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I understand that the three dimensions each have their own axes, for lines, planes and volumes, and that 4 dimension has an axis but it is complicated and hard to determine, but from what I understand is that it is there.

So I was wondering, disregarding their complexity, do Higher-Dimensions such as 5D to 10/11D have axes?

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    $\begingroup$ what do you mean by an axis for volume? $\endgroup$
    – Mirko
    Jul 19 '19 at 3:14
  • $\begingroup$ @Mirko as in three dimensional space $\endgroup$
    – C. Jordan
    Jul 19 '19 at 3:36
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    $\begingroup$ Heh heh. I am thinking about a higher-dimensional hero swinging an ax at the head of a Killing Form to defeat the Lie Algebra. $\endgroup$
    – puppetsock
    Jul 19 '19 at 13:42
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    $\begingroup$ I'd say that's pretty much the definition of a dimension - think of it as another set of directions you can go. 1D lets you go only left and right, 2D adds forward and backward, 3D adds up and down, etc. Each dimension is just like the previous, only with another axis at a right angle to all the previous axes. (Obviously this becomes difficult to visualize past 3, but the math doesn't really care about that, it works pretty much the same for any number of dimensions.) $\endgroup$ Jul 19 '19 at 14:29
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    $\begingroup$ @DarrelHoffman I don't see what makes it so hard to visualize. I just use a trick that my high school pre-calculus teacher taught me ages ago: if you want to visualize a four dimensional space, first just visualize an $n$-dimensional space, then take $n=4$. What could be simpler? :P $\endgroup$
    – Xander Henderson
    Jul 20 '19 at 13:09
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They sure do. In three dimensions,

  • the $x$-axis is the set of all points where both coordinates besides $x$ are zero,
  • the $y$-axis is the set of all points where both coordinates besides $y$ are zero, and
  • the $z$-axis is the set of all points where both coordinates besides $z$ are zero.

If you're looking at a $26$-dimensional space with coordinates $(a, b, c, \ldots, z)$, then

  • the $a$-axis is the set of all points where all coordinates besides $a$ are zero, and
  • the $b$-axis is the set of all points where all coordinates besides $b$ are zero, and
  • the $c$-axis is the set of all points where all coordinates besides $c$ are zero, and
  • ...
  • the $z$-axis is the set of all points where all coordinates besides $z$ are zero.
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  • $\begingroup$ Thank you, this will help, I am wondering if this can be somewhat similarly said for physics or theoretical physics as well $\endgroup$
    – C. Jordan
    Jul 19 '19 at 13:51
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    $\begingroup$ I feel like even though this answer technically provides the direct answer to the OP's question, since the question itself is aimed at better understanding higher dimensional spaces, this answer could be improved by mentioning that while those space do have axes, not everything about axes in 3d applies to higher dimensional spaces. Such as that in dimensions higher than 3, a single axis no longer uniquely determines a rotation. $\endgroup$ Jul 19 '19 at 14:59
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    $\begingroup$ @Shufflepants - The OP appears to be speaking strictly about coordinate axes. So mixing in the axis of a rotation (which has nothing to do with coordinate axes) is just sowing confusion. And it is only in 3D that you can associate rotations with an axis. In 2D, they are associated with points, not lines, and in 1D rotations don't even exist. More generally, a rotation is associated with an affine space of codimension 2. $\endgroup$ Jul 19 '19 at 16:58
  • $\begingroup$ What about the vector space $\mathbb{C}^2$ over $\mathbb{C}$? $\endgroup$
    – user5402
    Jul 19 '19 at 21:35
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Sure. In $n$ dimensions, the $k^{\text{th}}$ axis, for some $1 \leq k \leq n$, is the span of the vector whose $k^{\text{th}}$ component is $1$ and all other components $0$. The problem is that there's not a particularly satisfying geometric visualization of these axes and one largely has to settle for this algebraic abstraction.

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The first important term here is that of a vector space. This is a general term; vector spaces can be far more than simply some $\mathbb{R}^n$. Polynomials, for example, form a vector space of infinite dimension.

Now you need to understand what a dimension is. If you have a vector in three dimensions, you write that vector like $(4,1,7)$. Those coordinates make sense in the context of a base. A base is a set of linearly independent vectors, which has the maximal possible size for that vector space. This size is the dimension. A three dimensional space is defined by having maximally three linear independent vectors in a set.

Edit: On the recommendation of Kaj Hansen, let me at this point make clear that a base of a given vector space has always the same amount of vectors. This is important, because this defines the dimension of the vector space.

The coordinates are now in relation to your base. If your base is $\{b_1, b_2, b_3\}$, the vector $(4,1,7)$ is the vector you get by the linear combination $4b_1 + 1b_2 + 7b_3$.

Now realize that this base can change. You can take three linear independent vectors different from your current base and make them the new base, resulting in all vectors getting different coordinates.

To give you an example when this is used, consider a 3D image in computer graphics that depicts one of those wooden puppets that artists use. There is a global coordinate system for the whole image, and the puppet has coordinates within that. But the limps have local coordinate systems which are in relation to the joints of the puppet. Hope this helps you.

Now, for the axes, an axis is simply the line created when you follow the direction of a base vector. Nothing more. They are arbitrary, and they do not even have to be orthogonal to each other. They are not a fundamental principle of a space. Coordinates are just numbers how we model it.

