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Thread: a^2 + b^2 = c^2, the blue sky, and ice cubes floating in your glass of water.

  1. #1

    Default a^2 + b^2 = c^2, the blue sky, and ice cubes floating in your glass of water.

    These are simple, obvious concepts and constructions that we live with each and every day. The first is observable with blocks and introduced reasonably early on in education, and the last two are directly observable.

    Unpacking the how of both concepts comes significantly later, if at all.

    Consider our right-triangle friend on a Cartesian grid.

    The sum of the squares of the sides are equal to the square of the hypotenuse.
    Yes, we all learned this in school and filed it away somewhere. Along with the Quadratic Equation, except that the Pythagorean Theorem can help you figure out if a window frame is square or not (e.g. not tweaked), as you can decompose the shape into a rectangle diagonally-divided--itself two triangles. If these are found to be right triangles (e.g. the Pythagorean Theorem holds), then you've just made your window frame true and can start nailing it into its surrounding framing members.

    It's useful for this.

    But what is really going on here? It wasn't until my first year at university that I saw the diagram from Euclid's Elements posted on a faculty member's door, and thought it a brilliant illustration:

    http://en.wikipedia.org/wiki/File:Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg

    (Clicky-link to image here. I believe that Moo is looking to do away with IMG tags, anyway.)

    Aha! Certainly different than the recitation of the theorem and scribbling down of its elements.

    The other two questions are so simple that even a child can ask these without prior knowledge. Answering them, too, is seen as simple--but it's not. The first response requires knowledge of optics, and the second requires knowledge of chemistry.

    And it is here that the thread opens: what other questions seem terribly simple, are posed (or their answers used) routinely, and are not thought through? I'm curious to know these things, and this curiosity once had me scribble out 3 pages of equations in a high school physics exam. I only memorized a scant few equations, you see, and derived the rest. My answer went through kinematics, optics, E&M, and circled back in on itself--and in the end, I'd proven that, yes, p=mv. If I still have that exam result, I'll dig for it to post here--it was a blast to think through and write*.

    Oh. And the Fibonacci series. Specifically, the equation that generates the series. It's crazy to think that thing generates integers.

    *I received a zero on that exam question. Oh well.
    Attached Thumbnails Attached Thumbnails Click image for larger version. 

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  2. #2

  3. #3

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    How strange the picture link was working this morning.

    Oh I am going to answer this thread, I've just been trying to think of something.

  4. #4

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    Quote Originally Posted by Dreamaker View Post
    Hate to bring down the atmosphere but: Pictures broken.


    Quote Originally Posted by WoodlandWanderer View Post
    How strange the picture link was working this morning.

    Oh I am going to answer this thread, I've just been trying to think of something.
    The image is an attachment now. I took a screencap of the SVG image, converted, uploaded, and attached it here.

  5. #5

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    Quote Originally Posted by h3g3l View Post

    Consider our right-triangle friend on a Cartesian grid.

    Yes, we all learned this in school and filed it away somewhere. Along with the Quadratic Equation, except that the Pythagorean Theorem can help you figure out if a window frame is square or not (e.g. not tweaked), as you can decompose the shape into a rectangle diagonally-divided--itself two triangles. If these are found to be right triangles (e.g. the Pythagorean Theorem holds), then you've just made your window frame true and can start nailing it into its surrounding framing members.

    It's useful for this.

    But what is really going on here? It wasn't until my first year at university that I saw the diagram from Euclid's Elements posted on a faculty member's door, and thought it a brilliant illustration
    I'm not sure if it counts as a proof, but one of the most elegant illustrations I've seen of Pythagoras' Theorem was at a science museum in Glasgow.
    It consisted of a wheel stood vertically which was able to be rotated, on this wheel was the shape of a right angled triangle with the squares of the sides attached. The squares were joined to the other two only at their corners and were filled with blue liquid.
    One could turn the wheel so that all of the liquid was in the largest square (the hypotenuse) and precisely filled it; then one could turn the wheel so it was upside down, and all the liquid in the large square would flow into and precisely fill the smaller two squares. A beautifully simple way of showing it.
    On another note, I believe that theorem is actually the most proved in history.



    Quote Originally Posted by h3g3l View Post
    And it is here that the thread opens: what other questions seem terribly simple, are posed (or their answers used) routinely, and are not thought through?
    I know this is not the type of question that you are looking for, but it is one I've been curious to find an answer for.
    Why is it, that the sun should have an 11 year cycle of activity?


    Quote Originally Posted by h3g3l View Post
    Oh. And the Fibonacci series. Specifically, the equation that generates the series. It's crazy to think that thing generates integers.
    The Fibonacci sequence of F(n) is simply defined as F(n) = F(n-1) + F(n-2) where F(0) = 0 and F(1) = 1. If the next number is the sum of the previous two, and the first two are integers, then it follows the rest must all be integers.
    Hence you must be thinking of something different to me.

    Why the Fibonacci numbers should crop up in nature however is a very interesting question indeed.

  6. #6

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    Explaining ' how ' something behaves as it does is always easier than explaining ' why ' something behaves as it does.

  7. #7

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    Quote Originally Posted by WoodlandWanderer View Post
    I'm not sure if it counts as a proof, but one of the most elegant illustrations I've seen of Pythagoras' Theorem was at a science museum in Glasgow.
    It consisted of a wheel stood vertically which was able to be rotated, on this wheel was the shape of a right angled triangle with the squares of the sides attached. The squares were joined to the other two only at their corners and were filled with blue liquid.
    One could turn the wheel so that all of the liquid was in the largest square (the hypotenuse) and precisely filled it; then one could turn the wheel so it was upside down, and all the liquid in the large square would flow into and precisely fill the smaller two squares. A beautifully simple way of showing it.
    You're right--that's beautiful, and elegant in its simplicity.



    Quote Originally Posted by WoodlandWanderer View Post
    The Fibonacci sequence of F(n) is simply defined as F(n) = F(n-1) + F(n-2) where F(0) = 0 and F(1) = 1. If the next number is the sum of the previous two, and the first two are integers, then it follows the rest must all be integers.
    Hence you must be thinking of something different to me.

    Why the Fibonacci numbers should crop up in nature however is a very interesting question indeed.
    I'm pretty sure I'm thinking of the Fibonacci sequence here. The equation lets you specify the nth element in the series.

    What's the thousandth Fibonacci number? I've no idea without the generating equation. I'll hunt for it, but I remember a root-5 in there somewhere.

    UPDATE: I'm thinking of Binet's Formula, described here.

    Also, there is a book that I really thought brilliant. It's simple, elegant, and explains math to kids: The Number Devil. I think it should get a ... er ... well, whatever the award is for the best children's book.

  8. #8

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    Well, here's what I got...

    43,466,557,686,937,456,435,688,527,675,040,625,802 ,564,660,517,371,780,402,481,729,089,536,555,417,9 49,051,890,403,879,840,079,255,169,295,922,593,080 ,322,634,775,209,689,623,239,873,322,471,161,642,9 96,440,906,533,187,938,298,969,649,928,516,003,704 ,476,137,795,166,849,228,875

    What did everyone else get?

    (I found it on a web site...this is the one, evidently...or so I'm told. I'm not gonna check his math. )

  9. #9

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    Quote Originally Posted by h3g3l View Post
    I'm pretty sure I'm thinking of the Fibonacci sequence here. The equation lets you specify the nth element in the series.

    What's the thousandth Fibonacci number? I've no idea without the generating equation. I'll hunt for it, but I remember a root-5 in there somewhere.

    UPDATE: I'm thinking of Binet's Formula, described here.
    That is quite beautiful in a mathematical sense. I've always been convinced there was a way to find any single term of the Fibonacci sequence, now you have finally shown it to me.

  10. #10

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    When I was revising over some Maths during the Easter break, I was still puzzled at how Forier analysis worked in a mathematical sense, It worked. But how? It was a formula pluck the numbers in, calculate and you get the answer.

    I got some where, but it wasn't much. Other than it worked out how "cos-like" and "sin-like" a signal was. And you got some numbers back to which you could figure the over-all amplitude and phase such as the bottom (pulling from memory, most likely a mistake somewhere)

    A*Sin(x)+B*Cos(x) = (A^2+B^2)^0.5Cos(x+y)
    where y = tan (A/B)

    Now, I could draw it on a graph and go "Look!", However relating it back to Fourier analysis is some what harder. It's because I am flicking between, Time domain, Frequency Domain and some intermediate equations to glue the two together.

    Still need to also find out what the 's' domain is for Laplace transformations (similar to Fourier Analysis in a way). But I haven't looked at it much, and I failed that test. The lest said about that, the better.

    But I find it somewhat upsetting that I get taught on how to use the equations/techniques rather than learn how they work. As I think they should come hand in hand. Ok, So a racing driver might not need to know how the car works in as much depth as a mechanic. But if you want to drive it well, knowing how the car works, means you can manipulate the power some time and place in future.

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