Pi Day

This month's ΜΑΘ meeting featured some intense Π  discussion. One of the cool things about π is that it is an irrational, transcendental number and as such it can never be known to 100% precision. So the only way we can know the true value of π is to use methods of analysis that will approximate it with both great accuracy and great precision. One of these methods (Monte Carlo) was discussed during our meeting and was the focus of a recent Veritasium video:

To celebrate this, I coded my first javascript program on KhanAcademy:

PiApproxDarts

Made using: Khan Academy Computer Science.

There is much room for improvement, but it's pretty cool to observe how much variance there is in samples of 100 throws by clicking the 'Restart' button.

Week 4

PHYS120 - Physics I with Calculus

Lab

We concluded the lab from last week by working with the Data Analysis Toolpak in Excel. Our group's results were pretty strong, I think. We measured, and calculated, the gravity near the earth's surface to be about 10.2 m/s². Our regression line was strong ($r^2\approx.998$). We were supposed to hand in print outs on the spot, but I was using my laptop - which decided to give me battery-death stares after only 45 minutes. I had figured on it being good for at least the two hours that we would be in the lab so I didn't bring my charger with me. It wasn't.

Lecture

This week's lecture was on Newton's laws of motion, incline planes, friction, tension, and Atwood machines. Tension is one strange thing for me wrap my head around. It makes a lot more sense to think of forces as action-reaction pairs, but it still gives me a bit of trouble to set up pulley problems with the proper number and direction of tension forces acting within the system.

MATH213 - Calculus III

Well, the familiarity with derivatives and integrals was lost midway through the week. Now we have somewhat familiar concepts of tangent lines, only now they are vectors and direction matters (a lot). The notation for vector calculus is still a little strange to me. I understand magnitude and unit vectors and all that jazz, but I still get tripped up with the notation for the projection of one unit onto another.

Our first project was due on Wednesday and I'm honestly not sure if I did it correctly. We had to find the distance between skew lines which kept me burning the midnight oil from Tuesday night all the way until 6am Wednesday. For a moment, I thought I might pull an all-nighter but suddenly I had to give up and crash for two hours. Good thing I have a great alarm!

ENGL102 - English Composition II (Literature)

I stayed up on Monday night reading the texts, as we were supposed to, and I woke up scrambling to read the rest of the chapter Miró Miró from Let the Great World Spin. I didn't have time to finish before running out to class so I had a great idea. I'll just listen to it on my Kindle, robot voice and all. Brilliant! ^Brilliant! Except for the part about actually bringing the Kindle with me. *Sigh* Then... a miracle occured! I pull into the parking lot, with my automobile making college-kid-automobile noises, find a great parking spot for a change, get out of the car, go around to the passenger's side to gather my stuff and hear "You're in my English class, right?". I look up and say "Yeah?!" "It's cancelled." Wot waaaht waaaaaaaaaaaah. You didn't notice, but a slide whistle was just played.

This was actually not a good thing. I typically only carry the materials I need for the few classes I have each day, so I had to figure out what I was going to do for the next 3 hours. I decided to head to the library to print up the brochures and problem sheets I had put together for the math club. Then I went to the student activities office to copy a whole bunch of them so we would be good for Thursday.

I was upset that I didn't have literature class this week. I don't have the best memory when it comes to reading fiction, so I may have to review last week's readings again for next week. Plus, I spent a lot of time on the readings the night before when I could have been mathing around. Why won't you let me math? On the bright side, I wouldn't have to stress over materials for the Meet the Clubs gathering.

MATH215 - Linear Algebra

I think Tuesday's class was very productive and helpful. In preparation for the first test on Thursday, our professor handed out a sheet of statements about matrices, vector space, linear combinations, linear dependence, solution sets, etc. We had to form a small group to discuss whether we thought each statement was either true or false and give reasons for our answer. These types of exercises really get my brain working. For most of the statements I couldn't simply read it quickly and instantly give a response. I had to think deeply about what the statement was actually saying before I could answer with a level of confidence that was satisfactory. These first few weeks have introduced us to a type of mathematics that most of the class has not seen yet. This course has, thus far, focused heavily on the conceptual aspects of linear systems as opposed to the processes of solving and manipulating them. The latter (the how?) is much easier to understand; the rigorous attention to the former (the why?) is what is causing brain meltdowns in Sector 7-G.

