Tesseract with Time as Fourth Dimension

[Charlie G]

Hi forum, I was having some difficulty in understanding how time as a fourth dimension allows a cube of three spatial dimensions to obtain the properties of a tesseract.

Suppose we allow a cube to exist for some interval of time. At the start such a cube would have eight vertices as well as at the end of the time interval; sixteen vertices in total, just as the tesseract does.

My difficulty lies in understanding how the properties of thirty-two edges and twenty-four faces come about. I continue to get much smaller numbers when I try to use copy the method for vertices, I appear to be missing some subtle fact that allows these properties to become manifest.

Any help is well appreciated,

Thank-you

 

[DaveC426913]

I am not sure if this is the crux of your confusion:

A tesseract is an object in four spatial dimensions.

There is ambiguity when people talk about "4 dimensions" if they don't specify whether they mean 4 spatial (common) - or 3 spatials and 1 time (also common).

We definitely know that the 3+1 exists; we do not have evidence of a 4th spatial dimension beyond hypothetical.

 

[Charlie G]

Ah, man. I've been thinking about it all day haha.

Thanks for clearing that up for me Dave, so that I don't spend any longer thinking about it.

 

[Martin Rattigan]

Each vertex of the starting cube traces an edge giving 8 edges parallel to the time axis. Together with the 12 edges at the start and 12 at the end this gives 32 edges in total.

Each edge of the starting cube traces a (2 dimensional) face giving 12 such faces parallel to the time axis. Together with 6 such faces at the start and 6 at the end this gives 24 such faces in total.

Similarly the 6 faces of the starting cube each trace out a 3 dimensional (2 spatial + 1 temporal) cube parallel to the time axis which together with the start and end cubes give 8 cubic "hyperfaces" in total.

 

[Martin Rattigan]

Next question is, if you take special relativity into account, could the start and end positions of the far side of the cube be the opposite way about from the near side?

 

[Rasalhague]

Next question is, if you take special relativity into account, could the start and end positions of the far side of the cube be the opposite way about from the near side?

Wouldn't that require a vertex to travel further than one space unit in one time unit, i.e. faster than c?

 

[Martin Rattigan]

I was rather thinking about the vertices (which trace separate event lines) being far enough apart not to be in each other's absolute past or future. Presumably then the start and end could switch over for different parts of the cube depending on the observer.

 

[Rasalhague]

I was rather thinking about the vertices (which trace separate event lines) being far enough apart not to be in each other's absolute past or future.

Yes, that's what I was thinking, for the vertices of a given purely spacelike cube, i.e. the start cube and the end cube. (But a "cube" with any timelike edges would have vertices that were in the absolute past of future of others of its vertices.) In fact, for this object to merit the (frame-dependent) title of tesseract, I was thinking of the 8 vertices at the start as being all simultaneous with each other in the inertial reference frame where the tesseract is defined, and the 8 vertices at the end as being simultaneous with each other in this same frame, the edges of each cube, say, one light year long, and each vertex of the start cube one year's time from the corresponding vertex of the end cube. In flat spacetime, in the frame where the tesseract is defined, I don't think there can be any timelike trajectories (worldlines) connecting any vertex of the start cube with any vertex of the end cube except the one it's paired with, only lightlike trajectories connecting it to the three nearest vertices, the ones that are exactly one light year away. If there's no timelike path between two events in one inertial frame, there can't be in any other inertial frame; we can't make a noncausal path causal, and lightlike paths stay lightlike. That was my reasoning, anyway. But I think I maybe misinterpreted the question...

Presumably then the start and end could switch over for different parts of the cube depending on the observer.

If we're allowed to change frames during the history of the cube, the "observer" could just rotate 180 degrees, in that way distorting the tesseract by giving it a twist, and reversing the orientation of the end cube. That's easy way to do it! But could we rotate the coordinates of the cube this far by a combination of boosts, not all in the same direction? I need to read up more on Thomas rotation. I don't know enough to work it out yet. Is that what you had in mind?

(Obviously a start point for one inertial observer would be a start point for all, however its spatial location changed, and an end point for one an end point for all.)

 

[Martin Rattigan]

I think you were right in post 6. I was posting more to see what the experts might say about it than from some clear idea of what I wanted.

 

[SonyAD]

Suppose we allow a cube to exist for some interval of time. At the start such a cube would have eight vertices as well as at the end of the time interval; sixteen vertices in total, just as the tesseract does.

