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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