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Stephen wrote:
> What about xyz = the ratio of any circle's circumference to its diameter?
Define "circle". Now define "distance". Now we're getting somewhere.
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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>> 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.
> Now define "distance".
I actually can't find a definition for how to do this in a non-Euclidean
space.
> Now we're getting somewhere.
Well, yes... I'm sure somebody somewhere has long since worked all this
out in excruciating detail. ;-)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Darren New wrote:
> Stephen wrote:
>> What about xyz = the ratio of any circle's circumference to its diameter?
>
> Define "circle". Now define "distance".
Why?
> Now we're getting somewhere.
>
No we are here ;)
“Here and Now!” Quoth the mynah bird.
--
Best Regards,
Stephen
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Stephen <mca### [at] aoldot com> wrote:
> What about xyz = the ratio of any circle's circumference to its diameter?
If it's in Euclidean space, then it's pi, else it isn't.
--
- Warp
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Warp wrote:
> Stephen <mca### [at] aoldotcom> wrote:
>> What about xyz = the ratio of any circle's circumference to its diameter?
>
> If it's in Euclidean space, then it's pi, else it isn't.
>
Would you run that past me again, in English? :-)
--
Best Regards,
Stephen
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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.
>
>> Now define "distance".
>
> I actually can't find a definition for how to do this in a non-Euclidean
> space.
Oh, I forgot. Now you have to also define the length of a curved line with
no endpoints. :)
> Well, yes... I'm sure somebody somewhere has long since worked all this
> out in excruciating detail. ;-)
Well, yes.
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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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.
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Darren New wrote:
> Now define "distance".
I thought the underlying assumption in this thread was that we're
dealing spaces equipped with a metric tensor, do you automatically get a
definition of distance in the standard manner. How does this require
extra care to define?
I suppose you could argue that circles are a bit trickier to define, but
I think the normal one will work well, and should even be well-behaved
on genus-0 manifolds, or locally on any smooth manifold (like spherical
or hyperbolic spaces).
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Kevin Wampler wrote:
> should even be well-behaved
> on genus-0 manifolds
I take that back, it's obviously not quite well behaved on a sphere
(under some definitions of well-behaved) since there are no circles of
sufficient large radii.
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>> should even be well-behaved on genus-0 manifolds
>
> I take that back, it's obviously not quite well behaved on a sphere
> (under some definitions of well-behaved) since there are no circles of
> sufficient large radii.
Any possible line segment in elliptic geometry can be used as a circle
diammeter. (Since lines that are "too large" to define a circle won't
fit into the space in the first place.)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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