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.

prerotation

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:

geogebra-export(1)

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.

finalHorn

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.

Shameless Plug

An extension cord walks in on its son, a vacuum (three-pronged cord of course), plugging itself into an electrical outlet. Aghast at what it sees, the extension cord can only cry out: “You shameless plug, you’re grounded!”

 

I’m going on vacation this upcoming week. This original joke is the best I could do. The actual shameless plug is my friend Brandon’s review blog, as well as my competing review of his blog, where I verbally abuse and critique his writing even if I have not experienced whatever he is reviewing.

It’s all in good fun.

Changing Teams

I’m moving out to San Diego, and with that move comes a very important question: How wholeheartedly do I join the Padres’ fan-base, and how much do I keep following the Twins?

For any of you who don’t follow baseball, the key thing to know is that as far as future prospects go, this decision feels like a total wash. The Twins do have a bit of a larger group around them, I believe, but the Padres seem to be making a few moves to help things improve. They care about their fans, and are at ease with their current losing situation. Last year when I was in San Diego, they had deal going where you could pay a flat rate (it wasn’t too much) and guarantee at least 10 tickets, and you would get a ticket for every game after that until they won. That is a team aware of their losing, and willing to help bring in fans.

I also think they might have a better stadium. Don’t get me wrong, I love Target Field. I was there on its opening day, and have been to countless games there over the past nine years. But Petco Park (while being as horrifically branded as ours) has a certain distinct charm to it. First, it allows pets in a green area outside of right field. It seems larger, and it is in San Diego. However, the location, and traveling there, is not quite as good. Public transportation in San Diego is rather sparse, and I couldn’t determine a better way to get to the stadium than driving and pre-paying for my parking.

Over time, we’ll have to see how the teams develop. I am more entrenched in the history and culture of the Twins, and in my mind the Padres have none except for Tony Gwynn. Thinking about who to support, what games to go to, and the relative difficulty of going compared to my current experience in the Twin Cities, is a little stressful. Sports, especially baseball, has been a big part of my life since I could walk. Throughout college I had the freedom of disposable income (kind of) and transportation to attend them at my leisure, as long as I had the time. I am not sure if the San Diego sports scene is quite as accommodating.

 

This gets to a bigger question I’ve had in my mind. To what extent does changing states affect my life, my “loyalties” so to speak, and the bridges I have. In an interconnected world, it seems that physical barriers are not quite as important, but they do put forward some stress testing on friendships and what you know about where you are. I am so familiar with Minnesota culture, the Twin Cities and its surrounding suburbs, everything that is available to me. This information has been obtained through years of living here, driving around, growing up in it. How do I reach that level of comfort in a new place, when I don’t know exactly how long I’ll stay? What changes do I make in my activities?

I know there is so much to explore, but at what point does the awe of exploration turn into either familiarity, or complacency? When I was out there last summer, I quickly latched onto safe places where I could be safe bide my time: Starbucks, Panera, and a single hiking trail I walked about 10 times. It took friends and family visiting to go beyond those places, and even then the exploration was minimal. I spent many weekends feeling sick, watching movies in bed, or just going to a local theater in a mall. I was afraid to strike out by myself. I found a minimally comfortable zone, and wished to stay there.

This is what I need to change. I don’t need to change teams, or give up on what I love about Minnesota. I can still have that part of me, while appreciating the new things San Diego has to offer. I will always have a pain in my heart when the Twins lose, even if I support the Padres bandwagon for a while. That doesn’t make things less scary. It will be a long period of adjustment for me, but it something I know I must do.