Trevor Project Donations

I recently put a short story I wrote on Amazon. It’s called When You Come Back.

You can find a link to it here.

When I originally wrote it, mental illness was not necessarily on my mind. But, the majority of its readers have told me it resonates with them to a fairly strong degree. So, I’ve put it up for $1 and any proceeds I receive will go towards The Trevor Project, in support of mental illness assistance.

If you’re not interested in the short story, or in providing Amazon with some of the overhead that comes with buying a Kindle book, you can donate to them directly.

If you want the story in PDF form, you can contact me.

Mathbook: Introduction to Sets

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.

Continue reading “Mathbook: Introduction to Sets”

New Chapter

I started my first post-graduation job at Art of Problem Solving this past week. I was an intern here last summer, and I was lucky enough for that to lead to a job. My official role is “Curriculum Developer”. I work on developing their elementary school math curriculum, as that is their current focus. It’s an incredibly good job, with fantastically intelligent and caring people.

While it is a great company whose mission I am deeply invested in, and San Diego is a beautiful place to be, these first few days have been very tough. It was weird driving across the country, having a good time, then suddenly getting to work. I’m living on my own, in a room I’m renting (technically an AirBnB) from a nice lady. I know the area from last summer, but I’m still getting over a mental hurdle of actually going out and doing things. Although at the time I write this I’ll have only been here a few days, I’m already feeling antsy. It’s strange.

Perhaps the biggest reason for my feelings is that I spent my entire life in Minnesota. I am very rooted there, and despite many friends leaving who have also graduated, there are many people I’ve left behind. I’m leaving the comfortable world of academia to work at a place where I have no true connections. It’s a rather isolating feeling that I am working through.

Yet, this is something I correctly anticipated. I have been solidifying Operation: Have a Conversation and Comical Start as ways to keep in contact with people. I’ve reached out to people (or luckily have had them reach out to me) to stay in contact via phone calls or letters. And I also committed to myself that I would write weekly on this blog, and not worry about people reading it. It’s just a good thing to have on my schedule, both for the purpose of self-reflection, and to stop myself from falling idle after I do a day of work.

I already reached out to the San Diego Concert Band, a local community band that has fairly open policies for joining. I’ll be rehearsing with them regularly starting next week, which I am incredibly thrilled for. Although my percussion chops are not what they once were, they will improve and I will be better off for having the experience. I also plan on finding a group (or maybe just a person or two) to try and play tennis with. It’s an easy sport to play as long as you have another person, and I definitely enjoy playing it. Ideally I’d find a softball league as well, and I also have a long book list to get through.

Despite a touch of melancholy and some misgivings on traveling so far, I am excited for what is to come. Knowing that my time here is rather indefinite, I can feel more comfortable finding my place and joining new things. I have more opportunity to be involved and help myself as I go along. It’s an interesting time.

Movie Review: “Christopher Robin”

The other day my girlfriend and I went to see the film Christopher Robin, all about the titular character outgrowing his friends in the 100 acre wood, and slowly finding his way back to childhood. After leaving the movie close to tears (my girlfriend did cry multiple times), here was my one line review:

The entire movie was super predictable, but it was so well-done and moving that I didn’t even care.

Continue reading “Movie Review: “Christopher Robin””

Gabriel’s Horn

Here is one of my favorite “paradoxes” in mathematics, that many people will learn in a first year calculus course. It is called “Gabriel’s Horn.”

First, we take the function f(x) = \dfrac{1}{x}. This is a curve that we can imagine beginning at the point (1,1), then quickly sloping down towards the x-axis at which point it becomes nearly horizontal. A plot is shown below.


Now imagine taking that curve and rotating it around the x-axis, forming an infinitely long shape that looks like the bell of a trumpet, or horn. After doing so, we get a shape that looks roughly like this:


There are two things we wish to determine about this new three-dimensional figure: its volume, and its surface area. We can think of the volume simply as how much stuff we need to fill it completely, while the surface area is how much paint we need to coat the surface. As we will show, this figure has finite volume but infinite surface area. What this means is that we would need an infinite amount of paint in order to just cover the inside surface with a layer of paint. However, if we really wanted to paint the inside, we could take our finite amount of paint and just “pour it in” the horn. We could fill up this infinite horn with a finite amount of paint, and thus end up painting the inside.

