Re: Questions (Space)

Jonathan L Cunningham <spam@xxxxxxxxxxxxxxxxxxxx> wrote:
Tina Hall <Tina.Hall@xxxxxxxxxxxxxxxxx> wrote:
Jonathan L Cunningham <spam@xxxxxxxxxxxxxxxxxxxx> wrote:
Tina Hall <Tina_Hall@xxxxxxxxxxx> wrote:
Jonathan L Cunningham <spam@xxxxxxxxxxxxxxxxxxxx> wrote:
Tina Hall <Tina_Hall@xxxxxxxxxxx> wrote:

It's interesting that a^2+b^2+c^2=d^2, too, though.

I wish I knew why.

There's no "why" because it doesn't have to be true.

Why not?

Eh? The question seems meaningless to me. Why are't peanuts made of
chocolate? If you start asking why things are *not* true, we'll
never stop.

You said it doesn't have to be true. There should be a reason why it
doesn't have to be true. It is true as far as I know, after all. I'm
asking you for what I don't know.

Ok. Someone or something has convinced or persuaded you that it is
true.

No, as far as I know it's the _only_ option, and you say there is _more_
than one option. I ask what that other option is.

Since it doesn't have to be true, that "someone or something" must be
wrong. Before I can tell you why it is wrong, I have to know what it
is. (There are a million wrong explanations for everything.)

It would help if you told me what you are talking about.

It's like I said that trees are green, and you say it doesn't have to be
true. I ask why, and you just keep saying that trees don't have to be
green, but give no alternative. You don't say that plants are green,
too, or that trees can be brown or white, you just keep saying that
trees don't have to be green.

So I don't even know whether you're talking about tree or green. In fact
I wonder whether it's got anything to do with the three kinds of tree I
know (say oak, fir, and peach), because I know there are a lot of other
trees that I can't name (calculate).

If you can tell me why you believe it is true, I can say where the
explanation depends on some "fact" which is incorrect.

What fact?

If you can't remember it (which seems likely, most of the "proofs" are
quite hard to remember, I think) then we are stuck.

It would help if you told me what you are talking about. I learned
a^2+b^2=c^2, and here someone said that a^2+b^2+c^2=d^2 works, too.
That's a triangle, and a cube.

For the triangle, you draw a square at each line, and the a and b
squares have the same area-size as the c square. I guess the same should
be true for the cube thing, I haven't checked.

You say the formula doesn't have to be true, and I don't know what you
mean. I started out thinking that you know of some special occasion
where the formula doesn't apply to that triangle. Then I began to wonder
whether you just meant other shapes, which naturally have other
formulas.

Right now I wonder whether you want to bend the triangle. But then the
result wouldn't be a triangle. If you flatten it, one side isn't
straight but curved, that's not what the formula is talking about, and
has nothing to do with it.

Just for interest, though, it may depend on what is often called "the
parallel postulate" which I will call PP. This can be stated
approximately as:

(1) Draw a straight line
(2) Mark a point somewhere off to the side (not on the line).
then
(PP) you can draw *exactly one* other line, so that it goes
through the point from step (2) but never crosses the line from step
(1) in either direction.

That's parallel, nothing to do with triangles.

Many mathematicians, over about two thousand years, spent large parts
of their life trying to prove that statement (PP) must be true.

Huh? What's there to prove? If it isn't parallel, they cross somewhere.
You can draw one line on either side of (1), and that's it.

They failed, because it doesn't have to be true.

There's that again. Why wouldn't it have to be true?

It *is* true in Euclidean geometry, which includes it as an "axiom".

What's that got to do with triangles?

(An axiom is something which is "obviously" true. Dangerous if the
"obvious" is wrong: it can waste the entire life of a mathematician.)

You haven't yet said where it isn't true.

In "spherical" geometry, there are no straight lines you can draw:

So the whole thing doesn't apply. It's about straight lines, after all.

And if you count curves as straight, then you just slice your sphere at
different spots, say, horizontally. You get differently sized circles,
but they don't meet each other.

You can't have both.

all straight lines eventually cross somewhere.

Either there are no straight lines you can draw, or they don't cross.

In "hyperbolic" geometry, you can draw many straight lines through the
point.

?

That's just different angles, but no longer parallel.

And I don't know what it has to do with triangles.

On other geometries, it depends where you are, whether you can do it.

I still have no idea what you are talking about.

Think of anything you believe must be true about (Euclidean)
geometry. If it were not true, the geometry would not be Euclidean.

No idea what you mean.

To unravel that...

a^2 + b^2 = c^2

I wish I knew why. That means I wish I knew whatever intrinsic link
is there between the different sides of a (right-angle? I forgot)
triangle.

It's not true. It's only true for Euclidean geometry.

Whatever that is, and whereever it might not be true.

I can think of two other very simple geometries where the formula is
different. To avoid complications, suppose a is less than b (you can
swap them if it is the othe way around). Mathematicians would say,
WLOG, suppose a < b.

?

Then in "warehouse crane geometry
b^2 = c^2 (no a^2)

?

