- Aug 2, 2001Hi Neil

I wrote@> >> >Re the area/dia business - of course its obvious when you see them as

of

> >> >decimals. We're looking at whole number right-angle triangles, pairs

> >> >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!

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/perimeter

have

> >> 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 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.

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 the

what

> 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 modern

of

> >> science it would have dimensions of length, since area has dimensions

> >> length squared, and perimeter has dimensions of length. IMO modern

interests

> >> 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

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

Now 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 franca

Mike

> format is, which Excel 8 can save as, and that keeps formulae?

> Neil Fernandez

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