiration, check out Alex Kasman’s page about Math Fiction. If we have fun with this, maybe we’ll do it again in 2021.

Happy math reading!

]]>

Today, I live eight degrees farther north than where I grew up. As one would expect, the climate is a bit different. Latitude is part of that: on any given day, it is probably cooler here in Salt Lake City than at my parents’ house in Dallas. But altitude and distances to mountains and oceans play a role as well.

My Texan roots mean I like to be warm. People who think cold is better than heat say you can always put more clothes on, but there’s a limit to how much you can take off. I think they’re vastly underestimating both how tedious it is to put on layer after layer every day for months and the power of a cool shower and a ceiling fan in the summer.

My first winter outside of Texas was the warmest, mildest winter in Chicago’s history—and the worst winter I could imagine. Now, several years later, I am a much more sophisticated layerer, and I have discovered the joys of merino wool. Winter will never be my favorite season, but I have adjusted fairly well to the cold. (Please do not fact check this assertion with my spouse.)

What has surprised me more is the dark. As a kid, I was aware of the fact that days were longer in the summer than the winter. At that time, we still “fell back” from Daylight Saving Time the week before Halloween, and I remember that it was always a bit of a shock how early it got dark when we went trick-or-treating. But in my memories, the dark doesn’t seem to encroach nearly as strongly as it does now.

Last December, I was walking through my neighborhood one evening and had a disconcerting realization: I loved the houses with Christmas lights. I’m not a big holiday or decoration person, and I don’t usually feel warm and fuzzy to see big holiday displays. I wondered as I wandered: Were my Decembers always this dark?

Dallas gets almost exactly 10 hours of daylight this year on December 21. Salt Lake City gets 9 hours and 15 minutes. Eight degrees farther north means 45 fewer minutes of daylight. Obviously, the difference is not confined to the solstice. Every day in December is at least 40 minutes longer in Dallas than it is here. That’s more than 20 hours more daylight there than here in the darkest month of the year. That seems pretty substantial for just eight degrees of latitude. The summer, of course, has as much extra light as the winter has darkness, but I can’t bottle that and use it to get through December. The lights in my neighborhood are little capsules of cheer during the long nights of winter.

I know that my sudden enthusiasm for holiday lights is nothing new. Humans dwelling far from the Equator have probably always found ways to brighten the winter months with lights, decorations, and celebrations.

I doubt I’ll ever be the neighbor with an elaborate reindeer scene or choreographed light and music show on my house in December. But a shift of eight degrees has softened my scroogey tendencies, and I’m ready to celebrate the light.

]]>But perhaps you’re a bit of a Chidi, and furthermore you made the mistake of learning about the geometric mean at some point. The geometric mean is another measure of central tendency, as statisticians say. The geometric mean is like the arithmetic mean, but you change addition to multiplication and division to taking roots. To find the geometric mean of two numbers, multiply them together and take the square root of their product. (This operation can only be performed on two numbers that are either both positive or both negative. But the world is negative enough. Let’s just think about positive numbers!) True to its name, this mean has a nice geometric interpretation: it is the side length of a square with the same area as a rectangle having your two numbers as side lengths. To find the geometric mean of a lot of numbers—let’s say *n* of them—multiply them all together and take the *n*th root.

Knowing about both the arithmetic and geometric means, you are wracked with internal turmoil: Which mean will best represent your numbers?

The arithmetic mean is nice. It seems very balanced and equitable. But the geometric mean has its merits as well. It is a useful tool when you’re working with processes that work multiplicatively instead of additively, like interest rates. It pulls larger numbers closer to smaller numbers more than the arithmetic mean does, whether you are taking the geometric mean of just two numbers or many numbers. The geometric mean might be a better representative than the arithmetic mean or even the median for a data set that has a lot of smaller values and a few large ones—say, income distributions.

Which will it be? Decisions are so hard!

Why not both?

The arithmetic-geometric mean lets you find a number between your two favorite positive numbers that is a compromise between the arithmetic and geometric means, letting your inner Chidi rest easy.

Finding the arithmetic-geometric mean is an iterative process. Each step produces two numbers: the arithmetic and geometric means of the previous two numbers. So starting with, say, 1 and 2, the first step produces the two numbers 3/2 and √2. At the next step, you find the arithmetic mean of 3/2 and √2, which is approximately 1.457, and the geometric mean of 3/2 and √2, which is approximately 1.456. At that point, the two values you’re getting are already very close together, and subsequent iterations will produce two numbers that are arbitrarily close together. The limit of both the arithmetic and geometric means produced in this process is the same, so it is called *the* arithmetic-geometric mean. The arithmetic-geometric mean of 1 and 2 is 1.45679…; a bit disappointing in that it would be more fun if it started 1.4567*8*9, but a satisfying answer nonetheless.

The approximations of the arithmetic-geometric mean of two numbers get very close together very quickly, so the process has been used to find good approximations for irrational numbers, as in this paper about how to use it to approximate π.

What if you have more than two numbers? As far as I can tell, no one has ever defined the arithmetic-geometric mean for an arbitrary set of positive numbers, but that didn’t stop me. I’m not going to use the name arithmetic-geometric mean for the generalization to make sure nobody thinks it’s an “official” math term. Instead, I’ll call it the ditherer’s mean.

For the arithmetic-geometric mean of two numbers, we had an iterative process that gave us two numbers at every step. One way of thinking about it is that we replaced the smallest number with the geometric mean of the previous numbers and the largest number with the arithmetic mean of them. We’ll do the same thing for the ditherer’s mean.

