Math on For Your Consideration

From the Vault – Möbius Strips

Sun, 25 Jan 2026 00:00:00 +0000

I last edited this draft back in 2020. I vaguely recall wanting to write more here and probably revisit the diagrams, which were made entirely using Asymptote. But it’s a good post and worth publishing.

I made my first Möbius strip during my sophomore year of high school in math class. Since then, I have been fascinated by their construction and mere existence. I share them with every person I can, basing my own presentation on a mix of my math teacher’s introduction, provided so many years ago, and a wonderful talk given by Matt Parker at the Royal Society.

What is Good Mathematics? by Terence Tao

Mon, 30 Dec 2024 00:00:00 +0000

Terence Tao is a gift to the mathematical community. He is an excellent collaborator, a talented communicator, and one of the broadest and sharpest minds working today. Eighteen years ago, when he was just past thirty years old, he wrote the essay What is Good Mathematics?. I discovered this when Steven Strogatz invited Tao onto his podcast The Joy of Why to discuss how well this essay holds up.

Both of these are great, but start with the opening section of Tao’s essay, where he lists twenty-one ways to measure mathematics as being “good.” It displays his impressive clarity of thought and writing ability and evinces how the professional mathematics scene isn’t what one may have expected.

I Just Learned What BBCOR Means

Wed, 15 May 2024 00:00:00 +0000

In early high school, I remember all the hubbub about requiring metal baseball bats to align to the BBCOR standard. Given the pronunciation of this (“Bee-Bee-Core”), I always assumed it was a regulation about what specific materials must be used to make the bat. That’s only true insofar as the standard actually defines a material property.

BBCOR stands for “Batted Ball Coefficient of Restitution”. In other words, the standard tells you how elastic the collision between ball and bat is allowed to be. This standard was designed to dull metal bats in an effort to protect pitchers, the most likely players to be grievously injured by a batted ball. It was adopted by the NCAA in 2011, and most youth leagues that I’m aware of followed their lead. I used BBCOR-certified bats throughout high school, and continue to do so in my adult league.

Problem-First Thinking

Mon, 13 Nov 2023 00:00:00 +0000

I was in a position to provide some talking points for my company’s upper-level math textbooks. It was written in the aftermath of customer-induced pique regarding how we sell ourselves. While it’s focused on my company, the core idea of a problem-first approach extends beyond what we do in particular.

Rolling Every Number on a Die

Mon, 07 Aug 2023 00:00:00 +0000

Thanks to Mikhail for posing this problem.

On average, how many rolls would it take to see every face on a particular die?

2022 Blogmas Day 23 - Defensive Runs Saved and Defensive WAR

Fri, 23 Dec 2022 00:00:00 +0000

Hot on the heels of Ultimate Zone Rating is another tool with the same goal: measure a player’s defensive contributions in terms of runs saved. This other statistic, Defensive Runs Saved (DRS), doesn’t just have a better name, but in my experience is favored over UZR in most situations.

2022 Blogmas Day 21 - Fielding Percentage and Range Factor

Wed, 21 Dec 2022 00:00:00 +0000

Leading into the home stretch, let’s discuss some defensive statistics. As a category, these represent recent innovations in baseball, as defense is the most dynamic aspect of the game. Players can change where they are positioned, which affects their ability to get to a ball. As we gather more data — ball speed off the bat and launch angle — we can better analyze defensive ability. But it’s still messy.

2022 Blogmas Day 20 - Pitcher WAR

Tue, 20 Dec 2022 00:00:00 +0000

We’ve already addressed the concept behind WAR (Wins Above Replacement) in the abstract and for offensive players in particular. Today, we’ll consider how we quantify runs for pitchers and overview calculating their WAR.

2022 Blogmas Day 19 - Leverage Index

Mon, 19 Dec 2022 00:00:00 +0000

Relief pitchers can enter the game in very different situations, and that should be accounted for when evaluating their performance. Coming in to clean up a 12-0 victory (or defeat) is different than coming in during a close, tense game with a lot on the line. We measure this disparity using Leverage Index (LI).

