Mathematics is for everyone. Really.
This article ( and the succeeding ones in the series) aims to prove that point. That everyone has a mathematical brain. Specifically, I’ll concentrate on a certain area of mathematics in this article known as geometry, and then go to more advanced geometry (usually college or graduate level geometry). Don’t fret! There are no equations here which will make your eyes wander and do something else (at least while you’re reading the article). There are a lot of science articles around, but what you usually don’t get often are articles about math, how beautiful and useful it is, and how important it is to science and modern civilization.
A Short Love Story
I love math. That’s how I feel about it: I’m passionate about it, just as I am a passionate about science (particularly physics), sci-fi (Star Trek et al) and technology (programming, software et al). However I started out disliking math in grade school. Like many things in my life which started with me disliking them but ended up with me liking them instead, my love for math grew more and more as I progressed through grade school, then through high school and then college. I disliked math at first because I didn’t pay much attention to studying it, and considered it far less important and impressive than science at that time. The effect then was that I performed poorly in math. As I paid more attention to math however, I learned I was quite skilled at it, and that it was fun for more than one reason. In short, as the years progressed and I learned and matured along with math, I fell in love with it as well.
The Real Mathematical Deal
Unfortunately a lot of people (all over the world you’d be surprised, even in well-developed countries like the UK and the US) both young and old struggle with math. Oftentimes you’ll read/hear news about how kids from different countries around the globe, from grade school and so on, struggle with math. They are usually victims of many unfortunate circumstances which I don’t think , mind you, include not having what is commonly labeled as a “mathematical brain”. Humbug. These circumstances I refer to include a poor/ineffective educational system, poorly trained/ineffective teachers, lack of support from people around you, or combinations of those. Those of us who turned out to love and appreciate math for its beauty, purpose, and elegance, are the fortunate few I think. The exceptions to the rule perhaps. But like I said, humbug. Everyone is born with a mathematical brain, it’s just that we don’t realize it most of the time.
Most of us think about long, boring, frustrating, confusing (and perhaps frightening sometimes?) equations when we think about math. Our grade school teachers made us memorize the multiplication table from 1 to 10 (I heard some even go as far as 15). In high school we learned algebra, or perhaps some of us didn’t, unfortunately. Perhaps some of you later on in life cursed (or still curse) the Arabs for improving and making algebra very widely known and used.
The truth is, mathematics is a large collection of sub-areas of interest, and algebra is but one of them. Some,as you may already know, include geometry, trigonometry, discrete math, and so on. In this post, which is one among a series, I will show you that not only do you have a mathematical brain, but that you can grasp even advanced ideas, in this particular case, geometry. Now let’s get to it shall we? 🙂
Please observe the figure below (Figure 1), which includes objects (a) to (e). We first start out with an object with no dimension, which is a point in space (a).
Then, if you have two points, connect them together and you have a 1 dimensional object, a line (b). In fact, if you recall your basic high school geometry, a line is just a series of points. The only dimension in this case is of course the dimension of length. So far so good?
Now, if you have a line (b), and then another one similar in length as well as parallel to it, connect their ends and you get a square (c). Now you have a 2 dimensional object, which has both the dimensions of length and width. See where this is going?
Next, if you have two squares which are of the same size, you connect their edges and you get a cube (d). Obviously this is a 3 dimensional object, having width, length, and height/depth.
Finally, and usually this is tackled in more advanced math/geometry courses in college, we extend the 8 corners of the cube and get what is known as a hypercube, in this case a 4 dimensional hypercube (e). You can also imagine this as a smaller cube within a larger cube, and then you connect their edges.
You’d notice that, in order to go to higher dimensions of cubes, you’d just keep on extending and connecting their edges/corners. Easy as pie no? And you just had a 101 on advanced geometry. Many people don’t know that math, especially in advanced courses, involves a lot of imagination and creative as well as abstract thinking. Cool no? 🙂
Great, great. But what’s the use?
What’s the use you say, old chap? What’s the use of knowing how to create hypercubes? Why, quite a lot actually. Aside from the cool realization that you just used your imagination to waltz through an advanced math/geometry class, hypercubes such as the 4D one in the figure (e) have lots of uses.
One is aesthetic. As you can see in the figure below (Figure 2), the Grande Arche in Paris, France is quite a beauty, and one hell of a tourist attraction. It looks so futuristic to me too. It’s inspired by a 4D hypercube such as the one shown above in Figure 1 (e).
Another is that hypercubes are used in supercomputing, or computing that involves trillions upon trillions of data. Supercomputing is necessary and used in weather modeling and prediction, aircraft design, and other scientific and compute-intensive areas and applications. The interconnections of the computers are in 4 or more dimensional hypercubes.
And lots more. In fact, we may still discover more uses for hypercubes in the future. When hypercubes were first realized and imagined many many years ago, mathematicians didn’t really know what good they were for. It was just that it was their work to do math, they were curious, bored, or something in between. What they didn’t realize is that their work would pave for more wonders which our generation now enjoy and is rarely aware of.
So, still think math (or at least more than one area of knowledge in it) isn’t for you? Or that it’s hard to get into advanced math? If you don’t think so, or that you’ve slightly changed your opinion about the topic (about math not being for everyone) then my article has somehow served its purpose. If not, then keep reading my future posts on how everyone can do, use, and even love math as I do. Perhaps even more. 🙂
Resources, references, and further reading
- Wikipedia page on hypercubes: http://en.wikipedia.org/wiki/Hypercube
- Wikipedia page on The Grande Arche: http://en.wikipedia.org/wiki/Grande_Arche
- Wikipedia page on the use of hypercubes in complex computing: http://en.wikipedia.org/wiki/MIMD