4D translation to/from null-space.
Author
This isn`t strictly a math question, but its rather mathematical.
I`m working on a 4D engine right now, and my end goal is to have a demo where you can do null-space jumps.
Now, we know that in Euclid, d=sqrt(x^2+y^2+z^2+q^2)
So if you are movil along a vector in 3d, at point P move in the q component, and move back onto the original vector at point Q, you will have moved a greater distance relative to you in Euclid than if you had kept on the original vector.
This defeats the purpose of having a null-space jump, which is to get somewhere very fast.
So, I`m going to try and work out a relationship between null-space and real-space.
I have a web page with a 143kb pic on it illustrating this.
http://thefivelions.bugle4d/distance/index.html
Your translation point into null-space is P.
Your translation point out of null-space is Q.
Your distance traveled in null-space must be less than the distance traveled in realspace from those 2 coords.
Find a relationship that hold true in interval [0, inf] between each point in null-space and each point in real-space.
Currently, I think that the relationship should probably be non-linear; furthermore, I think that it should take longer for small jumps than for large.
So given that my func that describes distance/time in realspace is d(t)=t
And my null-space distance/time describing func is d(t)=t^2(t-1)+.2
At a given time t,
Each ordered pair has the form
(t,d(t))
So:
(t,t)
(t,t^2(t-1)+.2)
So in my null-space point structure, I need to have a counter for distance, which will determine real d
Lets try it out.
Given that we have a (x,y,z,q) tetra.
When q changes from 0, we move into null-space.
When it moves back to 0, we move into real-space.
Moving around in null-space is moving around in 4-space.
To translate back into 3-space with q=0,
we use this formula
x(d)=d^2(d-1)+.2
y(d)=d^2(d-1)+.2
z(d)=d^2(d-1)+.2
we set q=0, and we make sure no accidental translations occur with our code in case q happens to hit 0 when we are in null-space.
How does this look, people ?
Bugle4d
First of all i am confused by your use of the term null space without defining it. Null space is usually the "kernel" or space that gets mapped to zero of some operator.
But I am guessing your "null-space" jump is a to move between two points in a space via short-cut through a higher dimensional space. For example on a 2d-surface of a sphere you would need to move a distance 2*PI*r to get to an antipodal point, but could take a shortcut through the euclidean space the sphere is embedded in where you would travel a distance 2*r.
The relationship between "null" space and "real" space isn''t something you''d really work out, it just depends on spaces you''re using and the metrics defined on them. I don''t understand what you are trying to do with this "distance/time" function
But I am guessing your "null-space" jump is a to move between two points in a space via short-cut through a higher dimensional space. For example on a 2d-surface of a sphere you would need to move a distance 2*PI*r to get to an antipodal point, but could take a shortcut through the euclidean space the sphere is embedded in where you would travel a distance 2*r.
The relationship between "null" space and "real" space isn''t something you''d really work out, it just depends on spaces you''re using and the metrics defined on them. I don''t understand what you are trying to do with this "distance/time" function
Messy!
Well, you did ask!
Why is your realspace displacement given by d(t)=t?
You clearly state that d2 = x2 + y2 + z2 + q2
which is true for a 4-dimensional Euclidean space. For 3-d space, q=0 for all t.
Did you mean your realspace velocity was given by d(t)=t? In which case, your notation is very awkward.
Finally, I suspect that the mapping you seek does not exist between Euclidean manifolds. In fact, I''m almost sure it doesn''t since to obtain the affect you seek, the transformation would not be affine.
Of course, since I''m only human, I could be wrong.
Good luck,
Timkin
Well, you did ask!
Why is your realspace displacement given by d(t)=t?
You clearly state that d2 = x2 + y2 + z2 + q2
which is true for a 4-dimensional Euclidean space. For 3-d space, q=0 for all t.
Did you mean your realspace velocity was given by d(t)=t? In which case, your notation is very awkward.