2022 Blogmas Day 18 - Fixing ERA+ With a Minus

Sun, 18 Dec 2022 00:00:00 +0000

ERA+ is a funny, messy statistic. Unlike OPS+, it doesn’t tell us how much better a player is than the league: it instead tells us how much worse the league is than the player. This ignores our intuition and causes unnecessary confusion. Furthermore, it makes it harder to use as a tool for direct comparison: someone with a 200 ERA+ is not twice as good as someone with a 150 ERA+, while that would effectively be true with OPS+ (and similar offensive statistics.)

2022 Blogmas Day 17 - Expected FIP

Sat, 17 Dec 2022 00:00:00 +0000

FIP is a cool, clever, and simple statistic to try and normalize how we work with ERA in a way that simply ignores defensive factors. However, we can ignore, or normalize for, at least one other factor to adjust the comparison.

2022 Blogmas Day 15 - Fielding Independent Pitching

Thu, 15 Dec 2022 00:00:00 +0000

Remember many days ago when we talked about how pesky it is to have a ball in play? That’s as true for pitchers as it is for hitters. So, some enterprising folks struck out to roughly measure ERA in a way that ignores balls put in play, instead focusing entirely on the Three True Outcomes. Fielding Independent Pitching (FIP) is a defense-agnostic statistic to compare against ERA, and in some ways is an ideal “predictor” statistic for ERA.

2022 Blogmas Day 14 - ERA+

Wed, 14 Dec 2022 00:00:00 +0000

That’s right, the “+” statistics exist in the realm of pitching. Park adjustments and a percentage normalization is a natural extension of ERA, and makes it more meaningful for directly comparing pitchers, even across generations.

2022 Blogmas Day 13 - Walks and Hits per Inning Pitched

Tue, 13 Dec 2022 00:00:00 +0000

While ERA is a measure of overall pitcher effectiveness, there’s a more nuanced statistic that considers how frequently a pitcher allows runners on base. Walks and Hits per Inning Pitched (WHIP) is easy to calculate, has a cool acronym, and is another factor to consider when measuring how good a pitcher is at keeping hitters from doing their jobs.

2022 Blogmas Day 11 - Earned Run Average

Sun, 11 Dec 2022 00:00:00 +0000

Let us leave the world of sluggers and slap-hitters behind us, and move 60 feet forward to the pitcher’s mound. I’ve always enjoyed and valued pitching over hitting in my baseball life. As such, the first ten days of hitting required me to learn more than I anticipated, while I feel more prepared to handle this section of Blogmas.¹ We’re not starting as simple as possible, but we are starting with the most important of the original pitching statistics: how good are you at preventing runs from scoring?

2022 Blogmas Day 9 - Offensive Wins Above Replacement

Fri, 09 Dec 2022 00:00:00 +0000

Today we move onto one of the gems of advanced baseball statistics¹ that generalizes player value beyond concrete elements like hits and runs, and looks specifically at how many wins a player is worth.

2022 Blogmas Day 8 - Weighted Runs Above Average and Runs Created

Thu, 08 Dec 2022 00:00:00 +0000

Let’s dive back into the weeds. Hitting the ball and getting on base are important jobs for hitters. But what actually wins the game? Scoring more runs than the other team. Today, we’ll look at two ways of quantifying run production, both of which are weighted in the same way we saw with wOBA.

2022 Blogmas Day 7 - Batting Average on Balls in Play

Wed, 07 Dec 2022 00:00:00 +0000

Today we’ll look at a statistic that is interesting for both the offensive and defensive side of baseball. When a ball is put in play, how often does it go for a hit?

