cliche

Whilst many don't subscribe to the idea of time as being equivalent to a spatial dimension, it's something that got me thinking the other day.

If I have in my possession a square of 1metre side (yes, the 2-dimensional object and not some kind of square frame etc) - how long does it have to be in existence to be a cube?

My initial guess is that the speed of light in a vacuum defines the relationship between distance and time - so 1/299792458 of a second. Does anyone have any other theories? Could it be the delightfully SI answer of one second? I'd be interested to hear.

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RespectThat

2d is a 3d illusion

Slavakion

I'd think that the square would "cubize" after the smallest amount of time ever. Like 1/infinity seconds. After that, there would be a (small) dimension added to it.

And 2D is only an illusion because everything in our world exists as 3D.

cliche

Slavakion - I'm not sure 1/infinity really exists (see other topics) unless you're talking about the Planck time (about 10^-43 seconds I think). However, I'm not talking about the time to become 3d, but to become 3d and a cube. If the small dimension you talk about adding to it isn't equivalent to 1 metre, it's not a cube. Whatever the answer is, a 2metre square would exist for twice as long etc...

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antisuck

Maybe I don't get the question, but it seems to me that squares and cubes are nothing more than mathmatical models, and nothing will get one to change into the other.

Now if you're trying to visualize time as another dimension so as to draw it on a piece of paper, then I suppose one could say that after a certain amount of time the length along the "time axis" might be conveniently expressed as a segment a meter long, and that length of time is your answer. But that seems to me such an "abstraction of an abstraction" as to lose all meaning. And of course it depends on how you scale your graph, and what the actual "speed of time" is.

Maybe that's a different way of phrasing your original question: "what is the speed of time?" In that case, the speed of light is an attractive answer but I have no idea how one would come to such a determination. And even that might be too abstract a question to try and solve using conversational English as opposed to lots and lots of very hard math.

Slavakion

Oh, whoops. I misread it as when it became 3D.

mathmo

Whilst I don't really like considering time as a spatial dimension, I would hazard that the answer will be the delightfully SI answer of 1 second.

The dimensions of the cube are 1x1x1, a unit for each dimension, so 1 metre by 1 metre by 1 second makes sense to me. This is of course assuming you measure time in increments of 1 second...

Also, if you were to plot this in 3D, using time as a 3rd spatial dimension, it should look like a cube, so it would have to be one unit in the time dimension.

zen_tom

Is there any other way to describe distance, except in terms of time? Perhaps you are onto something cliche.

John Henry

I think you're probably right, cliche. Maybe we should consider the local speed of light, ie taking the material of the cube as the medium. However, since the square is immaterial, we can safely model it as being made of vacuum.

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adysav

How do you go about constructing a truly 2 dimensional object in the first place.

molloby

Time and distance are 2 different things, yes they are both dimensions but the size of one "unit" of time isn't linked to the size of one "unit" of distance. Any link that you ascribe will be entirely arbitrary.

If 1m*1m*1s is a cube then why isn't 1ft*1ft*1year?

Assigning the speed of light is just as arbitrary, it only applies in this geometry and is not a function of a normal euclidean geometry.

It might just be best to accept that the speed of time is 1 second per second and it is a fourth dimension in the sense that it is a fourth number that you can assign to any point in "spacetime".

cliche

I guess you take a cube and then rotate it 90 degrees through time...

molloby - I guess that's the point. Is the speed of light just arbitrary? As mentioned at the start, the whole question is predicated on the idea of time as being exactly equivalent to a spatial dimension so I'm not keen to take your 'it's just a fourth number' argument even though that's what I believe myself...

