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Thread: Fibonacci

  1. #1 Fibonacci 
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    I am not a trained mathematician, calculus or scientist ok so i might be talking rubbish and i won't be offended if you tell me so (but at least try to do it in a nice way

    I'm an artist and creative at heart and i like playing and making patterns with numbers and geometry and making pictures from them but i'm still a fresh faced newbie when it comes to maths!

    Anyway i put the fibonacci numbers in a Kamea in the circular motion and it doesn't matter how big or small the square is each line adds up to the same and with some there seems to be significance with similar numbers 6 and 42 particularly.

    I have tried to research (online) if there has been any other research of the same but i can't find anything.

    Does anyone know if this has significance and if so what and whether this explains anything or anyone else has come up with the same?

    Thankyou


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  3. #2  
    Forum Professor serpicojr's Avatar
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    Could you describe in more detail what process you're using or maybe post a picture of a final result? I'm trying to reproduce it myself and having a tough time of it, i.e. I'm not reproducing your results. I'm assuming kamea is synonymous with magic square, and the circular motion I'm using is, say, starting in the upper left corner, going down to the bottom, all the way right, all the way up, as far left as I can go, and then down again, etc.


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  4. #3 Kamea and Fibonacci image (i hope) 
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    No you've got to start as close to the middle as you can and imagine it like an unfolding snail shell.
    Yes it's like a magical kamea.
    You get the same part emerging around the square when you add the numbers up.
    I am going to attempt to put a diagram in, not sure if it will be succesful as i'm usually pretty naff at this, but there's only one way to find out!

    (well i have code for the image so i hope it works!)

    obviously A- you add each line
    Then B - you add up the numbers e.g 6+2+6=14 and so on.

    Interesting how you get the same pattern, and it works however big or small your kamea. It also works if you start with the second 1 and omit the intial 0 and 1 as i mistakenly did.
    But then if you play around with the numbers you have in various ways you get patterns and dominant numbers.

    In this Kamea i seemed to get the dominant numbers of 3 and especially 9
    I'm still yet to play further with this but if you come up with anything else i would be very interested.

    One of the things i do also is place patterns and geometric shapes on a square grid like a kamea, such as ones found in crop circles and deduce a number system from them. By playing around with these numbers i can produce further geometric shapes and patterns.

    I'm not really a scientist or a mathematician and i do it mostly for fun and curiosity, but then i guess that's how things were discovered in the first place.

    Well good luck, happy playing and best wishes[/img]
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  5. #4  
    Forum Professor serpicojr's Avatar
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    Experimentation is a vital part of mathematics--it reveals patterns and hints at general truths. And it's a lot of fun, too!

    Now that I see what you're doing, I have an idea of what's going on and might be able to get back to you with a good "reason" of why this pattern is emerging. I'm particularly interested in the numbers you get in the B column/row--in fact, I may suggest that you add more columns/rows so that you can keep adding the digits together until you can't do so anymore (e.g., so column C in your picture would read down 4, 7, 5, 5 and row C would read across 5, 5, 4, 7). You suggest that these two always contain the same numbers, and I would believe that this is true in general. I'm interested in which numbers get repeated, which numbers appear for different size squares, etc.
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  6. #5  
    Forum Junior DivideByZero's Avatar
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    I tried it with 6x6 squares.

    http://img236.imageshack.us/img236/4...bsquarelw5.jpg

    notice anything?
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  7. #6  
    Forum Professor serpicojr's Avatar
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    There's some sort of pattern going on, but it's hard for me to really describe or generalize it. I'll try generating a few and see what I can come up with.
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  8. #7  
    Forum Freshman Faldo_Elrith's Avatar
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    Taking the sum of digits of a number is the same as taking congruence modulo 9. So in the kamea, let , , , , and so on.



    So you want to prove that and so on.
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  9. #8  
    Forum Freshman Faldo_Elrith's Avatar
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    Okay, just calculate for each n and you get

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  10. #9  
    Forum Ph.D.
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    Quote Originally Posted by Faldo_Elrith
    Taking the sum of digits of a number is the same as taking congruence modulo 9.
    It’s also the same as taking congruence modulo 3 – which is easier.
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  11. #10  
    Forum Freshman Faldo_Elrith's Avatar
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    No, they are not the same! It's true that anything congruent mod 9 is also congruent mod 3 (since 3 divides 9). But the converse is not true! After all, 3 and 6 are congruent mod 3 but not congruent mod 9. The two are not equivalent. And since cong mod 9 results are more general than cong mod 3 ones, it's better to use the former.
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  12. #11  
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    Okay, right. Then I suppose the modulo-3 method is more applicable to DivideByZero’s example, in which there is apparently no pattern to the 6×6 kamea.

    Let’s put it this way. If the sums of digits of two numbers are the same, then the two numbers are congruent mod 9, and hence mod 3. Therefore (by the law of contrapositives) if the two numbers are not congruent mod 3, then the sums of their digits can’t possible be equal. Indeed, instead what DivideByZero did, you can fill up the 6×6 kamea with the Fibonacci numbers mod 3:



    The row sums are 0, 1, 1, 0, 1, 1 (top to bottom) while the column sums are 0, −1, −1, −1, −1, −1 (from left to right). Since there is no pattern in mod 3, we can be sure that there will be no pattern with the actual sums of digits.
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