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 
Euclidian space; you probably want the Poincaré disk model if you're 
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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clipka wrote:
> 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.

In this context a lightlike geodesic refers to a path in spacetime which 
  light would follow, and I meant to imply that defining the distance 
between points which could only be connected by going *slower* than 
light would also be hard to define uniformly.

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clipka wrote:
> 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...

I fail to see how stating that the Pythagorean theorem doesn't hold 
isn't just another way of saying that distances are measured differently 
in our spacetime?

Is you point that in space time there *is* a well-defined notion of 
distance which unifies both the spatial and temporal aspects, and thus 
we don't really need to use one set of units for space and another set 
for time?  Your comment about light-seconds would make more sense in 
this context.  If this is the case perhaps I've been misunderstanding 
your point form the beginning, since I'd surely agree with this.

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Kevin Wampler wrote:
> clipka wrote:
>> 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.
> 
> In this context a lightlike geodesic refers to a path in spacetime which 
>  light would follow, and I meant to imply that defining the distance 
> between points which could only be connected by going *slower* than 
> light would also be hard to define uniformly.

In light of one of your other comments I think I now understand what 
point you're making: we can treat space-time distances in a common set 
of units by treating distances as times via the speed of light.  In 
which case I'd agree, and it fact the necessity for this clearly falls 
out of being able to have a single number represent the distance at all.

I still don't see how it's relevant for my initial comment that the 
space and time coordinates are treated differently though, since they 
are most definitely factor into the distance function in different ways. 
  Stated another way, swapping the time axis with a space axis is *not* 
in the symmetries of Minkowski spacetime, but swapping any of the space 
axes *is*, and this there's something "different" about the time axis. 
Otherwise why bother saying (3+1)D instead of just 4D?

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Kevin Wampler wrote:
> 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.



-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> Kevin Wampler wrote:
>> 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.
> 
>  my head asplode


It was just a fancy way of saying that there's a sense in which the 
spacetime isn't "curved"; which I probably should have just said in the 
first place since the way I phrased it should really be made more 
precise to be accurate anyway.

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On 20-10-2009 12:41, Invisible wrote:
> Saul Luizaga wrote:
> 
>> I don't know if some day would be possible but would be great to visit 
>> a 4D world and meet 4D people :-D
> 
> I should point out that a rotation in 4D can leave your body in mirror 
> image. If you go to 4D-land, turn around the "wrong" way, and then come 
> back, your body will be inverted.
> 
> Why would you care? Well... certain biological molecules are chiral, 
> so... good luck assimilating your food. :-P
> 

Book reference: 'Doorways in the sand' Roger Zelazny

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On 21-10-2009 19:00, clipka wrote:
> 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?

Yes there is, but you have to keep in mind that the points here are 4 
dimensional points. i.e we are talking about two events that take place 
in two places that happen closer in time than the light needs to go from 
one place to the other.

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"Saul Luizaga" <sau### [at] netscapenet> wrote in message
news:4adf1059$1@news.povray.org...

> ANd nobody knows if we were or not 2D beings.

How do you know that?

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somebody wrote:
> How do you know that?
nobody has ever proved it.

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"Saul Luizaga" <sau### [at] netscapenet> wrote in message
news:4adfcffe@news.povray.org...
> somebody wrote:

> > How do you know that?

> nobody has ever proved it.

How do you know that?

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Kevin Wampler schrieb:

>>>> Well, AFAIK there's actually no fundamental reason to apply 
>>>> different "measuring tapes" to time and space: 
>>>
> Is you point that in space time there *is* a well-defined notion of 
> distance which unifies both the spatial and temporal aspects, and thus 
> we don't really need to use one set of units for space and another set 
> for time?  Your comment about light-seconds would make more sense in 
> this context.  If this is the case perhaps I've been misunderstanding 
> your point form the beginning, since I'd surely agree with this.

Well, yes, I guess that's one way to express what I said: We don't need 
different "measuring tapes" for time and space. How distance is measured 
"diagonally" in spacetime is another story.

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Kevin Wampler schrieb:
> clipka wrote:
>> 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.
> 
> In this context a lightlike geodesic refers to a path in spacetime which 
>  light would follow, and I meant to imply that defining the distance 
> between points which could only be connected by going *slower* than 
> light would also be hard to define uniformly.

Maybe you'd first need to define "point": Are you talking about a point 
in 3D space which extends along time - i.e. a line in 4D space - or an 
actual 4D point that only exists at a certain time T?

In the former case, this is actually a non-issue: Either you can find 
(true) points on that line that /can/ be reached at /exactly/ light 
speed, or you cannot reach that other line at all.


In the latter case, you can define the spacelike distance between the 
points as the minimum difference between the spacelike component of two 
intersecting lightlike geodesics running through each of the points 
respectively.

Or, to eliminate the problem of "far-away" spacetime being curved, you 
can have a look at all possible sequences of points that can be 
connected with lightlike geodesics, compute the vector sum of the 
spacelike components, and define the spacelike distance as the minimum 
length of any such vector sum.

In the end it will boil down to defining the spacelike distance as the 
length of the spacelike component of the shortest "direct" geodesic 
between the points.

(Note that of course this leaves the definition of distance subject to 
your frame of reference, as it defines the orientation of the spacelike 
components of spacetime, but that's a known effect in relativistic physics.)

