Fractals: The Up-Goer-Five Edition

As my first post, I’ll post my most recent (as of now) rambling.

If you’re reading this, then most likely you’re also a fan of xkcd. If you’re a fan of xkcd, then you likely know of the Up-Goer-Five post. And if you’re like me, and if you know of the Up-Goer-Five post, then you likely know of the Up-Goer-Five text editor at http://splasho.com/upgoer5/. This basically forces a user to explain a complex topic using only the ten hundred most used words in the English language. This way, they explain complex topics in layman’s terms. I’m surprised the site has no reference to xkcd #547, where Randall mentions in his title-text that the world would be much better off if math professors worked together to explain complex math topics in a simple way for Simple Wikipedia. Now of course this isn’t going to happen. I’m not a math professor–not yet, at least, though I do aspire to eventually be one–but that doesn’t stop me from doing such. And they’re right–it is MUCH harder than it seems! But I accepted the challenge…

So I proceeded to explain fractals and the Cantor Ternary Set using only the ten hundred most used words in the English language (you can verify this by copying and pasting the following into the text editor).

XKCD 547 Simple: I never actually have this problem. More often than not, I end up forced to talk like this to get my point across.


 

If we do something, and do that same something over again to that something, and we keep doing that something over and over again without stopping, we can do some crazy stuff, like make a line seem to disappear, even though it is still there.

Say that we take a line. Now we break it into three smaller lines that are all the same. Now we take away the middle line. Now we can see that we have two new lines that are just like each other, but is one-third smaller than the first line. We will call what we did a “something.” Now we do this “something” again to these two new lines. Remember, each time we do this “something,” we take each of these small lines that we had just made and break it into three new lines, and then take away the middle line. We’re left with four lines that are each one-third smaller than the line that was just broken, but is one-third of one-third of the first line.

But why stop here? Let’s keep doing this “something” over and over again. Let’s say that we keep doing this “something” to each set of tiny lines we make “so many” times. The more we do this “something,” the smaller these lines get. Each tiny line is like the first line. Say that we do this “something” three times. Then each tiny line is one-third of one-third of one-third of the first line. If we do this “something” four times, each tiny line is one-third of one-third of one-third of one-third of the first line. And so on!

We now notice that our tiny lines keep getting smaller and smaller and smaller each time we do this “something.” If we do this something “so many” times–and when we say so many, we mean so many that we forget how many times we did this “something”–our tiny lines seem to disappear! But that’s what happens! We keep taking away from each line. The more we do this “something,” the more we take away from each line–but we’re always leaving a little bit left. We’re always leaving two-thirds of the line to take away from next time we do this “something.”

But at the same time, each time we do this “something,” there is less and less for us to take away from in the tiny lines we have just made. So of course, the more we do this “something,” the less and less we have to take away from each line the next time we do this “something.” The more and more you do it, the less and less you have, even though you never really run out of tiny lines to do this “something” on (there is always a little bit left). But it still seems that our first line disappeared after you did this “something” “so many” times, even though we know that there’s still some left! That’s how to make a line seem to disappear without actually making it disappear completely!


 

Personally, I find laymanese to be harder to understand than jargonese. For those of you that are like-minded, I essentially stated that the process of subtractive iteration on a line of length L will cause L’s length to approach zero as the iteration depth approaches infinity. Whenever I say “something,” I imply the iterative function. Doing that “something” is the actual process of iteration.

But of course, one way or another, you’re likely more interested in the laymanese version: either that’s the only version you’d understand, or that you want to get a kick out of how silly it actually sounds. Or maybe both. Honestly, this version would confuse me even more. But then again, I’m not Captain Clueless, so if I’m trying to actually learn something seriously, and not simply get something for general knowledge, this wouldn’t be helpful. But if I was trying to learn something I know nothing about (popular culture, for instance, apart from Internet memes), I’d need a similar explanation… not that you’d need anything more than such to explain pop culture… really, the lack thereof. Culture, my ass.