2022 Blogmas Day 6 - Isolated Power

Tue, 06 Dec 2022 00:00:00 +0000

In another day of simpler statistics, Isolated Power (ISO) is derived from slugging percentage. It’s calculated in the exact same way, but ignores singles. Hence, it is isolating the power of a hitter, defined as the rate at which they get extra-base hits.

2022 Blogmas Day 5 - Three True Outcomes

Mon, 05 Dec 2022 00:00:00 +0000

Let’s take a step back today and consider an overall shift in the way baseball has been played over the last few decades.

2022 Blogmas Day 4 - Weighted On Base Average

Sun, 04 Dec 2022 00:00:00 +0000

You may have recognized that OBP and SLG each measure slightly different things. OBP is only a measure of whether you got on base, while SLG makes a big fuss over your hits in particular, and how “big” they were. OPS (and even better, OPS+) are a quick summary of these combined, which is cool. However, OPS treats OBP and SLG as essentially equal in their importance, while getting on base is demonstrably of higher value. If you want a more accurate measure of offensive production that is (relatively) simple to calculate, look no further than weighted on base average, or wOBA.

2022 Blogmas Day 3 - OPS and OPS+

Sat, 03 Dec 2022 00:00:00 +0000

BA, OBP, and SLG make up what is typically known as the triple slash line for hitting. You’ll often see something written like .275/.310/.382. That is BA, OBP, and SLG in order. However, we do like our summary statistics that reduce how many numbers we need to compare for making quick calls, so we use OPS: On Base plus Slugging.

2022 Blogmas Day 2 - On Base and Slugging Percentage

Fri, 02 Dec 2022 00:00:00 +0000

We move on from batting average to discuss a much simpler statistic, as well as a weighted version of batting average. These two new statistics, along with batting average, have over the last decade come to comprise the base-level portfolio of a batter’s hitting abilities.

2022 Blogmas Day 1 - Batting Average

Thu, 01 Dec 2022 00:00:00 +0000

Though not the most intuitive statistic, batting average was a gold-standard statistic for many decades, and is still considered important today.

2022 Blogmas Day 0 - Baseball Statistics

Wed, 30 Nov 2022 00:00:00 +0000

After a couple of years toying with the idea, I’ve finally decided to do a blogmas, my own take on creative challenges such as Advent of Code. In this case, you’ll be treated to 25 days of posts about the wonderful world of baseball statistics.

Agonizing Over Engagement

Mon, 10 Jan 2022 00:00:00 +0000

Applying mathematical topics and skills to the real world is an often-discussed tactic to engage students who are otherwise dismissive of, or frustrated with, mathematics as a whole. By connecting the math a student does with real life situations—particularly skills they will “need” in the future, but also more abstracted situations that pertain to real life—some people expect students to gain an appreciation for what mathematics can do, and feel inclined to study it so they can actively participate in these various applications.

Dangers of Mathematical Rules

Mon, 13 Sep 2021 00:00:00 +0000

Do you remember when you first learned how to round numbers? For example, to round 687 to the nearest hundred, we notice it’s between 600 and 700. Since 687 is closer to 700 than it is to 600, we round up to 700. Maybe you even learned a rule: Look at the digit to the right of the one you’re rounding. If it’s 5 or greater, round up. If it’s 4 or less, round down. Rules can be good, but they can also be dangerous.

I'm an Author (Almost a Year Late)

Mon, 02 Aug 2021 00:00:00 +0000

With everything that happened last year, I managed to never write about my co-authorship of the Level 3 Puzzle Book put out by my company. It was the largest project I’ve been on, and the content I’m most proud of producing. Not only did I get to write and review close to half the puzzles in the book, I managed to learn the basics of InDesign and help compile the book into its final form.

Evidence-Based Education Part 0

Mon, 12 Apr 2021 00:00:00 +0000

As part of a new role at work, I’ve been researching aspects of funding for education in the United States. A recurring phrase is evidence-based interactions. I’m going to explore some of the facets of this idea, from its definitions and origins, to potential ramifications of its increasing usage in education.