To take the ditherer’s mean of *n* numbers, we want an iterative process that gives us *n* numbers at each step. So at each step, we replace the smallest number from the previous list of numbers with the geometric mean of the previous numbers and the largest number with the arithmetic mean of the numbers.

Let’s take a look at a set of 4 numbers to get a feel for how the process works. We’ll start with the numbers 1, 5, 20, and 26. The arithmetic mean of these numbers is 13, and the geometric mean is approximately 7.14. So we replace the largest and smallest numbers in our first list with 13 and 7.14. Now we have the numbers 5, 7.14, 13, and 20. We repeat the process. The arithmetic mean of those four numbers is about 11.285. The geometric mean is about 9.82. Now our list is 7.14, 9.82, 11.285, and 13. The arithmetic mean is 10.31 and the geometric mean is 10.07. Keep going: 9.82, 10.07, 10.31, 11.285. Then 10.07, 10.31, 10.35, 10.37. Progress! A few more iterations, and it’s clear the numbers are getting closer and closer together, landing around 10.3.

The arithmetic-geometric mean of two numbers has been a useful concept for mathematics. The iterative process that produces it converges very quickly, so it has been used to compute approximations quickly and accurately, as in this paper about computing π using the arithmetic-geometric mean. As far as I can tell, mathematicians have not yet found use for the ditherer’s mean, but I hope it will help some indecisive people take an average and move on with their lives.

There you have it: Now you can find an average of a set of positive numbers without having to choose between their arithmetic and geometric means. Isn’t it wonderful the way math always gives you a tidy answer with no room for uncertainty or ambiguity?

Wait, what’s that? Harmonic mean? Heronian mean? Identric mean? Nooooooooo!

]]>Almost two years ago, a friend asked if I knew of any page-a-day calendars about math. I poked around a bit and could find various page-a-day puzzle calendars but nothing that was quite what my friend had in mind. We went to the AMS to see if they would be interested in publishing a math calendar that didn’t focus on puzzles or arithmetic but on a much broader range of mathematical topics. Yes, puzzles are fun, but there’s a lot more to math than that. What about math-inspired art, math history and mathematicians, and all the weird and wonderful shapes and spaces mathematicians dream up? We wanted to make a math calendar that reflects the breadth of the subject and the diversity of the people who have done it, today and throughout history.

Much to our delight, the AMS said yes, and now this object exists in the world! I did most of the writing, but about 50 mathematicians, artists, and poets contributed images, poems, coloring pages, activities and puzzles for your enjoyment.

You might know that I’m a total scrooge about holidays, especially Pi Day. Well, sometimes you’ve got to give the people what they want, and the people want Pi Day! To make up for my former scrooginess, I put a bunch of them in the calendar, and I even included some other important holidays: Swiss Cheese Day, e Day, World Snake Day, the International Day of Sign Languages, and of course Constitution Day in Nepal, to name a few.

Math is done by people, so I made sure to include a lot of them. Sure, you’ll see Gauss, Euler, and Noether on the calendar, but you’ll also meet Anna Julia Cooper, Chuan-Chih Hsiung, Dattathreya Kaprekar, Graciela Salicrup, and no fewer than four Olgas. (Rejected subtitle for the project: Four Olgas, No Mercy.) You’ll learn about mathematics from every continent except Antarctica (get with it, penguins!) and from thousands of years of history. There are jokes, riddles, puns, and a few days where I’m just trolling you. Sorry about that. Feel free to send me a video of yourself throwing the calendar across the room on April 16.

The calendar is not specific to 2020. You can use it any year! If you’re stumbling on this post well after I wrote it in November 2019, you should still be able to enjoy it. It will still be a steadfast mathematical companion as you face whatever new year it is. You can even drill some holes in the top and put it on a couple of binder rings and start using it at any point in the year.

I’m not used to asking readers to pay money directly for my writing. I love that so much of my work is available for free for anyone with an internet connection. But I really am proud of this calendar, and I think anyone who is interested in any aspect of math will learn something from this calendar—and perhaps even be entertained in the process.

]]>On our most recent episode of My Favorite Theorem, Kevin Knudson and I talked with University of Nebraska mathematician Judy Walker, who works in the field of coding theory. You can find the audio, transcript, and show notes at kpknudson.com.

Walker’s field, coding theory, is about how to transmit information over noisy channels and correct the errors that arise when we do. So it seemed very appropriate that our podcast itself was an example of transmitting information over a noisy channel. For our podcasts, we use a live video chat. Technology and internet connections are not perfect, but we could talk to each other despite the glitches that crept in. As Walker explained, the very error-correcting codes she studies are used in online communication. I was excited about the fact that the recording of the podcast itself was an example of the math discussed in the podcast.

Walker’s theorem is probably the most difficult theorem we’ve had on the podcast, at least in terms of being able to spell it based on hearing someone say it. It’s called the Tsfasman-Vladut-Zink theorem, and it is related to how efficient error-correcting codes can be. The basic idea is that for a long time, researchers knew about a lower bound for efficient codes, the Gilbert-Varshamov bound, that says that there are codes at least this efficient. For 30 years, no one could find codes more efficient than the Gilbert-Varshamov bound, and they thought perhaps it was also an upper bound. But with the Tsfasman-Vladut-Zink theorem, researchers showed that there were codes that were more efficient than the Gilbert-Varshamov bound and that realized a known upper bound for efficiency.

Of course, this big picture description hides a lot of the devilish details, like how exactly to measure efficiency. To learn more about those details and the ins and outs of error-correcting codes—and to find out why lemon zest is a perfect pairing for the Tsfasman-Vladut-Zink theorem—listen to or read the full episode here.

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