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2405Pythagorean triples/area-perimeter relations

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  • mikebispham@cs.com
    Aug 2, 2001
      Hi Neil

      I wrote@
      > >> >Re the area/dia business - of course its obvious when you see them as
      > >> >decimals. We're looking at whole number right-angle triangles, pairs
      of
      > >> >which will fit together to make whole number rectangles, so of course
      both
      > >> >diameters and areas will be either whole or whole and a half!

      > >> I am missing something here. The hypotenuse will be an integer, the
      area
      > >> will either be an integer or end in .5, but why does area/perimeter
      have
      > >> to be an integer or end in .5?

      > >Because the sides will be integers! Think of any right angled triangle,
      > >place another upside down on top, and you have a rectangle. With
      pythagorean
      > >triangles each such rectangle will have integer sides, thus integer area
      and
      > >perimeter. Any triangle will thus have isuch rectangle/2 area; whole
      number
      > >when the rectangle is even, whole and a half when its odd.
      >
      > I understand that far, but how do you get from this to prove that
      > area/perimeter (i.e. 0.5 * (area of rectangle)/(perimeter of triangle) )
      > is an integer or 'integer and a half'? Or in other words that area of
      > rectangle = either an exact multiple of the perimeter of the triangle,
      > or half of an exact multiple?

      Hmm. I thought I had that sussed. I'll think on that some more.

      > Arranging four triangles around a square of which the side length is the
      > hypotenuse (always my all-time favourite method of proving Pythagoras's
      > Theorem) may also give a proof.

      > /

      > >> I was going to post something about how area/perimeter doesn't give
      what
      > >> modern science would call a pure number, since according to modern
      > >> science it would have dimensions of length, since area has dimensions
      of
      > >> length squared, and perimeter has dimensions of length. IMO modern
      > >> science has got something wrong with regard to dimensions. It seems so
      > >> weird and maybe even 'arbitrary', for example, that a given physical
      > >> constant has, say, dimensions of mass squared times length times time.
      > >> Doesn't sound convincing. Too far away from the Pythagorean view of
      > >> numbers as basic. But unfortunately I have not reached the stage of
      > >> formulating a coherent alternative viewpoint! Very interesting, how the
      > >> discussion is going with regard to the relationship between area and
      > >> perimeter, expressed as a number.
      > >
      > >Yes, it had crossed my mind that we were playing with apples and oranges.

      > >But it feels perfectly valid in a pythagorean way, which is what
      interests
      > >me.
      >
      > Me too.
      >
      > The idea of the square is deeply inside this. First, for each triangle
      > it makes sense to grasp area as equal to the area of a number of
      > squares. Second, the whole thing is about p^2 etc.

      > What if the idea of the square could be usefully understood as more
      > profound than the idea of area?

      Are they separable, since area is read as squares?

      Certainly the idea of square-ing is very deep in pythagorean mathematical and
      mystical thought - although I couldn't easily back that statement up. The
      geometric lambdoma which is built on progressive squaring - and which
      accounts for the harmonic structure of the world, operates by raising the
      fundamental numbers 2 and 3 by powers of two. Again, the
      point-line-surface-solid account of 'being' is a progressive squaring of the
      primary value - line. And the inverse of squaring is tied to the mystery of
      root 2 - shown in the first diagonal - on the square. These are fundamental
      links between 'being' or 'substance' - three-dimensionality, and the
      arithmetic process of squaring. How much can these fundaments be
      *separated* from the pure science of pythagorus' theorum? Not at all
      probably, in my view, if fact its probably only modern science that has done
      so.

      That said, (thinking aloud), non-square areas, as opposed to pure squares,
      are different, though they must have carried the clutter associated with
      'surface' in the P-L-S-S ontology.

      > The file "Neils Pythagorean triples4" saved as an Excel worksheet opened
      > nicely with all formulae.

      Now if you can save as something I can open, complete with formulae, we, at
      least, will be off : ) Surely you can save in a microsoft format?

      > Could someone on the list please advise on what the best lingua franca
      > format is, which Excel 8 can save as, and that keeps formulae?

      > Neil Fernandez

      Mike
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