It is the amount of time it takes a crane with two motors, one moving
horizontally, one moving vertically, to change position. It takes as
long as whichever movement is longer.

Which has nothing to do with the thing we're talking about. (That's what
you get when you don't tell me what _other_ thing _you_ are talking
about; I stick to what I wrote; calculating the length of a side of a
triangle, or the distance between the two diagonal opposite points of a
cube. And it doesn't matter how long your crane needs, the distance is
still the same.)

If you don't want to specify that a is less than b, it is:
max(a,b)^2 = c^2

?

Another one is "city block" geometry:
|a| + |b| = |c|

?

This means that the distance is the sum of the north-south and
east-west distances, when you can't cut through buildings.

The distance (that, say, a bird could fly above it) still remains the
same.

(Hmmm. I originally used (a+b)^2, but that's not correct. The |x|
notation simply means ignore which direction it's going, and only look
at how far.)

Which isn't the straight distance between the begining of a and the end
of b.

All I can think of concerning what you say is

a^2 + b^2 != c^2

I want to know where and why a^2 + b^2 != c^2.

I've given a couple of examples above.

No, actually you didn't. You just measured other things than the length
of the line that connects a and b.

But they probably weren't useful to you (might have interested other
people, I hope).

I don't know what they've got to do with geometry, either.

The short answer to *where* a^2 + b^2 != c^2 is: almost everywhere.
Here on Earth, for example.

Maybe if you do it on the ground, where your triangle gets curves, but
on a flat piece of paper, I don't know how it couldn't be. And that's
the question.

But on Earth it's *almost* correct. If you put a one tonne weight
inside a two metre diameter circle, then the error would be in about
the 25th decimal place (approximately, I'm going on memory). Like
1000000000000000000000001 instead of 1000000000000000000000000. That's
(very much) less than the diameter of an atom.

But it's a circle, not a triangle. I don't even know in what calculation
your error would occur. Not the one we're talking about because it
doesn't apply.

As for "why" I'd need to know why you believe it is true.

As I said, I wish I knew what intrinsic link is between the two sides to
the third. I just know that it works.

If it is not true, it is not a Euclidean geometry.

That's not helpful. I don't know where it could not be true.

Your kitchen. Your living room. Everywhere you look.

Well, if I draw a triangle (I think it's a right-angle triangle, but am
still not sure) on the table here in my one room with kitchen corner,
it's true.

You haven't yet told me how it doesn't have to be.

Like Pi isn't 3.14 but with infinite numbers behind the dot?

The ratio of the circumference of a circle to its diameter (the
original definition of pi) will not be 3.14, yes. The mathematical
quantity calculated for Euclidean geometry won't change (it can't,
any more than 2+2 can stop being 4) but the mathematical pi need
not be the correct measure for a non-Euclidean geometry.

You've lost me.

I thought I was agreeing with you.

While I have no idea what you meant.

If you stretch a piece of string across a circle, you get a length. If
you wrap a string around a circle, you get a different (longer)
length. The ratio of those two lengths is called "pi". The value we
get for this ratio is about 3.14159.

I know that.

Mathematicians can calculate what they think this value ought to be.
They get the answer: 3.1415926535897932384626433 etc. (or whatever).

Yes, with infinite numbers behind the dot.

Trouble is, if they calculate it *that* accurately, it's no longer
true, because "space" (the space we live in) isn't that shape. Real
circles differ (a bit) from the calculated value.

It's either a circle (flat) or it's not. And you're saying it doesn't
have to be true for real, flat, unbent, circles. At least that's what
you sould like saying, except not about circles, but about triangles and
cubes. And that's where I ask why it doesn't have to be true. How could
it not be true.

That's partly why the orbit of Mercury around the sun doesn't make
a perfect ellipse (it's only part of the reason, because the orbit
is also disturbed by the other planets). Because circumference of a
circle around the sun is not exactly pi times its diameter.

It's not a circle, it's an ellipse. Don't know how you get from one
to the other. (Both ways. I don't know where your thougths go from
one to the other in that sentence, and I don't know how you
calculate ellipses.)

Circles and ellipses are similar enough, for purpose of this
discussion.

They're as similar as circles and squares. Or a d4 and a d10. (One is a
three-sided pyramid, the other two five-sided pyramids with their
bottoms glued together.)

Getting vague when you don't say what you're talking about in the first
place doesn't help.

The same thing would happen if Mercury's orbit were a
perfect circle.

And how?

It's the shape of the space around the sun which is different.

If the shape you draw the circle on isn't flat, then it isn't a circle,
certainly no perfect one.

You could measure it by measuring the diagonal of a square: the shape
you use doesn't matter. It's just that some shapes are easier to
calculate what the difference is.

Then explain how, with a flat square (half of which is just that
triangle with the a^2+b^2=c^2), the calculation is not true.

--
Tina
WIP: Space: 2666 words
WISuspension: Seasons & Elements trilogy | Magic Earth series
Posted to Usenet newsgroup rec.arts.sf.composition.

.

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