This post is a placeholder for those future investigations. The educational landscape is of interest to many people, and learning about some of the guiding tenets can help explain a lot of the situations states are finding themselves in. I’m excited to be diving in.

Socks and Underwear

Mon, 15 Mar 2021 00:00:00 +0000

Admittedly, this post is ripping off a conversation that happened on a work Slack channel the other week. Here’s the general idea: If you count in a particular way, there are 12 options for how to put on your underwear, and 12 (ordered) options for putting on socks and shoes. Since there are the same number of options, we can create a bijection between the two scenarios. In other words, each choice of underwear arrangement will correspond to exactly one choice for socks and shoes.

The question is what bijection is “natural”, for whatever definition of natural you choose to have in this strange scenario. Let’s explore it a bit.

Geometric Series: The Basics

Mon, 13 Jul 2020 00:00:00 +0000

I was just tutoring someone in their Calculus 2 course, and we did a problem on geometric series. When I was a student, I remember memorizing several facts about geometric series without really grokking the algebraic details of why the formulas you come up with are so nice. Since it is fresh in my mind, I figured I’d go through the standard approach.

The Gold Standard for Math Typesetting on the Internet

Mon, 15 Jun 2020 00:00:00 +0000

Through a wonderful little game called TeXnique, I became aware of the online typesetting system KaTeX. Unlike MathJax, which is immensely finicky, slow, and has limited support for proper LaTeX, the KaTeX system supports scaling of rendered images, automatically detects the text color of my site and adapts as necessary, and has a fuller implementation of LaTeX commands.

A Short Tribute to John Conway

Mon, 13 Apr 2020 00:00:00 +0000

John Conway passed away a few days ago. He was a favorite among mathematical educators and communicators. He was involved in a large array of interesting projects and results during his career, many that can easily fascinate even the most disinterested people when it comes to mathematics.

Here are a few things he worked on, listed roughly in the order in which I came across them in my mathematical life.

Sniped by Math History

Mon, 02 Sep 2019 00:00:00 +0000

As with most nerds, I have a predisposition to being nerd sniped. Sometimes it’s a puzzle, but more often it’s simply an idea or the opportunity for a new project. After a conversation with a few coworkers, I’ve latched onto the idea of developing a summer course focused on the history of mathematics.

A Small Open Math Project

Mon, 26 Aug 2019 00:00:00 +0000

The other week, a student posted on /r/math that they made a summary of multivariable calculus. As they point out, it’s an 80-page summary, but at least it’s shorter than the textbook it followed.

The student admitted they’d used Google Docs, as opposed to LaTeX, to make these notes, and the images they used were just taken from the internet. As a result, a Github project was opened up for people to turn it into LaTeX, and I’ve taken it upon myself to slowly recreate the figures using Asymptote.

Dynamical Systems

Mon, 05 Aug 2019 00:00:00 +0000

Earlier this year, I started working on a programming project. I recently added a little more to it, and worked out a handful of kinks. It’s been fun, but I think I’m winding down on it. I’m sure in some spurts of passion I’ll work on it some more, but I’ve moved on to another Python-related project, that also gets to the core of my interests: mathematics.

Algorithmic Thinking and Metacognition

Mon, 15 Jul 2019 00:00:00 +0000

I’m teaching a math camp for students entering 6th grade. It’s my first time being the teacher of record for a course, and luckily it only took a few minutes for my anxiety to subside. It’s a small group – only 9 kids – so it reminds me of my days being a camp counselor, except now it’s talking about math with very advanced kids for 3 hours a day. It’s been a blast.

Triangle Inequality for Altitudes

Mon, 08 Jul 2019 00:00:00 +0000

I’ve written a short post on the triangle inequality for altitudes. As usual, I put it over on the math blog I have with my company, just because the support for literally everything is much nicer.

