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ANd nobody knows if we were or not 2D beings.
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Invisible wrote:
> These are two seperate, unrelated geometries.
>
> 4D Euclidian space is interesting because it's a straight extension of
> 3D Euclidian geometry.
I should read more about this, sounds interesting.
> Somewhat weirder is hyperbolic geometry, where multiple "straight lines"
> through a single point do not intersect each other [except at that
> point]. You really need to play with this:
>
> http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html
I'll check it out.
I've been trying to figure out how to calculate hyperbolic geometry, so
I may create new "circle limit" Poincare disk tesselations. I figure it
must be possible to create such a thing with POV-Ray functions, but so
far all my attempts have been total failures :(
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>> These are two seperate, unrelated geometries.
>>
>> 4D Euclidian space is interesting because it's a straight extension of
>> 3D Euclidian geometry.
>
> I should read more about this, sounds interesting.
3D Euclidian space = the place where we (apparently) live.
4D Euclidian space = the place where objects such as the hypercube live.
4D non-Euclidian space = any space with 4 dimensions that *isn't* 4D
Euclidian space. This includes a system where time is the 4th dimension,
but also many other kinds of space as well.
>> Somewhat weirder is hyperbolic geometry, where multiple "straight
>> lines" through a single point do not intersect each other [except at
>> that point]. You really need to play with this:
>>
>> http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html
>
> I'll check it out.
>
> I've been trying to figure out how to calculate hyperbolic geometry, so
> I may create new "circle limit" Poincare disk tesselations. I figure it
> must be possible to create such a thing with POV-Ray functions, but so
> far all my attempts have been total failures :(
I believe you can use ordinary 2D coordinates for hyperbolic space, you
just need to remap them when plotting them in normal 2D space.
(Unfortunately, I can't find any formulas for doing this.)
Note that there is more than one way to project hyperbolic space into
trying to copy Escher's Circle Limit.
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clipka wrote:
> Well, AFAIK there's actually no fundamental reason to apply different
> "measuring tapes" to time and space:
It's not measured *quite* the same way... The distance between two events is
sqrt(x*x+y*y+z*z-t*t) Note the - sign.
--
Darren New, San Diego CA, USA (PST)
I ordered stamps from Zazzle that read "Place Stamp Here".
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Darren New wrote:
> clipka wrote:
>> Well, AFAIK there's actually no fundamental reason to apply different
>> "measuring tapes" to time and space:
>
> It's not measured *quite* the same way... The distance between two
> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.
Hence "non-Euclidian space".
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clipka wrote:
> Kevin Wampler schrieb:
>
>> It is worth nothing, however, that the time dimension is *not*
>> identical to the space dimensions (one would hope not!) and distances
>> are measured differently in time than in space.
>
> Well, AFAIK there's actually no fundamental reason to apply different
> "measuring tapes" to time and space: The constant vacuum speed of light
> can serve as a ruler for both, with the distance of a light second
> equating a second.
>
> It just happens that it's still more practical to use meters for
> space-like dimensions and seconds for time-like dimensions.
I think that this becomes problematic for measuring between points which
aren't connectible by a lightlike geodesic, but maybe there's some
clever way around that (although I don't see how).
In any case, I was referring to the fact that under special relativity
spacetime has a metric that differs from that of a 4D Euclidian space
along the dimension corresponding to time. For instance, the length of
a vector can be negative, which is impossible under a Euclidean metric.
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Invisible wrote:
> Darren New wrote:
>> clipka wrote:
>>> Well, AFAIK there's actually no fundamental reason to apply different
>>> "measuring tapes" to time and space:
>>
>> It's not measured *quite* the same way... The distance between two
>> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.
>
> Hence "non-Euclidian space".
Yes. I wasn't sure if that counted as a euclidian space or not. :-)
--
Darren New, San Diego CA, USA (PST)
I ordered stamps from Zazzle that read "Place Stamp Here".
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Darren New wrote:
>
> Yes. I wasn't sure if that counted as a euclidian space or not. :-)
>
It doesn't, but it does share the property that the metric tensor is
constant over the entire space.
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Darren New schrieb:
> clipka wrote:
>> Well, AFAIK there's actually no fundamental reason to apply different
>> "measuring tapes" to time and space:
>
> It's not measured *quite* the same way... The distance between two
> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.
It /is/ measured in the same way - it's just that the Pythagorean
theorem doesn't hold in our peculiar (3+1)D universe...
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Kevin Wampler schrieb:
> I think that this becomes problematic for measuring between points which
> aren't connectible by a lightlike geodesic, but maybe there's some
> clever way around that (although I don't see how).
Well, if two points cannot be reached from one another - is there /any/
way to assign a distance to these points at all?
So I think this is a non-issue.
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