2405Pythagorean triples/area-perimeter relations
- Aug 2, 2001Hi Neil
> >> >Re the area/dia business - of course its obvious when you see them asof
> >> >decimals. We're looking at whole number right-angle triangles, pairs
> >> >which will fit together to make whole number rectangles, so of courseboth
> >> >diameters and areas will be either whole or whole and a half!area
> >> I am missing something here. The hypotenuse will be an integer, the
> >> will either be an integer or end in .5, but why does area/perimeterhave
> >> to be an integer or end in .5?pythagorean
> >Because the sides will be integers! Think of any right angled triangle,
> >place another upside down on top, and you have a rectangle. With
> >triangles each such rectangle will have integer sides, thus integer areaand
> >perimeter. Any triangle will thus have isuch rectangle/2 area; wholenumber
> >when the rectangle is even, whole and a half when its odd.Hmm. I thought I had that sussed. I'll think on that some more.
> 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?
> Arranging four triangles around a square of which the side length is thewhat
> 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
> >> modern science would call a pure number, since according to modernof
> >> science it would have dimensions of length, since area has dimensions
> >> length squared, and perimeter has dimensions of length. IMO moderninterests
> >> 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
> >me.Are they separable, since area is read as squares?
> 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?
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
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 openedNow if you can save as something I can open, complete with formulae, we, at
> nicely with all formulae.
least, will be off : ) Surely you can save in a microsoft format?
> Could someone on the list please advise on what the best lingua francaMike
> format is, which Excel 8 can save as, and that keeps formulae?
> Neil Fernandez
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