Another point of view, let's say that you are to determine the axes of reality, of the three-dimensional space we live in. In which direction would you point the first axis? Arbitrary. You decide on some direction for the first an then for the others, and then you write things in relation to those (or, better to say, the base vectors, as you also need a length). There is not the one objective way to create your axes.

Edit: Inserted from the comment of Kaj Hansen, note that usually one uses the standard base $\{e_k\}^n_{k=1}$ for $\mathbb{R}^n$, which consists of vectors $k_i$ that are all zero except for the $k^{th}$ component which is one. With those, a coordinate evaluates to a vector with the same numbers. Now, I brought this here and not above for a reason, and that is because we just talked about the "axes of reality", which do not exist. In the real world, there is no objective coordinate (1, 0, 0). For that reason, even the standard base is not "objective" or anything, but bound to some definitions we made. When we compute things in math, (1, 0, 0) is an objective thing, though, because in math, we live only in definitions and axioms.

With more dimensions, it is not different. You have a base and you work with it, and you can totally change the base.

Things might get difficult when you talk about physical spaces, though. Time behaves different as a dimension, and we are now in a Minkowski space, and things are getting complicated.

If you are interested in examples for vector spaces which aren't R^n, like the polynomial vector spaces or that of other functions, I can explain that too, but as this is long enough already, I'll stop for now.

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  • $\begingroup$ That is alright, I think I get the picture now, sounds like dimensions do have at least one axis for each one, even for 4 dimensional and beyond $\endgroup$
    – C. Jordan
    Jul 19 '19 at 13:53
  • $\begingroup$ I salute you for posting this. The word "dimension" is probably the most misunderstood mathematical term that's entered lay vocabulary, and ultimately something like this is what's needed to explain it properly. $\endgroup$
    – Kaj Hansen
    Jul 20 '19 at 1:39
  • $\begingroup$ Two recommended additions: explicitly point out the facts that 1) any two bases have the same number of vectors; and 2) we generally, unless otherwise stated, use the standard basis $\{\mathbf{e}_k\}_{k=1}^n$ for $\mathbb{R}^n$, where $\mathbf{e}_k$ is the vector whose $k^\text{th}$ component is $1$ and all others $0$. Hence, writing $\langle 2, 5, 3 \rangle$ is implicitly saying $2 \langle 1, 0, 0 \rangle + 5 \langle 0, 1, 0 \rangle + 3 \langle 0, 0, 1 \rangle$. $\endgroup$
    – Kaj Hansen
    Jul 20 '19 at 5:03
  • $\begingroup$ @KajHansen Inserted your recommendations. Continued on your second one with some more "philosophical" aspects of math, thought they belong nicely in a post about the basic idea of a dimension. $\endgroup$
    – Aziuth
    Jul 20 '19 at 8:37
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In three dimensions the three axes each perpendicular to the other two tell you that you identify a point by giving its coordinates $(x,y,z)$. The three axes are the lists of coordinates that have just one entry that's not $0$.

Then you can do geometry by doing algebra with the coordinates. For example, the eight points

(0, 0, 0)   (1, 1, 0)
(1, 0, 0)   (1, 0, 1)
(0, 1, 0)   (0, 1, 1)
(0, 0, 1)   (1, 1, 1)

are the vertices of a cube. (Those are all the lists of coordinates you can make using just $0$ and $1$.)

If you just allow yourself more coordinates you can study higher dimensions. So the $16$ ways to make all the lists of four coordinates each of which is just $0$ and $1$ are the vertices of the analogue of a cube in four dimensions. You can study its properties. Actually "seeing" it as a geometric object is pretty much beyond our human abilities. Here's the best we can do: https://en.wikipedia.org/wiki/Tesseract .

Clearly there's no need to stop at four coordinates/dimensions.

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Each dimension will, by definition, have an axis that is 90 degrees askew to all other dimensions. Since our everyday world is 3-dimensional, this may be difficult to envision/imagine for axes beyond the basic x-y-z, but mathematically this is how it works.

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  • $\begingroup$ Use MathJax please $\endgroup$
    – Aqua
    Jul 19 '19 at 17:39
  • $\begingroup$ Thank you very much, this is what I wish to hear $\endgroup$
    – C. Jordan
    Jul 19 '19 at 18:27
  • $\begingroup$ From where do the 90 degrees come from? Any subspace is a proper space, no matter how it is angled to some axis. Plus, you make it sound as if a "dimension" is something particular, rather than a number of independencies. $\endgroup$
    – Aziuth
    Jul 20 '19 at 8:41
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Yes. Forget everything you previously saw with the cartesian coordinate system, etc.

Now, imagine a ball of yarn with a bunch of needles sticking out of it. Those dimensions each have a needle (an axis for that dimension). Now, when you have a bunch of dimensions, each thing you add to your system needs to have a thread of yarn that touches each needle somewhere along its length.

Each thread is a vector, or the position of that thing you added to your system.

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Proof: Suppose $n$ is the highest dimension for which we have axes. Let $(x_1,x_2,\ldots, x_n)$ be any arbitrary point in this dimension. By assumption there are no axes in $n+1$ dimension, so there must be infinitely many points with the exact same coordinates $(x_1,x_2,\ldots, x_n,666)$ i.e. unique points in $n+1$ dimension do not exist which is a contradiction to the definition of a dimension.

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