The reason this becomes fun is because this is an infinite object. The skinny part of the bell goes on forever, toward +\infty. But this is no obstacle for some tools in calculus. The main idea of calculus is to split your object into arbitrarily thin or short pieces. In our case, we will take an arbitrary vertical slice out of our horn at some horizontal point we’ll call x_1. The disk we get as a result will be circular, and have some radius r. Consider the figure below.


The radius is the distance from the x-axis to the curve f(x) = \dfrac{1}{x}. But by definition, this is just the y-coordinate, f(x_1) = \dfrac{1}{x_1}. So, for any arbitrary vertical slice taken out of our horn, we can find the radius. This allows us to easily find the circumference (2\pi r) and area (\pi r^2) of each disk.

At this point, we’ll have to assume some knowledge of calculus. Consider taking every possible arbitrarily thin disk in the horn, infinitely many of them! If we add their areas over and over again, \pi r^2 = \pi \left(\dfrac{1}{x}\right)^2, multiplied by their tiny piece of width we call dx, we will obtain the volume of the horn overall. We can do similarly with the circumference, multiplying each tiny bit of circumference, 2 \pi r = 2\pi \dfrac{1}{x}, with that same bit of width dx. This will give us our surface area.

We do this formally by using integration. In particular, we use an “improper” integral, and technically have to take a limit (of course, once I finished calculus I got too lazy to write out those steps normally, but I will now for completeness.)

For volume, our integral is

\displaystyle \text{Volume} =  \pi \int\limits_1^\infty \dfrac{1}{x^2}dx = \pi \lim\limits_{b\to\infty} \int\limits_1^b \dfrac{1}{x^2}dx

We can evaluate this integral directly, getting

\displaystyle \text{Volume} = \pi \lim\limits_{b\to\infty}\left. -\dfrac{1}{x} \right|_1^b = \pi \lim\limits_{b\to\infty} \left( -\dfrac{1}{b} + 1\right) = \boxed{\pi}.

That -\dfrac{1}{b} goes to 0, so that term disappears. As we stated earlier, the volume is finite. Choose your units, and the volume of that infinite horn above is just \pi. Neat!

We can do similarly for the surface area, but our integral will diverge.

\displaystyle \text{Surface Area} = 2\pi\int\limits_1^\infty \dfrac{1}{x}dx = 2\pi \lim\limits_{b\to\infty} \int_1^b \dfrac{1}{x}dx = 2\pi \lim\limits_{b\to\infty} \left(\ln b - \ln 1\right) = \boxed{\infty}.

Since \ln 1= 0, we just get \ln b going off to infinity, hence the surface area is infinite.

There are many cool problems like this, some with significantly less background necessary to formulate. I’m hoping to mix these in with some other types of posts.

Abducted: A 24-Hour Musical

This past weekend I had the fantastic experience of playing drumset in a musical put together in only 24 hours. My friend Tim, along with his friend Adam, wrote the entirety of the show. We showed up at Friday on 7pm, with nobody having seen the script or music except the writers. We then performed the musical — lines memorized, music rehearsed, choreography and blocking complete — at 7pm (and 9pm) Saturday evening.

I had an extremely good time. The music was engaging and written with some interpretation allowable, as all the members of the pit were experienced in this musical scenario. We had a lot of fun putting things together quickly, and were quite successful in performing our parts within a few hours.

The show was broadly a satirical take on the characters from Scooby-Doo. In addition to the normal gang (whose names are never explicitly stated at any point in the show), there is the scapegoat Brian, who is Daphne’s current boyfriend. He is verbally abused throughout the show, with some light slapping. In addition, Scooby-Doo is just a man in a Scooby-Doo outfit (naturally), although an old Hermit we meet at the beginning addresses this fact:

Velma: Oh, that’s our anthropomorphic dog. He loves food, and hates ghosts. So, we keep him on a leash and force him to solve crimes!