I don't think you can take time as a spatial dimension. I think you would have 8 vertices · number of state samples, not 16 vertices, as you would be sampling the same cube over all the time slices in the duration. You'd have a chain of squares instead of a hypercube:


[PLAIN]http://img28.imageshack.us/img28/2301/dimensions.png


To get a hypercube you take one cube, duplicate it and link the homologous vertices. You'll get different shapes in higher dimensions depending on how many vertices are kept in common during the dimensional extensions.

For instance, two triangles can define a tetrahedron by having the 2 triangles share an edge, a square pyramid by having them share a vertex (hopefully, the base vertices are coplanar) or a prism.

Two lines can define a triangle or a square in 2D, depending whether they share an endpoint or not.

 

[Rasalhague]

I don't think you can take time as a spatial dimension. I think you would have 8 vertices • number of state samples, not 16 vertices, as you would be sampling the same cube over all the time slices in the duration. You'd have a chain of squares instead of a hypercube:

The construction Charlie described in #1, and Martin in #4, has 16 vertices in the following sense. Points in spacetime are events. There are the eight events defined by the locations of an ordinary, purely spacelike 3-dimensional cube's corners at the start, plus 8 events defined by the locations of that cube's corners at the end. It seems like the most natural way to try to define a spacetime analogue to the tesseract. Of course, the true tesseract lives in Euclidean 4-space, \bbmath{E}^4, whereas this thing we've made is a denizen of Minkowski spacetime, \mathbb{E}^{1,3} (or \mathbb{E}^{3,1} depending on your sign preference), with all the peculiarities that entails. Good diagrams, by the way!

 

[SonyAD]

But if you move the cube at any instant during the time interval it is allowed to exist you wouldn't have a hypercube anymore.

Also, if you take time to be ultimately discrete, you'll have a string of cubes instead of a hypercube, would you not?

 

[Rasalhague]

We don't exactly have a hypercube anyway ;-)

If we "moved the cube" part way through its life, wouldn't that be analogous to putting bends or corners into some of the edges of a cube of some dimension in Euclidean space (and thus changing it from a cube into some other shape)? I think the idea of our shape is that its timelike edges are straight worldines, i.e. worldlines of a possible particle with constant velocity in an inertial reference frame frame, and each vertex event is connected to its neighbouring vertices by a spacetime separation of magnitude 1 (in units where c is 1).

Whether time is really discrete or not, the mathematical model we're talking about here, Minkowski space, like Euclidean space, is continuous.

 

[SonyAD]

By moving the cube you'd basically be adding vertices and edges even if you don't concur that space and time are discrete.

Intuitively speaking, no dimension can be continuous. That would mean you have infinite complexity in a finite space or duration, would it not? I think that's what the Greeks had in mind when the concept of atoms was first pondered: quantised space. I don't think for a second they were thinking of particles of matter but of the issue of the discreteness or continuity of space itself. Of course, maybe they didn't make a distinction between space and matter. Maybe there isn't one, at the fundamental level, heh?

Zeno's paradoxes also played to this philosophical issue though most people today hastedly mistake or misconstrue the conclusions, if any, as being supportive of the idea of continuous space or they neglect time entirely.

The Greeks would have been thinking of all these things because Pitagora's theorem and Archimede's approximation of Pi forced them to.

How the discreteness of space and time at the fundamental level translate into the world we see at the macroscopic level, where rectilinear uniform movement stays so, is anybody's guess but this might actually be the underlying cause of much of the funkiness of quantum mechanics, probability clouds, the "popping" in and out of existence of subatomic particles, Heisenberg's uncertainty principle, etc.

Myself, I don't see irrational numbers as actual numbers. I see them as shorthand for algorithms or ideals.

That's my Weltanschauung, anyway. Maybe it's because I think in pixels and voxels. :)

Food for thought: you can move a point across the screen by successively changing the colour of the various pixels that come into contact with the "world line" of the point. Now, you could do that abruptly, or you could do that gradually. Gradually bleeding the colour out of one pixel and "pouring" it into the adjacent pixels that come into contact with the point's world line next.

Maybe this mirrors some aspects of nature at a fundamental level. Unfortunately, it raises the need for an underlying level of existence more complex/fundamental in order to "compute" the world line in the first place and store the path as the point jumps from pixel/voxel to pixel/voxel aproximating it.

So there's probably some other mechanism at work. We just need to figure out what it is that enables the pixel to approximate the world line of the point without actually computing the world line.