Anyways… I do have a knack for putting things in layman’s terms. I just don’t usually have the patience to do so. Honestly, on EP, I’m known as the site’s math tutor. But oh, I don’t get asked for help in Algebra 2 topics, Stats topics, Calc (up to AB calc, since that’s where I am currently at, although I can explain harder concepts but not its process) topics, or math-they-don’t-teach-in-high-school topics. I get asked questions like “How do I do long division?” True story. That person was 22 years old and had no apparent learning disability. I always get asked the stupid questions. I don’t spend time with those people–unless they really do have a learning disability, and I’m more than happy to teach them (unless they ask me what a decimal point is… again, it’s not them. It’s me. I just don’t have the patience for such).

BUT I’d be more than happy to start from Algebra and explain things in layman’s terms. I especially like explaining definite integrals in layman’s terms, because calculus is something people associate with “very hard math,” and I explained it to them my 10 year old self would have understood (because I did understand the concept of an integral when I was 10… I just didn’t know it at the time).


 

“We have this graph of a curve here. Now we draw some rectangles underneath it. The top of the rectangle touches the curve, and the bottom of it touches the x-axis. Now we want to find the area underneath this curve from here [point a] to there [point b]. Now we draw a bunch of thin little rectangles under the curve in between those two points, and then find the area of each little rectangle, then add up all the areas together. Now we make the rectangles even thinner, and do the same. More rectangles can now fit, so the curve can be approximated more accurately. Think of a picture on a screen. The more pixels in a picture, the clearer it is. So the thinner and thinner we make the rectangles, the more accurately we obtain the area underneath the curve. At some point, we can see a trend of how reducing the rectangles’ width affects the curve’s area, and we can then extrapolate what it’s area would be if the rectangles’ width was really, really, really tiny–so tiny, that it’s just barely has any length. And that’s what a definite integral is, basically.”


 

While “actual’ math is hard, “school” math is not hard. “School” math is easy. It’s tedious, but easy. If you get the right person to teach it to you, it’s very, very easy. Just don’t be like me and overthink a question, or blank out on a test because you couldn’t remember if the Product Rule is (f’g-g’f)/g’² or (fg’-gf’)/g’², and despite working through a proof mid-test to identify which one is correct, decide to go with the other one anyways because you assumed you did something wrong with the proof. Or procrastinate on your math homework by doing math–just not your math homework’s math. What can I say–fractals are just so fascinating! Rolle’s Theorem and the Mean Value Theorem are cool, too, but not nearly as cool as dynamic systems are! And differentiable stuff aren’t nearly as fascinating as figures F that are non-differentiable along every point along F! And don’t get me started on sexy, sexy proofs. If only they asked us to prove stuff. Oh I really love proofs. I really, REALLY love proofs.

But that’s for another post. Literally. Mathematical fetishism is 100% justifiable and is not an atrocious feeling towards the subject, and I’ve written several things which I plan to repost to here that justify it!

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Fractals: The Up-Goer-Five Edition

3 thoughts on “Fractals: The Up-Goer-Five Edition

  1. Frank Truth says:

    Greetings Earthlings!

    If you like logic, AI, math, physics, cosmology, calabi-yau manifolds, thermodynamics, engineering, thermography, sailing, cats, nature and wildlife; if you like lessening human and animal suffering; if you have insatiable curiosity and a wild imagination; you might want to get in touch with me. My aol email address is UltraEmpathy.

    Healthy skepticism is good, pernicious cynicism is not.

    Many people confuse high IQ with genius, when the two qualities really have little in common. IQ measures the ability of humans to learn new things. GQ measures patience, curiosity, open-mindedness, imagination, and diligence.

    I do not drink often, but when I do I prefer monstrous moonshine.

    Why do humans use the word “energy” when they really mean exergy?

    I trade futures. I’ll swap your future out for a better one.

    “Think outside the tesseract and the hypersphere.”

    Like

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