Finally, I suspect that the mapping you seek does not exist between Euclidean manifolds. In fact, I''m almost sure it doesn''t since to obtain the affect you seek, the transformation would not be affine.
Of course, since I''m only human, I could be wrong.
Good luck,
Timkin
Author
Yes, my notation is fairly awkward.
I need to clean it up some.
d(t) was distance(time).
No, I was not aware that null-space has a defined meaning. Sorry.:/
Essentially, I think that mapping I seek would be between a 3d Euclid space(real-space), and a 4d euclid space(null-space), but all points in real-space are further apart then all points in null space.
Looking at Mathworld`s definition of affine mapping, I don`t think that I want an affine mapping.
As my pic shows, I think a non-linear mapping would be best.
-edit-
And if my stuff is messy, then its messy and you can say so.
Bugle4d
[edited by - Vlion on March 19, 2002 11:39:54 AM]
I need to clean it up some.
d(t) was distance(time).
No, I was not aware that null-space has a defined meaning. Sorry.:/
Essentially, I think that mapping I seek would be between a 3d Euclid space(real-space), and a 4d euclid space(null-space), but all points in real-space are further apart then all points in null space.
Looking at Mathworld`s definition of affine mapping, I don`t think that I want an affine mapping.
As my pic shows, I think a non-linear mapping would be best.
-edit-
And if my stuff is messy, then its messy and you can say so.
Bugle4d
[edited by - Vlion on March 19, 2002 11:39:54 AM]
I''m not sure this is what you want, but you could try working in 4D with a non-Euclidian metric. E.g. the metric for 4D space-time, or Minowski space, is
d = sqrt (t^2 - x^2 - y^2 - z^2)
[using units where the speed of light c is 1]
d measures the proper time of a journey, e.g. the time experienced by an object (such as a clock) as it completes the journey. The proper time is only the same as t when x, y and z are unchanged: in laymans terms objects in motion experience time dilation, and the faster they move the greater the effect.
What''s interesting is a roundabout trip can be shorter than a straight one. E.g. travelling from
{0, 0, 0, 0} to {1000, 0, 0, 1}
gives d = sqrt(999999), or about 1000. This makes sense as going at 1/1000 the speed of light gives minimal time dilation.
But doing the journey in two parts
{0, 0, 0, 0} to {500, 499, 0, 0.5}
then
{500, 499, 0, 0.5} to {1000, 0, 0, 1}
gives d = sqrt (500^2 - 499^2 - 0.5^2) = sqrt(999.25) = 31.6 for each part of the journey, or a total d of about 63. So going this way the traveller''s own clock runs at only 6% the speed of the direct route, a much more dramatic time dilation.
You cannot travel faster than light this way (try it and you get square roots of negative numbers) but it does give you a way of making journeys seem faster, and has the benefit of being mathematically very solid as it''s just a description of how our universe works, at least on a small scale.
There are many other non-Euclidian geometries and metrics, it''s hopefully just a case of finding one which matches what you have in mind.
d = sqrt (t^2 - x^2 - y^2 - z^2)
[using units where the speed of light c is 1]
d measures the proper time of a journey, e.g. the time experienced by an object (such as a clock) as it completes the journey. The proper time is only the same as t when x, y and z are unchanged: in laymans terms objects in motion experience time dilation, and the faster they move the greater the effect.
What''s interesting is a roundabout trip can be shorter than a straight one. E.g. travelling from
{0, 0, 0, 0} to {1000, 0, 0, 1}
gives d = sqrt(999999), or about 1000. This makes sense as going at 1/1000 the speed of light gives minimal time dilation.
But doing the journey in two parts
{0, 0, 0, 0} to {500, 499, 0, 0.5}
then
{500, 499, 0, 0.5} to {1000, 0, 0, 1}
gives d = sqrt (500^2 - 499^2 - 0.5^2) = sqrt(999.25) = 31.6 for each part of the journey, or a total d of about 63. So going this way the traveller''s own clock runs at only 6% the speed of the direct route, a much more dramatic time dilation.
