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Darren New wrote:
> I think I was more making the point that "distance" isn't a simple
> definition when you're talking about space that curves in different
> directions in different places.
Sure. I get that.
> Indeed, isn't it possible to have spaces where the distance from here to
> there is different than the distance from there to here?
No. The definition of "metric" demands that it is symmetric (and
reflexive, and transitive).
You can, of course, define a function that represents some sort of
"nearness" which never the less lacks this property...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Darren New wrote:
>
> I think I was more making the point that "distance" isn't a simple
> definition when you're talking about space that curves in different
> directions in different places.
I suppose I normally view distance in Euclidean space from the same
definition that applies to non-Euclidean spaces, rather than the
sqrt(dx^2 + dy^2) form, so I didn't really consider this. You're right
though, if you're starting from the pythorgean theorem view of distance
it does bear some thinking about how it generalizes to the non-Euclidean
space.
That said, I'm not sure it's necessary to actually understand the proper
definition of distance in order to talk about circles in other spaces --
particularly if we limit ourselves to spherical and hyperbolic spaces
which are more or less easy to visualize.
> Indeed, isn't it possible to have spaces where the distance from here to
> there is different than the distance from there to here?
Sure, but the standard interpretation of "non-Euclidean space" that I'm
aware, ie "Riemannian manifold", uses a definition of distance which
doesn't allow such things. All distances are proper distances in that
they obey symmetry, the triangle inequality, etc. I suppose, however,
that you could argue that "non-Eucliean" should be colloquially taken to
include pseudo-Riemannian spaces too (so as to incorporate Minkowski
spaces), but even here you have a pretty nice definition of distance,
although "circles" become a bit nastier to talk about.
It's possible other people mean different things when they say
non-Euclidean space though, so I don't want to pretend that it's the
only way to read that term. Still, that's what had interpreted to be
implicit in the conversation.
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Orchid XP v8 wrote:
>
> and transitive
>
I don't think that's what you intended to say. You probably meant to
say "triangle inequality".
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>> and transitive
>
> I don't think that's what you intended to say. You probably meant to
> say "triangle inequality".
Actually I meant that d(x,y)=0 implies x=y, but even so, that's not the
right term...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Orchid XP v8 wrote:
>>> What about xyz = the ratio of any circle's circumference to its
>>> diameter?
>>
>> Define "circle".
>
> The set of all points at distance r from the point c.
>
In how many dimensions? Imaginary or real?
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On Tue, 12 Jan 2010 11:09:05 +0000, Invisible wrote:
> ...but Toyota is Japanese...?
>
> More importantly, I gather that most American cars are horrifyingly
> inefficient, so beating that shouldn't be too hard I'd guess. I don't
> really know.
Not much worse than cars in the UK, remember that the US gallon is
smaller, so 34 MPG here (which is about what I get out of my 10 year old
Saturn coupe) is about 7 L/100 KM (or about 40 MPG in UK Gallons if I've
done my math correctly).
The Prius owners I know (a couple) typically get 55-65 MPG (in US
gallons). That'd be 66-78 MPG in UK gallons.
Jim
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Orchid XP v8 wrote:
>>> and transitive
>>
>> I don't think that's what you intended to say. You probably meant to
>> say "triangle inequality".
>
> Actually I meant that d(x,y)=0 implies x=y, but even so, that's not the
> right term...
"identity of indiscernibles" is the phrase you want. And at any rate,
that conflicts more with your use of the term "reflexive", rather than
"transitive". But hey, you got "symmetric"! (and the gist of the
others, I'm just pulling your leg).
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Kevin Wampler wrote:
> Stephen wrote:
>>
>> What about xyz = the ratio of any circle's circumference to its diameter?
>>
>
> The main problem here is that xyz is not a constant, but is rather a
> function. In spherical and hyperbolic spaces it can be defined as a
> function of the radius so you need to say something like:
>
> xyz(r) = the ratio of the circumference to the diameter of a circle of
> radius r
>
> If your space doesn't have a constant curvature, then it's a function of
> yet more parameters.
>
> The main issue isn't only that pi has a pre-existing meaning, it's that
> the value in these non-euclidean spaces isn't even a constant.
Too much information.
La la la la la. ;)
--
Best Regards,
Stephen
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> I'll be waiting over 42 American MPG with just enough room for 5 people
> and some luggage (OTOH most of the time just me, secondarily me and
> Sonic) - even with the warming needed at winter - still having
> 150bhp/320Nm (upgradeable to 175bhp/370Nm with just a software) with
> pretty decent equipment.
Aren't all cars upgradeable with just software nowadays? In my car the
"1.6L", "1.8L", "2.0L" and "2.3L" versions all actually have exactly the
same engine. You are just choosing whether you want the 116, 143, 177 or
204 BHP software installed when you buy it :-) I heard some people got up
to 265 BHP out of that engine, but to me that seems really stupid as all the
other components will not have been reliability tested at that power level.
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Darren New wrote:
> The Megane seems to be a diesel car, which tends to get better mileage
> anyway, and almost as good as a Prius running on gasoline.
My Megane is petrol, although I gather there is a diesel version of most
of these cars now. And yes, diesel has completely different performance
characteristics.
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