Despite the progress made on Tuesday, I don't feel so great about my performance on the first test. There were a lot of questions where I felt I may have been conflating two concepts such as span and linear independence. I'm sure I confused them at some point and contradicted the reasoning for a response to a previous question. After the two hours was up and I was one of two students still taking the test, my ego was sufficiently depleted so I tapped out. I reviewed some of the homework when I finally got home and I think it helped hammer home some misconceptions I had. We'll see how I did when class reconvenes on Tuesday.

ΜΑΘ

We had a great table location at the Meet The Clubs gathering on Thursday. When students arrived in to the student theater, they saw us looking right back at them. So we were the first to receive looks of fear and subsequent aloofness when they saw a sign that contained the word math on it. Many of them asked for a picture with us; Not because we were awesome (I mean we are, but..). No, they wanted a picture as proof that they attended so they won't fail their Foundations For Success class. Oh, how I remember that class... On one hand, I understand that some of the forced activities required to pass the course are suspect to say the least; but if I wasn't forced to go to Meet the Clubs last year, I may never have learned about the Mu Alpha Theta club/society. So as far as MTC is concerned, I think it can be a very good activity that all SCCC students should attend.

Anyway, back to the table... I put together a list of interesting problems I had found from various sources (Brilliant.org, World Wide Center of Math,...) as well as a puzzle from the YouTube channel Scam School that had stumped me last spring. I hope people enjoyed that as much as I did.

We got a few people interested in joining, but I don't think it was quite the success I had hoped it would be. If we don't get new members signed up and potential new officers in the house, this society is going to sink fast when the spring semester concludes. I'm going to fight as hard as I can to not let that happen.

Notes and Bibliography

Week 3

PHYS120 - Physics I with Calculus

Lab

The purpose of this week's experiment was to measure the gravitational acceleration near the earth's surface. We got to play with stopwatches and vacuum cleaners that had "gone from suck to blow". Eye protection was quasi-required for the first time. After many technical difficulties with our equipment, we finally got an excellent process for quickly gathering good data for our experiment. If you've never worked seen an air track before (and I mean, come on, who doesn't have one of these in their basement) basically it's a long aluminum prism with a tube through it and holes evenly spaced along its length so that a cushion of air is produced. It's like an air hockey table, except you'll catch some discontented looks from your professor if you try to hit the air car with a paddle. That's another thing. The air car. It's just this bent slab of aluminum without any wheels or Batmobile-esque exhaust system. Expectations were not met.

Anyway. the first issue we had was measuring the dimensions of this thing. We have to prop this, normally level, air track up on an angle using a block of wood. The reason for this is that we need to find the velocity and accelerations vectors so we can measure the force acting on the $y$ component. So we had to measure the block of wood, then measure the length of the air track from support to support. I don't know why, but trigonometry is hard. I'm not sure who decided to make trigonometry difficult, but it was probably just to screw with me. The hypotenuse is the distance from support to support of the air track and the height is the block of wood. Why did that take me 10 minutes to wrap my head around?

So we get everything set up, do a few test runs, figure out how to operate the stopwatch, and get a few measurements in when our track just decides to stop working. The vacuum... ummm let me use more technical terms. Our blower wasn't blowing. Well, it actually was blowing, it just wasn't blowing the way we wanted it to blow. The air car simply would not move once we put it on the track. After some quick handy work from our professor, we were back in business.

We needed to take five time measurements at various distances so we can later plot distance as a function of time (actually time-squared, more on that next week). Since we had a group of three, one of them suggested that we each take two measurements per distance so a total of six measurements could be recorded per distance. This was a fantastic idea. This will make the upcoming lab report and discussion on sources of error and bias much easier. With our system in place, we got all the measurements recorded in about 15-20 minutes.