__________________
I can't understand why people are frightened of new ideas. I'm frightened of the old ones. -- John Cage (1912 - 1992)

fckm

Not quite. Time is simply a mathematical degree of freedom. In relativistic cases, energy, momentum, and mass can be distributed among the four degrees of freedom, 3 spatial and 1 time, by taking some tensor transformations. While the temporal spatial degree of freedom is orthogonal to the spatial, you can think of the time basis vector as being imaginary (while the three spatial basis vectors are pure real). To make the units equivilent, you always measure time in units of ct. time and space are not exactly equivilent.
A "cube" is in 4-space is not a physical object. It's not the same thing as thinking about a cube in 3-D. It doesn't suddenly gain special properties because it has "lenght in time".

cliche

Hence the 'assuming that they were'... section at the top. I think you may be taking the question a little literally. I'm not sure that science can, will, or is prepared to provide an answer to the question but the question exists. Like the synaesthetes who see sounds as colours and shapes I'm wondering what unit of time is equivalent to a metre (or any other unit for that matter). It's always struck me that we use time and distance in a very interchangeable manner: in the highbrow concept of spacetime, the standard confusion as to why a lightyear is a measure of distance, and the everyday 'he lives about ten minutes down the road'.

So I guess another answer may appeal to our culture. In early times, a metre would be about a second - that being the pace at which I might walk to the next village. The addition of horseriding to our repertoire might bring this closer to 1/15s. The invention of the telegraph brings us closer to the 1/299792458s mentioned earlier. And perhaps research by Enders and Nimtz has already started to take us even lower (approx 1/1400000000s in one experiment).

__________________
I can't understand why people are frightened of new ideas. I'm frightened of the old ones. -- John Cage (1912 - 1992)

stingc

Actually, science does give an unambiguous relation between space and time. However, it is clearly not true that space and time are exactly the same thing. By rejecting this, you have left the realm of physics, and are talking about pure mathematics. There, you can simply write down whatever (non-contradictory) set of axioms you feel like to get the answer you want.

I don't think that's the answer you wanted, though. Let's go back to physics: While it was always possible to specify events by tacking on a time to the position coordinates, relativity taught us that this is essential. In everyday experience, time is absolute. Once an origin is set, everyone agrees on what 5 seconds into the future means. However, it was eventually discovered that this is not correct. Different observers who are naive to relativity will measure different times. This is not normally noticed because the conditions under which the times will differ significantly are fairly extreme. In any case, the fact remains that time is not absolute in this sense.

However, this still does not justify time a 4th dimension. The reason that this is done is that the observers who disagree on the passage of time will also disagree on lengths. In fact, comparing the (classes of) natural coordinate systems for different observers will show that they are related by something that is essentially a rotation of the temporal and spatial axes. In Newtonian physics, different observers could disagree on their spatial coordinates by a rotation (plus translation), but now we see that different observers must compare their coordinates through a more general sort of rotation that unavoidably involves time. It is therefore very natural to associate time as a 4th coordinate (and not simply an absolute parameter).

Now I'm getting to your question. Rotating spatial axes is easy to imagine because they have the same units. In rotating a time axis into a spatial one, there needs to be conversion factor. That conversion factor happens to be the speed of light, c. It is best to think that light moves at the speed c because of the structure of spacetime, rather than thinking that spacetime is the way it is because of light.

If you don't like all of that, there is an equivalent statement that is more compact. In Newtonian physics, the time difference and the distance between two events are both invariants. They are the same to everyone. Define L^2=x^2+y^2+z^2.

It has since been discovered that L can be measured differently by different people. The quantity that has been shown to be well-defined is L^2=x^2+y^2+z^2-c^2t^2. Again, the speed of light shows up as the natural conversion factor between space and time.

Thermopyle

Dont mix time and space. As written above, time is a mathematical "dimension" and has nothing to do with the other space dimensions. A 1m*1m*1s is not a cube. It's a square that existed for one second.

zen_tom

[Thermopyle] Since the question has been asked in such a way so as to cross the boundaries of both physics and mathematics - so must the answers.

A cube is both a mathematical concept and something that can exist in the real world. Space and time are inexorably linked, they may not have the same properties, and they certainly don't act in the same way, but it is interesting to think along these lines.

However, to properly answer the question, we all need to agree on what a cube is, what space is, and what time is. And until we've got that covered, we really don't have a chance at providing a mutually acceptable answer.