# length,area,volume then ?

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If 1 dimension has length.

2 dimensions has area.

3 dimensions has volume.

what does 4 dimensions have?

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Higher-dimensional volumes are also called "volume", usually. Perhaps also 4D volume or 4-volume. But you can also call it "xurbit", if that makes you feel better.

EDIT: It's amazing that Google returns 2000 hits for a term I just made up.

Edited by timo

Or hypervolume.

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"Hypervolume" is good, or you could call it "Superspace". Or "Moreroom". How would it be expressed in Mandarin Chinese?

Probably as "Roomroom", but is that as expressive, no wonder the Long March rockets haven't yet reached Mars!

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Or hypervolume.

Right. Forgot about that. Makes one feel better due to it Sci-Fi touch and is actually used.

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Or hypervolume.

According to Wikipedia you are right and it gives the formulae:- V = 1/2 pi^2 R^4.

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As I hope has been sufficiently pointed out, names are just names. In relativity it is pretty common to refer to a 3D volume + time as a "four-volume". A 4D spacial volume may also be referred to as a "hypervolume." But just as length is a 1D volume and area is a 2D volume, higher dimensional volumes are most practically referred to as "volumes." Making distinctions between length, area and volume is useful, but when you add additional dimensions the distinction becomes less practical.

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If 1 dimension has length.

2 dimensions has area.

3 dimensions has volume.

what does 4 dimensions have?

Zero dimension is a point. (0D)

A moving point is represented by a line: you have length (1D)

A moving line is represented by a surface (2D)

A moving surface is represented by a volume (3D)

A moving volume is (difficult to represent) 4D

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You can also go like this:

begin with a point again (0D)

move the point (1D)

Accelerate the point; that is the multiplication of the point by square time= 2D (a surface)

accelerate the surface (3D)

accelerate the volume (4D)

Accelerate the 4D = 5D

and so on.

There is no end to acceleration. Have a look here.

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Zero dimension is a point. (0D)

A moving point is represented by a line: you have length (1D)

A moving line is represented by a surface (2D)

A moving surface is represented by a volume (3D)

A moving volume is (difficult to represent) 4D

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Is a moving volume a temporal displacement (4D)?

Edited by derek w
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A moving volume is 4D.

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You need 4 coordinates to fully describe a point on a moving volume.

You may note that my previous post about acceleration is not so obvious.

When you square meters, you obtain a surface because meters in one direction can be orthogonal to meters in some other direction.

On the other hand, when you square time (like in acceleration) we use to consider that there exist only one "direction

of time". So square seconds are usually considered as regular seconds squeezed (or extended) along the time line.

There is no time perpendicular to some other time because we consider that there is only one time. That is common sense.

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As a consequence, on an accelerating volume you need only 4 coordinates to fully describe a point. Not 5 coordinates.

Example;

Take a box (a 3D volume)

Make the box slide. You have a 3D volume sliding along a time axis, that is 4D

Now, while the box is sliding, push from the side so that the box constantly changes direction. In this case you can draw an orthogonal axis of time. This second axis inserts seconds that multiply the seconds of the first axis, and you obtain square seconds : Acceleration.

And indeed a change in direction corresponds to acceleration.

The thing is that the seconds on the first axis are considered the same as the seconds on the 2nd axis because there is only one time.

Edited by michel123456
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Just the term "volume" is standard.

...or you could call it "Superspace".

I would avoid superspace as it already has some specific meanings:

1) John Wheeler used the term superspace to describe the configuration space of general relativity. This usage is probabily not common today.

2) A "space" with commuting and anticommuting coordinates is the more common usage. Such thing are found behind supersymmetry and quantisation methods.

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