Lecture

Wednesday was the lecture for chapter four in our book, Motion in Two and Three Dimensions. Having a bit of experience with the two-dimensional ballistic problems from last spring, I didn't add much to my notes. I did perk up when the wedge product was mentioned, as well as the two basic models of drag ($f=-bv$ and $f=-cv^2$). I still have a lot to learn. I started this semester off liking all of my course topics, yet with a feeling of complacency. I expected this to be a fairly uneventful semester, yet I find myself enthralled with the prospect of going deeper into these subjects than I ever have before. I feel like my passion for actual science has rekindled to a point not seen since my MOOC endeavor two winters ago.

MATH213 - Calculus III

All of my mathematics-based classes have started off dealing with some application and/or conception of vectors. A vector is a directed line segment, so it has a length (magnitude), and a direction. Depending on whether you're guiding a point on the number line, or you're Mr. Square in Flatland or you're just a boring human, a vector will have 1, 2, or 3 components; more accurately a vector contains as many components as dimensions in the coordinate system. In calculus, vectors are used in many ways, particularly to calculate distances between geometrical shapes such as planes and cylinders. So most of my class thought we knew what a cylinder was. It turns out, it can mean any "surface generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line."^[1] So far, we've covered everything through dot product, cross product, lines and planes in space, cylinders and quadric surfaces. Next week we delve deeper into vector-valued functions and motion in space. After starting the course off with confusing new nomenclature, we finally get to see things familiar to us, derivatives and integrals.

MATH215 - Linear Algebra

I got an A on my first project in which we had to set up augmented matrices to solve 3 different systems of equations and explain the process of row operations along the way. I was one of the few students to type mine. Somehow, I forgot to do part (d) for the 3rd system (reduced row echelon form). It would have made it easier to do translate into parametric form. Over the summer, I learned a bit of programming for the TI-Basic language that my TI-84+ uses. It really helped in efficiently completing this assignment and it only took about 15 minutes of code. Well, at least it did after the calculator lost all of its memory after I replaced the batteries to assuage its annoying low-battery warnings. Much arrgggh was had.

We have just about finished the first chapter of David C. Lay's fourth edition of Linear Algebra and Its Applications. This stuff is mildly excruciating for someone like me, who is familiar with the kind of pure mathematical arguments that this first chapter has covered rigorously. For some of my fellow students who have neither studied nor seen these types of proofs, it must be near impossible to follow. Some of them are already expressing their dismay with the style. Even I am cautiously optimistic of how I'll fare with this course.

By far the coolest thing I learned was how to use a set of linear equations to balance a chemical equation. I would love to write a post about this if I find the time to do so. Using a matrix and row reducing makes problems like this so much easier. The only alternative I ever learned involved a lot of trial and error. For some simple chemical equations it may be easy to intuitively find an answer quickly, but for the really complex ones, the application of linear algebra is so much simpler.

I found a series of videos on YouTube from Gilbert Strang that help further explain some of the things we're learning right now. This in conjunction with Khan Academy is going to be practically necessary for me to succeed in this class. I really do have a lot to learn.

ENGL102 - English Composition II (Literature)

What's fun about general education courses is that the female:male ratio is immensely larger than my STEM classes. I think I'm one of five or six males surrounded by about twenty young women. Oh, to be young again!

At the beginning of each class, we write about the stories we have read for a few minutes in a journal. The professor gives us a topic such as a certain character's emotions or motivations, then we discuss what we thought about it. I quite a few books over the summer, none of which would have been on a liberal arts major's list of must-reads. Still, I can appreciate the feelings and thoughts that literature studies hope to evoke.

The first few classes revolved around some short stories but now we have moved on to reading Let the Great World Spin by Colum McCann. Every year, my college picks a few novels to be voted on to become The College Novel for the following school year. I read ahead after receiving a cheap copy via Amazon a few weeks before the semester started. One thing I'm not liking is this tired trope of having many disjoint story lines that converge in the end. You read a few of these types of novels and it gets old really quickly. That's why I really liked reading Cormac McCarthy's The Road on one summer day. It was just two main characters with a simple premise. None of this complicated interweaving of stories that takes you out of the flow of one of the plot lines.

