Topology is this week’s endeavor. This math is absolutely insane, but let’s just look at a funky shape and call it a day.

This is doughnut. For you nerds out there, this shape is a torus. While it looks simple, this shape can be fun to play with. First let’s investigate the structure of the shape. How does Dunkin manage to make torus foods? It’s pretty simple! The torus is made up of two perpendicular circles.

The smaller circle follows the path of the larger circle. You can imagine that as this circle moves, it extrudes like a cylinder, eventually getting back to its starting point and making the donut shape. You can do this by taking anything remotely cylindrical and putting one face onto the other, making a loop.

Now, the shape is pretty cool to look at, but the structure is cooler than looks. Since the torus is basically constructed out of a cylinder that makes a loop with itself, we should look at cylinders. Specifically, what’s the map of a cylinder like? The context of map here is pretty analogous to the surface area of the 3D shape. For example, the mapping of a cube looks like this:

You can see all 6 faces of the cube all laid out, allowing you to figure out how the cube is put together. Cubes are more complex than cylinders. Here’s the surface of a cylinder all laid out:

That’s right, this is just a plain rectangle. This should make sense. Grab a piece of paper and roll it. You’ll make a tube aka a cylinder! Go on and make the paper cylinder right now because we’re gonna check out this rectangular surface. When you secure the top end of the paper to the bottom, you notice that point on the top is exactly the same as that point on the bottom.

Cool, right? Don’t worry, we’re not done. Since the torus connects the two circular faces of a cylinder together, we get the same phenomenon with the sides of the rectangle.

The conclusion we can draw from that is moving to an edge of the rectangle makes you pop up on the opposite edge. This is pretty neat and allows us to do something cool to the torus. Take for example this map of the torus:

When you origami this bad boy, your torus looks like this:

What’s interesting to note is how the lack of intersection between lines remains constant on both the map and the model (torus). This is known as a simple curve because no intersections take place. While that’s cool and all, I’m really fascinated by the process of this 2D rectangular map producing this 3D torus model. I want to play more with this, but the think juice is boiling. Play around with this stuff with objects like paper and pool noodles. The torus is pretty cool as is mapping and modeling.

 

Stay Soupy.

All of this blog has been dedicated to topics that have blended my gray matter. Well, the smart mush isn’t working too well, for it was scorched by not a conversation but a tooth! This Brain Soup is physical soup.

My innocence was killed Sunday when I woke up with a minor toothache. There was just the slightest throb in my bottom left molar. Just enough that I could recognize and say, “man, this kinda sucks.” I was naive. My prediction was that my wisdom teeth were simply too stupid to grow in properly and assaulted my molar’s nerves. I also assumed the pain would fade away throughout the day, but this tooth was dead set on persisting its attack on me. The throb lasted all of Sunday.

By night time, it was just annoying enough that I could not fall asleep. I tossed and turned from 1am to 2. No such luck at being graced by the sweetness of sleep. 2 turns to 3 pretty slowly. I’ve become annoyed by just laying there, and my darn tooth hurts. I used my roommate’s ice cubes to ice my jaw so that it would numb away the pain. Unfortunately, this would yield a short interval of about 5 minutes of peace. It was no use at 3: I would not sleep at this hour.

I didn’t want to play on my phone, so I instead did homework, assuming the annoying task would make me drowsy. It did not. 5 am struck, and I had all of my seminar work done alongside some chemistry PLAs. I tried again to sleep. 6 am hit, and I was very much so conscious. My roommate’s alarm went off a few times, and I was awake for every last one of them. Fortunately, I passed out sometime after 6 until 7. A very long, fulfilling rest, I know. That was all the sleep I got for the day, but the adrenaline of this damned toothache kept me awake for the day.

Fast forward through some days of pain and misery, and I’ve reached peak suffering! My jaw barely opened, and pain shot from that molar all throughout my jaw, ear, and neck. It hurt to talk, eat, and swallow, and yogurt became my best friend due to it being the easiest food to consume. To make matters worse, I took two exams in this pitiful condition. I don’t remember much from the chemistry exam, but ibuprofen did its duty during my physics exam. I’m too tough a cookie to avoid the tests over some stupid bacteria.

Before you panic, do know that I’m dealing with the issue. My dentist viewed old x-rays of my mouth and believes infection is haunting me. So, I’m to start antibiotics tomorrow. This will get cleaned up, then I get to have a very fun and not at all terrible root canal over break.

So, there’s my brain soup. I know it is no scary math lesson, but I hope my tale was at least a bit entertaining.

 

Stay Soupy.

To preface, this topic goes very much so into crazy topology that hurt my head a lot, so this post is more about some fun geometry than the meaning of what I’m discussing. Anyway, just like real life, boundaries exist in the realm of mathematics. Unlike those silly little barriers, mathematical boundaries describe the edges of something. Let’s start with 1D:

The formula shown is the fundamental theorem of calculus, and the part we care about is the f(b) – f(a). Below the formula is a line segment from point a to point b. We can call this an interval, I, and have it be from a to b. This isn’t very interesting on its own, but let’s look at that expression from the fundamental theorem:

f(b) – f(a)

Specifically, we care about the minus. We can rewrite this segment by giving a negative sign to the a, and the fundamental formula can be achieved by adding negative a to b

You should notice the arrow I drew on the interval. This direction is arbitrary, but I’m establishing that arrows point from the negative to the positive of an interval. I know this is weird, but bear with me. Since these arrows are lines, we can make shapes by putting arrows head-to-tail. Here’s a square which is made up of 4 arrows. Note the direction of each arrow.

The square has an area which is determined by squaring the length of one of the arrows. This value is arbitrary, but it isn’t 0! I’ll call it A for area. Keep the fact that this area exists in the back of your mind. Time to focus real hard on the arrows. This square is made of a horizontal arrows pointing right, a vertical point up, horizontal going left, and a vertical going down. You’ll notice that we effectively have two pairs of arrows: up/down and left/right. Within these pairs, one arrow is the opposite of the other. For example, the opposite of up is down. So, if we add the arrows together we get this:

The claim we can make is that the perimeter of the square adds up to zero since all the arrows cancel each other. But, as you recall, the square’s area is NOT zero. This means that adding up these intervals gave us an area but not a perimeter. This perimeter is known as the boundary of the square. You can see it physically since it wraps around the area of the square.

The boundary of that initial interval is seen in the end points. Visibly, they’re where the interval line segment ends, so they’re the boundaries of the thing! To take this one step further: our interval came from the fundamental theorem of calculus. Specifically, the theorem says that integrating the rate of change of some function f(x) will yield the total change: final position f(b) – initial f(a)

You can interpret the process of the integral as moving from one endpoint (boundary) to the other. With the square, the integral would cover the space within the perimeter in the same manner since the perimeter is the boundary. This space ends up being the area. Unsurprisingly, actually calculating the integral of a perimeter with functions will yield the area. I won’t dive into the calculus but here’s an example with a circle:

Stay soupy within some boundaries.