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Kevin Wampler schrieb:

> In light of one of your other comments I think I now understand what 
> point you're making: we can treat space-time distances in a common set 
> of units by treating distances as times via the speed of light.  In 
> which case I'd agree, and it fact the necessity for this clearly falls 
> out of being able to have a single number represent the distance at all.

Yes...

> I still don't see how it's relevant for my initial comment that the 
> space and time coordinates are treated differently though, since they 
> are most definitely factor into the distance function in different ways. 
>  Stated another way, swapping the time axis with a space axis is *not* 
> in the symmetries of Minkowski spacetime, but swapping any of the space 
> axes *is*, and this there's something "different" about the time axis. 
> Otherwise why bother saying (3+1)D instead of just 4D?

Well, your original statememt was that "distances are measured 
differently in time than in space", not "distances are computed 
differently in spacetime than in space", so I took this as talking about 
"measuring tapes", not the math to combine space and time components of 
distance into spacetime distance.

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somebody schrieb:

>>> How do you know that?
> 
>> nobody has ever proved it.
> 
> How do you know that?

"Can you hear me, bomb?" :-)

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somebody wrote:
> "Saul Luizaga" <sau### [at] netscapenet> wrote in message
> news:4adfcffe@news.povray.org...
>> somebody wrote:
> 
>>> How do you know that?
> 
>> nobody has ever proved it.
> 
> How do you know that?

Oh you really think that 2D only beings discovery might have gone 
unnoticed? I don't think so, I think is perfectly safe to say that if 
that kind of world exist it has not been discovered, or do you know 
otherwise?

I think your question is pointless, anyone would had pointed out that 
the 2D beings actually do exist, such a news would be common knowledge 
by now. Otherwise people wouldn't be wondering:

http://www.unexplained-mysteries.com/forum/index.php?showtopic=163191
http://www.physicsforums.com/archive/index.php/t-185785.html
http://teamikaria.com/4dforum/viewtopic.php?f=5&t=1103

How do I know that? looks like you are the only one that it doesn't.

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"Saul Luizaga" <sau### [at] netscapenet> wrote in message
news:4ae01d8b$1@news.povray.org...
> somebody wrote:
> > "Saul Luizaga" <sau### [at] netscapenet> wrote in message
> > news:4adfcffe@news.povray.org...
> >> somebody wrote:
> >
> >>> How do you know that?
> >
> >> nobody has ever proved it.
> >
> > How do you know that?
>
> Oh you really think that 2D only beings discovery might have gone
> unnoticed?

No, but that's not what you said. You said "ANd nobody knows if we were or
not 2D beings." Well, if lack of disclosure of discovery is proof (faulty
reasoning), well, everybody knows that we were not 2D beings (probably a
correct statement, at least more sensible than its negative).

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clipka wrote:
> Maybe you'd first need to define "point": Are you talking about a point 
> in 3D space which extends along time - i.e. a line in 4D space - or an 
> actual 4D point that only exists at a certain time T?

In all cases I'm referring to a point as a 4D point in spacetime since 
there's no intrinsically unique way to extend a 3D point along time.


> (Note that of course this leaves the definition of distance subject to 
> your frame of reference, as it defines the orientation of the spacelike 
> components of spacetime, but that's a known effect in relativistic 
> physics.)

If you fix a reference frame there's certainly plenty you can do to 
define spacetime distances in a consistent way.  If you want to 
consistently measure distances independent of the reference frame, then 
you're pretty much stuck with what the Minkowski metric tells you.  Of 
course this metric gives you a single value for the distance (not 
independent time and space distances) and thus can't really be used to 
measure pure-spatial distances.

Not that this is necessarily inconsistent with what you said, I just 
thought I'd point out (in case you weren't already fully aware) that if 
you're going to measure spacetime distances independent of the reference 
frame then the way to do it is to use the Minkowski distance and forget 
about trying to determine what the "spatial" distance is at all.

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somebody wrote:
> "Saul Luizaga" <sau### [at] netscapenet> wrote in message
> news:4ae01d8b$1@news.povray.org...
>> somebody wrote:
>>> "Saul Luizaga" <sau### [at] netscapenet> wrote in message
>>> news:4adfcffe@news.povray.org...
>>>> somebody wrote:
>>>>> How do you know that?
>>>> nobody has ever proved it.
>>> How do you know that?
>> Oh you really think that 2D only beings discovery might have gone
>> unnoticed?
> 
> No, but that's not what you said. You said "ANd nobody knows if we were or
> not 2D beings." Well, if lack of disclosure of discovery is proof (faulty
> reasoning),

I deed, because in all fairness we don't know this for sure, I stand 
correct.

> well, everybody knows that we were not 2D beings (probably a
> correct statement, at least more sensible than its negative).

sensible? I don't think so since it has the same exact meaning.

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"Saul Luizaga" <sau### [at] netscapenet> wrote in message
news:4ae0df6a$1@news.povray.org...
> somebody wrote:

> > well, everybody knows that we were not 2D beings (probably a
> > correct statement, at least more sensible than its negative).

> sensible? I don't think so since it has the same exact meaning.

If a statement and its negative have the exact same meaning, they are both
nonsense :P. But, I'll compromise: I'm pretty sure many, many more people
know that we were not 2D beings than those who know that we were (whatever
either statement means).

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somebody wrote:
> If a statement and its negative have the exact same meaning, they are both
> nonsense :P. But, I'll compromise: I'm pretty sure many, many more people
> know that we were not 2D beings than those who know that we were (whatever
> either statement means).

Yeah this is more proper, thanks for understanding, but anyway not 
necessarily true.

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