Home > Math, Science > Four Dimensions: A Simple Proof

Four Dimensions: A Simple Proof

This is a proof that four dimensions is sufficient to contain all possible spaces and times — ten are not necessary.

First, let us remember what a dimension is. Mathematically, a dimension is an axis across which something can change. Space has three dimensions, because you can change along length, width, or height. Time has a dimension because you can change through time. Saltiness also has a dimension, although it’s not particularly useful for most purposes; but even so, my science teacher informed me that oceanographers treat saltiness as a dimension.

One dimensions represents every possible value of some particular thing. A dimension could be represented with a line, which contains every possible location in one-dimensional space. It could be represented as moving through every possible shade of red, or every possible level of saltiness, or any other measurable thing that you can think of. You can also use this line to represent every possible combination of this thing: you can represent one shade of red along the redness dimension, or you can use two points to represent two shades of red, or three points to represent three shades of red, and so on.

What about when you introduce a second dimension? Now you can represent every combination of two things and every possible set of combinations. The most obvious example is length and width. On a spacial plane, you use a single point to represent length and width, and a set of points to represent many different lengths and widths.

You can also extend this into three dimensions, for example with three-dimensional space. Using the axes of length, width, and height, you can represent any point in space.

Next, bring it into the fourth dimension. If the first three dimensions are space and the fourth is time, then we are now able to represent every combination of space and time that can possibly exist. This four-dimensional construct contains all possible universes and all possible timelines.

I have an objection to this proof, as well as a refutation. It is much more complex, so I will be addressing it in an upcoming post. If you still have a problem with my proof, stick around.

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Categories: Math, Science
  1. Matt
    April 25, 2010 at 1:48 am

    You could do it with one dimension if you represented every single point in space and time at a different point on a line.

    • April 25, 2010 at 6:22 pm

      But how can you match every point in four dimensions to every point on a line?

      Edit 12/19/12: This is indeed possible.

    • April 26, 2010 at 4:40 pm

      Although I don’t see any explicit way to do it, I’m sure you’re right: it must be possible to compress four dimensions into one (because there are aleph one points in a line and aleph one points in four-dimensional space). It’s not very meaningful or useful, though.

      • n
        October 29, 2012 at 10:23 am

        There isn’t a way to do this; both sets are countably infinite. However, cardinality of 4D space = aleph_1^4 = (2^aleph_0)^4 = 2^(4aleph_0) = 2^(aleph_0) = aleph_1 = cardinality of 1D space.

  2. March 6, 2012 at 2:13 am

    “This is a proof that four dimensions is sufficient to contain all possible spaces and times — ten are not necessary.”
    All you did here was say what a 4 dimensional space is. The ten dimensional video you wrote about has no factual basis as far as I’m concerned. The theory that does say that 10(11) dimensions are required for our universe to make sense is a superstring(M) theory. And if you want to prove that theory and many many of the top theoretical physicists in the world, good luck xD

  3. Tak
    June 16, 2012 at 5:27 am

    I have read some information about 4 dimensions on the internet. And I there are 2 different interpretations, one is time as the 4 dimension, the other is spatial 4 dimension. I am interested in the later one, because I think the time dimension is used to describe the space-time universe, and dimensions of space and time might be related, but they are not necessary in the same category (excuse my non-scientific term). Just like I can say their is a space-color universe that smell is the 4 dimension. And also I think space and time are basically very different: in space (3d) you can go up and down, forward bcakward, but in time you can only go forward.

    In the spatial 4th dimension, all I can find is speculation, using the relationships of 1d, 2d and 3d to describe the 4th d universe like they are actually real, but I cannot find any proof. It is like talking about ghost or god.

    Can anybody enlighten me?

  4. Tak
    June 16, 2012 at 5:30 am

    Sorry, just find some mistakes in the earlier post. Make the corrections here:

    I have read some information on the internet about 4 dimensions. And I there are 2 different interpretations, one is time as the 4th dimension, the other is spatial 4th dimension. I am interested in the later one, because I think the time dimension is used to describe the space-time universe, and dimensions of space and time might be related, but they are not necessary in the same category (excuse my non-scientific term). Just like I can say their is a space-color universe that color is the 4 dimension. And also I think space and time are basically very different: in space (3d) you can go up and down, forward backward, but in time you can only go forward.

    In the spatial 4th dimension, all I can find is speculations, using the relationships of 1d, 2d and 3d to guess the 4 d universe like they actually exist, but I cannot find any proof. It is like talking about ghost or god.

    Can anybody enlighten me?

  5. December 19, 2012 at 9:55 pm

    Michael Dickens :
    But how can you match every point in four dimensions to every point on a line?

    Crazy as it sounds, but there actually is a way of doing this, and it is by the use of a fractal. You can actually have a fractal that extends a single 1D line into as many dimensions as you want. See http://en.wikipedia.org/wiki/Moore_curve
    In such fractal, for every point in a 3D or 4D or whatever-D space you want, there exist a point that belongs to the 1D line of the fractal and can be addressed by a real-valued number. So you can describe any point in 3D space using 1D address. The caveat is that there may be more than 1 1D address for a 3D point, but that doesn’t break the principle.

  6. David Kokic
    December 10, 2015 at 3:48 am

    Michael, A matrix can be viewed 4D (h, w, d & t). See my facebook post of 12 August 2015. If you can not view my post send me a friend request. Best wishes, David

  1. April 24, 2010 at 12:19 am

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