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Thread: Cyclic number 142857 and Enneagram

  1. #1 Cyclic number 142857 and Enneagram 
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    1/7 = 0.(142857)
    2/7 = 0.(285714)
    3/7 = 0.(428571)
    4/7 = 0.(571428)
    5/7 = 0.(714285)
    6/7 = 0.(857142)

    This series of numbers has strange properties
    Above, it seems these numbers follow a continuous loop
    A better way to visualize this is the "fourth way enneagram"

    124578.jpg9 points for the 9 numerals (except 0) you can use to write any real number.

    Looking at the values of the fractions, one can trace all 6 the same direction in this closed path.
    I find this very interesting, what's the significance of this? Are there other patterns that make such loops?


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  3. #2  
    Forum Professor Wild Cobra's Avatar
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    I have something better. They make ellipses:






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  4. #3  
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    Quote Originally Posted by Wild Cobra View Post
    I have something better. They make ellipses.
    That is pretty cool.
    Using that first image, I made two congruent triangles.

    coord-cyclic.jpg
    Not as good as the ellipses, but it relates.

    So, I'm assuming the points were plotted from every 2 digits in each sequence.
    The sequences from 1/7, 2/7 and 4/7 corresponded to the red triangle.
    The sequences from 3/7, 5/7 and 6/7 correspond to the blue triangle.

    These two triangles are congruent.
    Approximate side lengths: 3.6, 6.3 and 4.1
    Approximate area of figure: 6.8 squares
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  5. #4  
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    It was this number 142857 that got me introduced to The Science Forum yesterday. It just popped into my mind.
    I knew about it from way back in the 1960s. I got it from my favourite number 7 by simply doubling and limiting to 2 digits.
    Thus 14 28 56, then the 1 of the next double 128 gets added to the 6 to make 142857. Multiply by original 7 and get 999999, so must be reciprocal of 7, if repeated.
    I also knew it was very quick to get the other 'sevenths' (2\7, etc.) by just starting the sequence at a different point.
    I also knew then about the enneagram and could construct it by drawing a circle, dividing the circumference into 9 parts
    (goes exactly 40 - thanks to those ancient Babylonians!). Then number 1 to 9 every 40 and join the points using 142857.
    The other three digits 3, 6, 9 - join them with a triangle to get a nice logo!.
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  6. #5  
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    Quote Originally Posted by Elterish View Post
    [What you said]
    It also loves the number 9:

    1+4+2+8+5+7 = 27 (9 x 3)
    27 huh? 2+7 = 9

    14+28+57 = 99

    14+28+5+7 = 54 (9 x 6)
    54? 5+4 = 9
    And you get the idea

    Using addition with these numbers, you will run into so many 9's and multiples of 9.

    Nine itself is also a magical number. 9^2 = 81.

    1/81 = 0.012345679012345679012345679... (skips 8)

    8/81 = 0.098765432098765432098765432... (skips 1)
    And there's more but I'm done there.

    So, yeah. Everything in math is intertwined and connected in a multitude of ways we'll never fully understand.
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