You cannot travel faster than light this way (try it and you get square roots of negative numbers) but it does give you a way of making journeys seem faster, and has the benefit of being mathematically very solid as it''s just a description of how our universe works, at least on a small scale.
There are many other non-Euclidian geometries and metrics, it''s hopefully just a case of finding one which matches what you have in mind.
quote:Original post by Vlion
Essentially, I think that mapping I seek would be between a 3d Euclid space(real-space), and a 4d euclid space(null-space), but all points in real-space are further apart then all points in null space.
Not possible. Your ''real-space'' as you call it must, by definition, be a sub-manifold of your 4d space. Since you want your 4-d space to be Euclidean and you want the sub-manifold to be Euclidean, then you cannot define a transformation between the two such that distances on the Euclidean sub-manifold are longer than distances on the Euclidean manifold. It goes against the whole definition of Euclidean space.
Indeed, the only way to produce wormholes, which is really what you are trying to do, is to use curved space. I highly recommend you consider this as an alternative...
... and please use the correct name; Euclidean, not Euclid! As I said, Euclid was the mathematician! It only confuses other people reading this forum.
quote:Original post by Vlion
As my pic shows, I think a non-linear mapping would be best.
Your page doesn''t display... I get a page not found error... could you check the URL please.
Cheers,
Timkin
Author
apologies.
This illustrates the distance I want to travel in null-space vs real-space.
http://thefivelions.tripod.com/bugle4d/distance/
will post later gotta run
Bugle4d
This illustrates the distance I want to travel in null-space vs real-space.
http://thefivelions.tripod.com/bugle4d/distance/
will post later gotta run
Bugle4d
Okay, makes sense now...
(BTW: Check your HTML source file... you've used a local refernce to the image rather than a global reference... browser cannot resolve your local file reference... thankfully it was pretty easy to determine the global reference...)
I should have picked this up earlier... my apologies for not doing so...
... you've already answered your own question...
Your 4th dimension is not a spatial one but a temporal one. In fact, what you actually have is two 4-d spacetimes with different temporal scales, but identical spatial scales. You have also provided the mapping between the two!
You write that y = mt and y' = at2 + bt (here I'm using y' to denote distance in your real space and y to denote distance in your other space).
Clearly, t = y/m and so
y' = a(y/m)2 + b(y/m)
This is the mapping you asked for in your original post.
Timkin
[edited by - Timkin on March 20, 2002 1:39:54 AM]
(BTW: Check your HTML source file... you've used a local refernce to the image rather than a global reference... browser cannot resolve your local file reference... thankfully it was pretty easy to determine the global reference...)
I should have picked this up earlier... my apologies for not doing so...
... you've already answered your own question...
Your 4th dimension is not a spatial one but a temporal one. In fact, what you actually have is two 4-d spacetimes with different temporal scales, but identical spatial scales. You have also provided the mapping between the two!
You write that y = mt and y' = at2 + bt (here I'm using y' to denote distance in your real space and y to denote distance in your other space).
Clearly, t = y/m and so
y' = a(y/m)2 + b(y/m)
This is the mapping you asked for in your original post.
Timkin
[edited by - Timkin on March 20, 2002 1:39:54 AM]
Author
Ahh....
So I pull time out of my real-space and put that equation into my null-space equation, and that will give me how much time i need to travel to go a certain distance.
So in real-space I select the point I want to move to, i come up with the point in null-space I need to go to at a given speed, then I proceed to translate to nulls-space, go to that point at the given speed, then translate to real-space...
Bugle4d
So I pull time out of my real-space and put that equation into my null-space equation, and that will give me how much time i need to travel to go a certain distance.
So in real-space I select the point I want to move to, i come up with the point in null-space I need to go to at a given speed, then I proceed to translate to nulls-space, go to that point at the given speed, then translate to real-space...
Bugle4d
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