But, in the off chance you want it, I have a PDF. I’ve played around with a wonderful program called Typora, which I’ve been using for a few months to write any blog posts when I’m not in my iPad. It has great Mathjax support (miles better than Wordpress), and so it’s easy to switch between the blog I use and this program, and I can export cleanly to PDF, as well as EPub and other formats that are really difficult to work with if I were using straight up LaTeX.

Are You Ready?

Mon, 10 Jun 2019 00:00:00 +0000

Assessments are a tricky business. Writing an exam that successfully tests a person’s knowledge or abilities, without inadvertently giving preference or advantage to certain demographics, is very difficult. The examinations I’ve written so far for my job fall into the category of testing whether a student has mastered a certain curriculum. After a couple of months of class, we give them an exam to check if they learned all that they were supposed to. Everybody is used to such tests, and everybody has experienced them.

Birthday Index

Mon, 15 Apr 2019 00:00:00 +0000

It was my older sister’s birthday yesterday, and it had me thinking about how we talk about birthdays. There’s a peculiar inconsistency in the language we use which I vaguely noticed for the first time a couple of years ago, but never really pursued the thought.

We index our birthdays from 0.

A Few Math Games and Puzzles

Mon, 04 Mar 2019 00:00:00 +0000

Up front, here is a link to my math blog that has the meat of this post. Also, there will be a PDF version attached.

I’ve finally gotten around to writing another math post. There’s a mix of reasons why I’ve been writing about a lot of other things. First, there have just been other topics in the front of my mind that I felt like writing about in the moment. Second, there were not any particular pieces of math I felt inspired to write about. And third, preparation of a math post is significantly more work for me, as I care a lot about being accurate, precise, and clear in my presentation.

That's a Bad Pirson

Mon, 07 Jan 2019 00:00:00 +0000

I was on one of my favorite subreddits today, /r/learnmath (of course I sort by New), and a question came up that ultimately came down to understanding order of operations. These always frustrate me because they stem from a poor understanding of how the order of operations work; furthermore, any problem involving order of operations can be fixed and cleared up with an additional set of parentheses.

Consider this math problem below – the type I see making its way around Facebook.

Holes in a Paper Towel Roll

Mon, 24 Dec 2018 00:00:00 +0000

At work the other day, we had a bit of a break to celebrated the month’s birthdays. During this time – my coworkers sitting around a large table, munching on pie and ice cream – a conversation broke out. How many holes are there in a paper towel roll: 1 or 2?

Of course, there is the joke that if you claim 1, then there is only one hole of consequence in the human body: that which leads from your mouth to the escape of your digestive system. At that point you’re just talking out of your ass.

Notes On Future Mathematics Posts

Tue, 30 Oct 2018 00:00:00 +0000

Updated November 23, 2022

I’m going back through my posts and recategorizing them, and noticed this one. My plan for having a secondary spot for math posts didn’t take off after I moved my blog to WordPress properly, and discovered the excellent $\KaTeX$ plugin for rendering math.

Original Post

I recently remembered my company, AoPS supports blog creation for their users. In particular, it has the full functionality I’m used to on their message board. In particular, they have native $\LaTeX$ support, in addition to support for the Asymptote vector graphics language. This makes writing math significantly easier on my end, and significantly easier to interact with on the reader’s end.

Mathbook: Functions as Sets

Tue, 02 Oct 2018 00:00:00 +0000

In this little tutorial, we’ll expand on what we’ve learned about sets and functions. Specifically, we’ll double-down on the claim that sets are vital to everything we do in mathematics. Functions are not just a way to describe interactions between sets: functions are sets!

Mathbook: Introduction to Functions

Tue, 18 Sep 2018 00:00:00 +0000

(Edit, 10/29) This post has been ported to a math-focused blog here. The goal of the change is improved typesetting for me, and improved readability and interaction for you. See this post for more information.

This is the next post in the series of posts inspired by my brief work for Mathbook. Previously, we talked about sets. Once we have this basic object, we can start to create, define and communicate relationships and patterns between sets. Mathematics uses the language of a function to describe these relationships.