Hermit (Cooper): That’s not a dog! That is clearly a man in constant pain!

Fred: (Firmly, maintaining eye contact.) It’s a dog.

The fact that the gang keeps a man in a suit on a leash, and is either indifferent to this fact or somehow unaware of its humanity, is repeated throughout the show. Every time a character (normally Brian) goes to take his leash, Scooby screams in terror. At one point, a completely silent scene opens with Scooby alone on stage. Scared by the audience, he slowly stands on both legs, moves forward, and proceeds to intently say Help me! to various members of the audience.

What I hope to show with these descriptions, as well as the plot description to follow, is the creativity and fun that Tim and Adam bring to the shows they write, as well as some interest in watching their other shows online.

Now for the plot. The aforementioned hermit opens the show, describing to Fred and Velma how he was abducted and probed by aliens. Once all the characters were quickly introduced, they were subsequently abducted. During the musical sequence, a small creature, reminiscent of the chest-bursting alien from Alien, kills Fred during the “probing” procedure. We then meet the head of the ship, Marvin.

Shaggy accuses him of working too hard, so he takes Marvin away to “relax”. We later learn their natural high is Captain Crunch.

Daphne falls into a motherly love, with a bit of sexual tension, with the alien in Fred’s chest, much to Brian’s dismay. Velma proceeds to look for clues, until she meets another female alien with the exact same disposition and actions. Some innuendos occur.

Brian sings his heart out to the audience, beautifully I may add.

We finally learn the Hermit is on the ship. His plan was to run everybody out of town with talk about aliens, so he could have the oil deposit he discovered entirely to himself. Of course, he is now abducted by aliens so there is not much to do. Brian and Scooby were within earshot, and attempt to tell the rest of the gang.

Daphne appears, crazed, yielding a gun, rounding up the aliens and, as Marvin put it, “Yelling lines from Alien 3.” Of course, the alien in Fred’s chest talks her down, at which point another chest-bursting alien emerges from Daphne. Those two aliens go to town.

Brian explains to Velma that he solved the crime, and attempts to remove the mask (as is customary). Of course, being Brian, he takes some other alien and nearly chokes it. At this point, Scooby goes up to the Hermit, who is wearing a very obvious mask, and removes it. It is revealed the culprit is none other than D.B. Cooper.

At this point, we have “A case with no loose ends”, as Shaggy puts it. Of course we then realize the characters are still on an alien spaceship. Brian asks Marvin if they can be dropped off on Earth. Marvin declines, but makes an allowance that they can stay another day, until they are eaten.

Due to the inevitability of their demise, Brian removes Scooby’s collar. Scooby rises to his foot, removes the dog head from his head, and embraces Brian. The show ends on this scene.


It was an incredible show. Things weren’t perfect, but they were as imperfect as the musicals I did pit for that had weeks of rehearsal. It was a great experience, and while Tim and Adam have just graduated college, making another musical unlikely in the near future, I am excited to hear about whatever creative endeavors they have moving forward.

Dull Edge

The cutting edge of technology is particularly awesome these days. Cars are doing more on their own, phones are surpassing some current computers in their performance, and VR is coming into its own finally. I listen to a lot of tech podcasts, and love messing around with technology, but due to my status as a recent college graduate, I am definitely not maintaining a collection of cutting-edge devices. And that’s okay.

First, let’s talk about cars. I recently purchased a post-lease 2015 Honda Civic, LX trim (in other words, the only model more basic comes with a manual instead of a CVT.) The disparity between that car and other higher-end cars from the same year is rather shocking. Sitting in it, I feel super cool. It’s a big upgrade from the 1998 EX-L Honda Accord I had been driving. There’s a “cockpit” feeling to it, good Bluetooth connectivity, and a back-up camera. It’s relatively zippy for a cheap car, and the gas mileage gained by the CVT cannot be beat.