This class meets only once a week and I like to read the stories the day/night before the class, so it's nice to have a 24 hour hiatus from my intense math courses. I'm not sure how much I'm going to like the essay writing part of this class. I struggled mightily last year getting my ideas together for Comp I. This year does feel different though. I think I have grown quite a bit as a writer and have a much better general framework to begin writing assignments. Indeed, this course has reignited my motivation for writing on this blog. Hopefully I can keep ahead of my assignments and still find time to write up a weekly summary. I do enjoy rereading my past posts to reminisce about the things I've learned over the past few years. I think it helps to solidify all of those concepts I studied so intently to understand.

ΜΑΘ

Being secretary of my college's new mathematics honor society is proving to be quite the task. I understand my role much more clearly now than I did when I agreed to accept it. I am essentially the conduit that facilitates the induction of new members and keeps records of the meeting proceedings. It wasn't very obvious to me what scholastic-club government was all about last year. I am still the very definition of learning by doing. My main goal is to at least leave this club in better shape than when I joined it. Hopefully, my efforts will allow the next group of officers to grow the chapter's recognition on campus. This is a role I have been taking very seriously lately.

Notes and Bibliography

1. Maurice D. Weir and Joel Hass, Thomas' Calculus: Early Trancendentals, Twelfth Edition (Boston: Addison-Wesley, 2010), 714.

Science shows from my childhood

     In my discrete mathematics class, we had a lecture on discrete probability and my professor was talking about factorials. "4 factorial is 4 times 3 ... 2 ... 1 ... " and I said "Contact!" because ... well... that's the answer. As a kid, I used to watch a lot of PBS after coming home from school. Yet, my memory of these shows has faded drastically over the years. So, I had a difficult time remembering the shows that were on during that time.  I remember 3-2-1 Contact but there was some other show about a detective duo that I couldn't quite pull up from my memory banks. Fortunately, I live in an amazing time period and have access to YouTube so I can nostalgerbate to my hippocampus' content instead of letting this memory evade me for the rest of my life...

     The name of the actual show was called Square One and this was a short segment of each episode called Mathnet which ran from 1987 to 1992. I remember watching and loving this show while I waited for Where in the World is Carmen Sandiego to come on. Someone has been awesome enough to compile a playlist of Mathnet episodes that can be found here. This made my Friday just a little bit more fun and little less productive.

JFLAP and the Sierpinski Gasket (aka Triangle)

      One of the coolest things that we get to work on in MATH201: Discrete Mathematics is recursive functions and fractals. From pine cones to coastlines to the tree in your backyard, fractals are found everywhere in nature. I'm not going to get into specifics on what a fractal is at this moment (there are tons of resources out there that can explain them better than I can), but it's basically this:

1. Make a basic shape
2. Use that shape to make the same shape on a larger or smaller scale
3. Repeat step 2 ... 

      The results that a simple recursive algorithm produces can be amazing. You can generate fractals with a program called JFLAP. On a recent quiz in class, one question asked students to create Wacław Sierpiński's famous gasket known as the Sierpinski Triangle. There are many ways to construct this triangle. One method involves starting with a large filled-in triangle and cutting out smaller triangles inside the existing triangles. The Wikipedia page has a good illustration of this:

Courtesy of Wikimedia Commons

      Another way of doing this is to use a recursive function to draw the outline of the triangle. In JFlap this can be accomplished by using something called an L-System or Lindenmayer System. An L-System uses an axiom which tells the system where to start and calls other defined functions.

      To make a Sierpinski Triangle in JFLAP, two functions are required in addition to the axiom. We'll label these functions A and B, while g tells the program to draw a line forward, + and - tell it to turn right or left.

Axiom: A

A -> B g - g A g - g B

B -> A g + g B g + g A

angle = 60

      On the second iteration of this system, a shape is drawn:

n = 2

      We use this shape again in the next iteration and the general pattern shows up for the first time:

n = 3

      You may be able to see a triangle emerging in the middle:

n = 4

      

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

      That's pretty cool, right? Well prepare to have your mind blown in the next minute. You can see that as the number of iterations increases, the finer the structure of the triangles become. Look back at the early iterations and find that no two lines ever actually touch each other. So, it can be shown that as the number of iterations goes to infinity, the perimeter of this shape tends to infinity while the area enclosed is precisely zero. Wait, what?!? That's crazy. How can something have infinite perimeter and no area?

      This is the beauty of fractals. Veritasium has a great video about the fractals and the coastline paradox: