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Math Help - Bizarre Triange Surface Area Anomaly

  1. #1
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    Bizarre Triange Surface Area Anomaly

    What in the world is going on here?

    Can you explain to this very novice math guy why this happens?

    http://kotinetti.suomi.net/jussi77/bizare.jpg

    The individual pieces are exactly the same. The surface area for each component is exactly the same. The area of the composite triangle is exactly the same. Yet if the two triangles are reversed, the irregular pieces form a new empty spot. It's boggles my meager math-oriented mind.

    Obviously I'm less than no math expert. My exploration was limited to drawing the shape on a piece of graph paper and calculating the area of each piece as well as the triangle as a whole. What I came to realize is that the surface area sum of the individual pieces is still somehow less than the surface area of the whole. But that's basically demonstrated by the drawing, and it only conflates the issue.

    Can someone please help me understand the math at play here?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by decadyne
    What in the world is going on here?

    Can you explain to this very novice math guy why this happens?

    http://kotinetti.suomi.net/jussi77/bizare.jpg

    The individual pieces are exactly the same. The surface area for each component is exactly the same. The area of the composite triangle is exactly the same. Yet if the two triangles are reversed, the irregular pieces form a new empty spot. It's boggles my meager math-oriented mind.

    Obviously I'm less than no math expert. My exploration was limited to drawing the shape on a piece of graph paper and calculating the area of each piece as well as the triangle as a whole. What I came to realize is that the surface area sum of the individual pieces is still somehow less than the surface area of the whole. But that's basically demonstrated by the drawing, and it only conflates the issue.

    Can someone please help me understand the math at play here?
    The upper figure is not a triangle, there is an angle where the red triangle
    meets the green triagle.

    RonL
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  3. #3
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    hmmm....

    I understand what you are fundamentally saying, but I don't yet see it or quite agree. Can you explain further?

    I have drawn both figures out very carefully on graph paper to see the results myself -- obviously the line for the hypotenuse on the upper figure I created with a straighedge (no angle created or involved).

    My recreation recreates the results as I see them. Unless I am missing something (which, respectfully to your answer is totally possible), I don't yet see yours as an explanation. I'd welcome and thank any further explanation or clarification you have or other ideas...

    thanks again!
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  4. #4
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    i do see...

    thanks, again.
    i checked more carefully, and now I do in fact see what you are saying. the angle is slight.

    im curious about how my attempt to carefully recreate it with an actual triangle failed (or succeeded), then. I guess I'll need to do it again, even more carefully.
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  5. #5
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    Hello, decadyne!

    This is a classic "paradox" . . . quite baffling until explained.

    The small right triangle has legs 2 and 5.
    The smaller acute angle is: \arctan(0.4) \approx 21.8^o

    The large right triangle has legs 3 and 8.
    The smaller acute angle is: \arctan(0.375) \approx 20.6^o

    The two triangles are not similiar and hence are not interchangeable.
    Code:
                                              * B
                                          *   |
                                      * 21.8 | 2
                                  * - - - - - *
                            *     |     5
                      *           | 3
                * 20.6           |
        A * - - - - - - - - - - - *
                       8
    Although the diagonal AB is supposed to be a straight line,
    . . but there is an imperceptible "dip" in the middle.
    There is a very narrow triangular strip missing.
    . . And that triangle has an area of a half square unit.

    If the two right triangles are switched, the diagonal "bulges" in the middle.
    . . The extra area is a half square unit.
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  6. #6
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    Here is a picture greatly exagerrated of this situation.
    Attached Thumbnails Attached Thumbnails Bizarre Triange Surface Area Anomaly-picture5.gif  
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  7. #7
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    thanks to all the replies

    thanks folks for indulging this non-math guy. i'm learning about the world of fibonnaci numbers, the golden ratio, and so on. it's captivating stuff for someone like me, although i know you people are much more familiar with it.
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  8. #8
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    Quote Originally Posted by decadyne
    thanks folks for indulging this non-math guy. ...
    Hello,

    I've attached a diagram which shows a square of 8X8 = 64 and below a rectangle (which is constructed by exactly the same pieces as the square) of 5X13 = 65.

    Much success looking for the missing link.

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails Bizarre Triange Surface Area Anomaly-recht_quad.gif  
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  9. #9
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    aha -- i figured that one out!

    well, since ive quickly learned what to look for, i knew right away how this one works. i've also found some other interesting examples, and have moved quickly on my way to trying to believe that .999 is equal to 1. as a philosopher, it's easy enough, but if you cut a .999 repeating inch piece of wood for a 1-inch slot, no one can ever tell me it will fit perfectly....

    do you folks have any other interesting example of properties of numbers or other curiosities like the one I got interested in?
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  10. #10
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    Quote Originally Posted by decadyne
    . i'm learning about the world of fibonnaci numbers, the golden ratio, and so on.
    There is much stuff about fibonacci numbers in the "Number Theory" section.
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