Mathbook: Introduction to Sets

Mon, 10 Sep 2018 00:00:00 +0000

(Edit 6/7/2020) No more Mathbook. I finally discovered the KaTeX plugin for Wordpress, allowing beautiful typesetting once again.

Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I’ll put down in two parts on this blog. It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator’s idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons. Moving forward, I’ll occasionally add a new post here to that effect. The people in my life don’t always understand the math I learned, so this is part of my effort in showing it.

Gabriel's Horn

Mon, 20 Aug 2018 00:00:00 +0000

(Edited 6/7/2020 for improved typesetting)

Here is one of my favorite “paradoxes” in mathematics. Many students learn it in a first year calculus course. It is called Gabriel’s Horn.

Card Game Simulation

Mon, 23 Jul 2018 00:00:00 +0000

I had another busy week, so I’m taking advantage of old stuff I can recycle. A month or two ago, I was playing a Solitaire variation my parents taught me when I was younger, and I realized that it was a completely deterministic game once the deck was shuffled. That is, unlike traditional solitaire, there was no element of choice by the player. As such, it made it very easy to write a simulation of it and analyze the details. The very brief report I wrote up is here, and the simulation code (which is also linked in the report) is here. The short version, is that it is a break-even game on average, which is pretty interesting. Furthermore, the overall result is normally distributed around breaking even. I’m trying to include a more well-rounded amount of content here, since math is still very close to my heart and I’d like to only maintain one sight for everything. It will continue to be a mix of things, so that we’re all on the same page.

Purchasing Pizza

Mon, 09 Jul 2018 00:00:00 +0000

Suppose you are at a pizza restaurant with your friends. You all agree you want to buy pizza to maximize your pizza-per-dollar. There’s an easy way to make comparisons between pizza sizes and figure out what the best deal is.

My Senior Thesis

Sun, 17 Jun 2018 00:00:00 +0000

Just to have this out for people to look at, here is my honors thesis from my degree. It is broadly about some of the mathematics behind a particular phenomenon in quantum mechanics.

Thesis

Thesis Formatted as a Book

Fun Coin Flips

Sat, 18 Feb 2017 00:00:00 +0000

I learned an interesting fact in my Stochastic Processes class the other day, and I managed to come up with an easier way to present it than using Markov chains (which are really cool, but not conducive to making a good blog post).

Monty Hall and Gambler's Ruin: A Third Small Step into Mathematics

Wed, 30 Nov 2016 00:00:00 +0000

Today we will be looking at two problems in probability: the Monty Hall problem and Gambler’s Ruin. These are two common probability “brain teasers”. For the Monty Hall problem, it feels paradoxical when you first learn about it, while the Gambler’s Ruin is instructive and important as you go forward in life. With that brief introduction, let us begin! There was a game show (so I have been told, though I am too young to have watched it) hosted by Monty Hall. One of the main elements of the show was a classic three-door setup, where behind two doors there were goats, and behind the third door there was a brand new car waiting to be won. The way this would go is that Monty Hall would ask you, the contestant, what door you wished to select. You picked door one, two or three. Then Monty Hall, bold and clever, would open up one of the two doors that you did not pick, only to reveal one of the goats! At this point in the game he would come back to you with a smirk, asking if you wished to stay with the door you picked at first, or if you wanted to switch to the other closed door. The problem is: in order to maximize the probability that you win the car, do you switch or stay (or does it even matter)?

"Oh, You're Studying Math?" A First Small Step into Mathematics

Sat, 12 Nov 2016 00:00:00 +0000

“That sounds hard. I used to like math, but then I had a bad teacher and realized I just wasn’t any good at it anymore. I think it’s really cool that you like it though, and that you can teach other people about it. That’s what you want to do, right? Become a math teacher? That sounds really great. I couldn’t be a teacher, especially for math, that sounds way too difficult.