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Generalization of Pythagoras by Haight.

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  • Floor en Lyanne van Lamoen
    Dear friends, Is there anybody on the list who can tell me how the sequences http://www.research.att.com/~njas/sequences/A004253
    Message 1 of 2 , Apr 29, 2006
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      Dear friends,

      Is there anybody on the list who can tell me how the sequences
      http://www.research.att.com/~njas/sequences/A004253
      http://www.research.att.com/~njas/sequences/A004254
      are derived in the paper
      F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C.
      Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

      Kind regards,
      Floor.
    • Ken Pledger
      ... That book was a Festschrift for the 80th birthday of H. G. Forder. I m sorry I don t know who Frank A. Haight was. Haight starts with a right-angled
      Message 2 of 2 , Apr 30, 2006
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        At 5.44 PM +0200 29-4-06, Floor en Lyanne van Lamoen wrote:
        >Dear friends,
        >
        >Is there anybody on the list who can tell me how the sequences
        >http://www.research.att.com/~njas/sequences/A004253
        >http://www.research.att.com/~njas/sequences/A004254
        >are derived in the paper
        >F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C.
        >Butcher, editor, A Spectrum of Mathematics. Auckland University
        >Press, 1971....


        That book was a Festschrift for the 80th birthday of H. G.
        Forder. I'm sorry I don't know who Frank A. Haight was.

        Haight starts with a right-angled triangle with sides a, b, c
        (the hypotenuse), and defines a succession of squares outside it.
        He refers briefly to each square by the length of its side, e.g.
        "square b" means the square in the diagram having side b. He writes
        a_1 = a, b_1 = b, c_1 = c, then (in his own words):

        "For j = 1, 2, 3, ...
        x_j is constructed on the segment joining adjacent corners of b_j and c_j
        y_j is constructed on the segment joining adjacent corners of a_j and c_j
        z_j is constructed on the segment joining adjacent corners of b_j and a_j

        For j = 2, 3, ...
        a_j is constructed on the segment joining adjacent corners of
        y_(j-1) and z_(j-1)
        b_j is constructed on the segment joining adjacent corners of
        x_(j-1) and z_(j-1)
        c_j is constructed on the segment joining adjacent corners of
        x_(j-1) and y_(j-1)

        In the above definitions the 'adjacent corners' referred to are those
        'exterior to the diagram' (see Fig. 1), which are not already
        connected, and such that the connection would not pass through the
        figure."

        Haight then names some angles and derives several formulae
        leading to the sequences which you asked about. The ratios 1 : 4
        : 19 : 91 : ....
        = a_1 : a_2 : a_3 : a_4 : ....
        = b_1 : b_2 : b_3 : b_4 : ....
        = c_1 : c_2 : c_3 : c_4 : ....

        and the ratios 1 : 5 : 24 : 115 : ....
        = x_1 : x_2 : x_3 : x_4 : ....
        = y_1 : y_2 : y_3 : y_4 : ....
        = z_1 : z_2 : z_3 : z_4 : ....

        Of course it's easier to follow with the diagram. If you'd
        like to send your snail-mail address I can easily post you a
        photo-copy of this little article.

        Ken Pledger.
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