Then, I read a Golf R Review by Casey Liss. He is one of three car enthusiasts on the Accidental Tech Podcast. Not surprisingly, the only one I can identify with is John Siracusa, who to my knowledge has mostly driven manual Civics and Accords for his entire life. Casey though, he complained about the lack of assisted driving and automatic parking. The car needs to be fast, it needs a sunroof, and of course Carplay! This was absolutely baffling to me. I just cannot get my head wrapped around why some of these things are important. These three men have attempted to address it on their podcast, but it still does not click with me. Cars can be purchased only so often to be at all reasonable, and so one cannot even stay on the cutting edge.

The other issue is that he recently began working from home exclusively, yet sounded very hesitant to become a one-car family. Oh well, to each their own.

Now, obviously cars are a very special case of not staying on the cutting edge. I’ve only owned my own car for a few months, and it’ll be a number of years before I can even begin to think about getting another. But phones: now that’s another matter.

I’ve had my Galaxy S7 for two years now. Previously, I had a Galaxy S4 for two years, and then some random LG (I think) phone for 4 years throughout high school. In my mind, until I can afford the “every year” phone upgrade, two years is a reasonable time in these days of mostly non-replaceable phone batteries. So, with my S7 really slowing down and the battery life starting to tank, it was time to figure out what to do. Was the newest Galaxy S9 worth the incredible price tag? Did I want to save some money and get a Pixel XL or a new LG phone, each containing last year’s processors, despite them being the newest in the lineup?

I ended up choosing a Galaxy S8+. Due to the release of the S9, I got a very good deal on it, and the processing difference (and battery life) between the S7 and S8+ is much larger than the S8+ to a comparable newest generation phone. Once again, I opted to stay on the duller edge of technology. And I am happy with that decision. I tried the S9 in stores, and it truly did not impress me anymore than the S8 does. The S8 was the revolutionary phone (just like the iPhone X, and whatever comes out next will not be quite as lauded).

This is what is interesting about technology. So many people are excited to get the newest and best thing. The hype is always there, but the price-to-performance normally isn’t. I spent all of last year in school working on a 4-year old Ideapad and a 5-year old refurbished Thinkpad. They performed admirably for me, because like most people, I’m not doing much heavy-lifting.

Being on the dull edge, and looking out at what is available and what others have, can be fun. I don’t think there is anything wrong with living life on some sort of delay with technology. Perhaps as I grow older and make a bit more money, that will change. But for now, I am happy with scouring the internet for good deals, and getting what I actually need for the best price.

Side Projects (Part 1?)

I think it is important to have a variety of projects capturing one’s attention. The breadth and depth of these will vary by individual, but they should be there nonetheless. Someone who is incredibly invested in one particular field or interest will be more aware of the branching-off points, and can thus develop projects related to the disparate branches of that field. Others may be interested in many topics, and have projects related to each.

I fall into the latter category, as do a good chunk of my friends. I have become widely interested in many topics throughout college, and this was one of the main reasons I did not immediately pursue higher education. While I love mathematics, I cannot see myself devoting a majority of my life to only studying it for the next five years, and wanted the opportunity to do many things I did not do in college, or double-down on some of the projects I started then.

In addition to this blog that I am trying to keep up with better, there are the podcasts Operation: Have a Conversation and Comical Start. There was the joke-blog I announced a couple of weeks ago. I’m trying to read more, and still keep up with doing some math so that I can be more effective at my new job I am starting soon. I have been playing tennis more, and joined a softball league while I’m still in Minnesota.

These projects keep me busy, and keep me happy. I like to have a variety of things to work on, because I’ve always loved each subject I’ve been introduced to. My passion for them may diminish at times — I’ll always  be more invested in math than in biology — but being able to have conversations or read a few articles about new ideas is exciting. Writing this blog is exciting, and talking with my friends and editing podcasts is invigorating. Playing newer and older sports to me is always a good time, because I like to stretch the muscles I’ve worked all my life, but also pick up new skills. The internal growth I want to achieve is being reflected in the growth in new activities I’m participating in. There will be more to come about that last sentence.

Card Game Simulation

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.