## Re: Pythagorean triples/area-perimeter relations

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• Hi Neil 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
Message 1 of 4 , Aug 1, 2001
Hi Neil

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! Still, might
lead on to an interesting look at the distribution. We'd really need a way
of generating *all* pythagorean triples to do that.

calculates them? I can do radians but not sine etc on my spreadsheet, and I
don't know how to work radians for trigonometrical purposes.

In a message dated 31/07/01 13:42:31 GMT Daylight Time,
neil@... writes:

> In message <8f.e0ca206.2897f077@...>, mikebispham@... writes
>
> >Hi Neil
> >
> >Fascinating as it sounds, (I've more to tell you about those angles when
I
> >can see them), I can't open your spreadsheet. Could you save as
something
>
> >non-excel and re-post? I suspect anything.ss will do. I use works for
> >windows. Is there a generic file extension we can all use?
>
> Hi,
>
> here's hoping that there is!
>
> '.ss' isn't in the list of what comes up when I try to 'Save As...', and
> the following message comes up when I try to save as a .wks file:
>
> "A formula in a cell (Cell:B3) could not be converted because it contains a
> function that is not available in the file format to which you are saving.
> If you continue the save, the formula and result will be saved, but the
> function will be converted to an error value."
>
> ...and the formula in Cell B3 is:
>
> =IF(AND(INT(SQRT(B\$2^2+\$A3^2))=SQRT(B\$2^2+\$A3^2),B\$2^2+\$A3^2<10001),"*","")
>
> I haven't had any experience with Works, so can't home in on the problem
> here. Maybe it is something minor?
>
> Here is the full list of available 'Save As...' options in Excel 97 (aka
> Excel 8).
>
> xls
> wks
> xlt
> prn
> txt
> csv
> xlw
> wk4
> wk3
> wk1
> wq1
> dbf
> dif
> slk
> xla
> (html)
>
This opened ok, but I only have numbers, not calculation. Is that right?
But...

> I've saved it as csv (comma separated variables), attached below, hoping
> this works. If this is the generic format we can all use, this will be
> great, the file is only 16kB in size. Not sure what 'ss' is - is it the
> 'generic' spreadsheet format used by Works?

I can't save as csv. My options are:

works ss
works for windows 3.0 ss
works for windows 2.0/works for dos ss
text & commas
text & tabs
text & tabs(dos)
excel ss
lotus 1-2-3
works 3.0 for macintosh ss (*.ss)
works 4.0 for macintosh ss (*.ss)

I've attached a 'text & commas' file (which saves as csv - loses 'special
formatting' - bold, italic, don't know what else, but at least we can still
open it. Does this work for you? (If any mac users want a copy, please
request one)

I've added columns to the triples for:

AREA
SUM (perimeter)

I can see some interesting and resonant numbers emerging, which might attach
particular significance to those triples that generate them - especially in
the sexagisimal framework. So far my favourite is:

18 24 30 216 72

A very sixy number!

Mike

> Also attached is a screenshot of one part of the spreadsheet, saved as a
> .jpg file.
>
> >PS does this contain the 1-100 square item you've been discussing?
>
> Yes! :-)
>
> >Lastly, below, is a GS calculator you can use to discover where your
navel
> >should be, and other useful things. Input your height, ta da! Maybe we
> >should do a survey of slandscapers, to see how close our average navels
are
>
> >to the 'ideal'!
>
> ...and how many of us are hi-phi and how many are lo-phi! :-)
>
> Best regards,
>
> Neil
>
• In message , mikebispham@cs.com writes ... I am missing something here. The hypotenuse will be an integer, the area will either be
Message 2 of 4 , Aug 1, 2001
In message <cb.1443eee7.2899075a@...>, mikebispham@... writes
>Hi Neil
>
>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?

>calculates them? I can do radians but not sine etc on my spreadsheet, and I
>don't know how to work radians for trigonometrical purposes.

This is all in the spreadsheet I'm trying to post. The formula for
conversion from degrees to radians is R = (pi/180) * D where R is the
angle size in radians and D is the angle size in degrees. The
spreadsheet calculates the angles using the formulae ASIN(B109/D109)
where B109 contains the shortest side and D109 contains the hypotenuse,
and ASIN(C109/D109) where C109 contains the middle side. The ASIN

>> Here is the full list of available 'Save As...' options in Excel 97 (aka
>> Excel 8).
>>
>> xls
>> wks
>> xlt
>> prn
>> txt
>> csv
>> xlw
>> wk4
>> wk3
>> wk1
>> wq1
>> dbf
>> dif
>> slk
>> xla
>> (html)
>>
>This opened ok, but I only have numbers, not calculation. Is that right?

No, it should be the whole thing.

>But...
>
>> I've saved it as csv (comma separated variables), attached below, hoping
>> this works. If this is the generic format we can all use, this will be
>> great, the file is only 16kB in size. Not sure what 'ss' is - is it the
>> 'generic' spreadsheet format used by Works?
>
>I can't save as csv. My options are:
>
>works ss
>works for windows 3.0 ss
>works for windows 2.0/works for dos ss
>text & commas

'text & commas' may be the same thing as csv, not sure.

>text & tabs
>text & tabs(dos)
>excel ss
>lotus 1-2-3
>works 3.0 for macintosh ss (*.ss)
>works 4.0 for macintosh ss (*.ss)
>
>I've attached a 'text & commas' file (which saves as csv - loses 'special
>formatting' - bold, italic, don't know what else, but at least we can still
>open it. Does this work for you?

Couldn't find the file.

>I've added columns to the triples for:
>
>AREA
>SUM (perimeter)
>
>I can see some interesting and resonant numbers emerging, which might attach
>particular significance to those triples that generate them - especially in
>the sexagisimal framework. So far my favourite is:
>
>18 24 30 216 72
>
>A very sixy number!

:-)

In message <LAW2-F53e1GTtBXO84t0001057a@...>, Terrie Halprin
<thalprin@...> writes

<snip>

>Neil, I am taking a look at these grids, the micro and macro in context,
>this morning AND I notice that these 14 prime triples, originally listed
>in/as micro prime, in the macro grid, have two more triples added to them:
>48 55 73 and 65 72 97 and that the 12 35 37 hasn't been pulled to
>the side -Ooh- so, you can in fact get to the 16/ square -man, that is just
>so FANTASTIC!

Terrie, thanks for this correction. Interesting that 16 appears in this
way when one works with a maximum side length of 100, another square.

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.

We are going away for a week.

Best regards,

Neil
--
Neil Fernandez
• ... My mistake. Apologies. Apparently csv does lose the formulae. Neil -- Neil Fernandez
Message 3 of 4 , Aug 1, 2001
>>This opened ok, but I only have numbers, not calculation. Is that
>>right?

>No, it should be the whole thing.

My mistake. Apologies. Apparently csv does lose the formulae.

Neil
--
Neil Fernandez
• Hi Neil I wrote@ ... of ... both ... area ... have ... pythagorean ... and ... number ... Hmm. I thought I had that sussed. I ll think on that some more. ...
Message 4 of 4 , 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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