How do you call higher dimensional axes?

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How do you call higher dimensional axes?

Postby daemonflower » Mon Mar 26, 2007 10:07 am

In 3D we've got the x, y and z axes. The fourth axis is usually called the w axis, or the t axis if you're talking about time.

Is there any wide-spread convention how to call the fifth, sixth, ... spatial axis? Or do I have to invent one myself?
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Postby bo198214 » Mon Mar 26, 2007 10:54 am

in the n dimensional case the coordinates are usually called x<sub>1</sub>, ..., x<sub>n</sub> ;)
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Postby daemonflower » Mon Mar 26, 2007 12:48 pm

bo198214 wrote:in the n dimensional case the coordinates are usually called x<sub>1</sub>, ..., x<sub>n</sub> ;)

Hm, not so beautiful... but I guess I could live with x, y, z, x<sub>4</sub>, ...
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Postby bo198214 » Mon Mar 26, 2007 12:52 pm

XD *I dont have seen this!*
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Postby daemonflower » Mon Mar 26, 2007 2:56 pm

Yeah, but calling the axes "x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>" in the three dimensional case, which is what users would normally see, would be nonstandard and confusing... anyway, thanks, I'll think of something!
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Postby wendy » Tue Mar 27, 2007 7:41 am

w,x,y,z is the usual set.

z is height, and y is forward. Everything else is across.

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Postby PWrong » Tue Mar 27, 2007 9:28 am

I've used x, y, z, w, u, v before, in that order. If I had to, I might use x, y, z, w, u, v, p, q, r, s. I'd order them that way because that's the way the alphabet tends to get divided up in maths, for whatever reason.
But if I got that high, I'd prefer to use x<sub>1</sub>, ... ,x<sub>n</sub> or x<sup>1</sup>, ... ,x<sup>n</sup>.
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Postby bo198214 » Tue Mar 27, 2007 9:46 am

but x,y,z,w,u,v is not lawful, rather x,y,z,w,v,u

I mean it makes anyway only sense (to use non-indexed variables) if you want to describe things that are particularly valid in that 4,5 or 6 dim space (for us humans it makes sense to have different words for depth, breadth and hight, but if you dont live in 6 dim space, why would you give the directions independent names/variables?). Usually in 5 or 6 dimensions one anyway tends to formulate things that are valid also in n dimensions and this is quite easier to formulate with indexed variables (for example matrices etc).
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Postby houserichichi » Tue Mar 27, 2007 1:40 pm

It's just semantics at this point...anybody versed in what you're talking about will get the gist of whatever you decide to label the axes. After three, however, it is most aesthetically pleasing and less cumbersome to use x<sub>i</sub> for Euclidean axes.

x<sub>1</sub> = x
x<sub>2</sub> = y
x<sub>3</sub> = z
x<sub>4</sub> = w (or whatever letter you'd like to choose)
ad infinitum.
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Postby daemonflower » Thu Mar 29, 2007 9:34 am

houserichichi wrote:It's just semantics at this point...
That's true.
houserichichi wrote:anybody versed in what you're talking about will get the gist of whatever you decide to label the axes.
With my program I aim at people who do not necessarily know what we're talking about here... educational purposes, if you wish.

I've decided on the following scheme. In the three-dimensional case, I'll just label the axes x, y, z. In the higher dimensional case, I'll have a different set of labels: x<sub>1</sub> (x), x<sub>2</sub> (y), x<sub>3</sub> (z), x<sub>4</sub> (w), x<sub>5</sub>, x<sub>6</sub>, ... This seems pretty clear to me.

It's mainly academic for dimensions higher than four, as I guess everybody will get lost anyway... :)
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Postby bo198214 » Thu Mar 29, 2007 10:11 am

daemonflower wrote:With my program I aim at people who do not necessarily know what we're talking about here... educational purposes, if you wish.


So what is your program?
At least you audience seems sufficiently educated to know x,y,z, otherwise I can barely imagine how to convey the idea of higher dimensions ...
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Postby daemonflower » Thu Mar 29, 2007 2:30 pm

I am working on a three dimensional Go game (http://sourceforge.net/projects/go-3/), which I plan to extend to n dimensions - simply because I can, because I have found an elegant solution to create a n+1-dimensional Go board from an n-dimensional one.

I don't expect many people to want to play Go in more than three dimensions (it's hard enough in two, after all). But I want to give them the possibility, and I want some consistency there.

The project of going from three to n dimensions is still in its early stages. I was coding some user interface elements when I stumbled across the naming problem.

I will post a link to my n-dimensional Go game on this forum when it is ready to be reviewed - which may take some time, I have other things to do too :roll:
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Postby pat » Thu Mar 29, 2007 8:15 pm

Care to describe your "elegant solution to create a n+1-dimensional Go board from an n-dimensional one"?

And, for my vote... I use x, y, z, w, u, v. If I need u or v for something else or need more than six dimensions, I switch to x<sub>1</sub>, x<sub>2</sub>, ...
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Postby pat » Thu Mar 29, 2007 8:19 pm

wendy wrote:w,x,y,z is the usual set.

z is height, and y is forward. Everything else is across.


For me, I use x is forward, z is height. The reason that I use x is forward is that there is always a forward (or, more to the point, I want there to always be a forward), even in one dimension.

My raytracer does x<sub>1</sub> is forward, x<sub>2</sub> is left-to-right, x<sub>3</sub> is top-to-bottom, x<sub>4</sub> is left-to-right across slices of x<sub>1,2,3</sub>, x<sub>5</sub> is top-to-bottom across slices of x<sub>1,2,3,4</sub>, x<sub>6</sub> is left-to-right across slices of x<sub>1,2,3,4,5</sub>, x<sub>7</sub> is top-to-bottom across slices of x<sub>1,2,3,4,5,6</sub>, etc. I recently added an option though that will start x<sub>4</sub> as top-to-bottom instead of left-to-right. This makes it easier for me to make stereo-pairs of renderings of 4-d scenes.

For some n-dimensional games that I've started working on, I've used x is forward, y is left-to-right, z is the other left-to-right, and w is top-to-bottom with gravity pushing things in the -w direction.
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Postby Keiji » Thu Mar 29, 2007 10:35 pm

Eh, I always use y as the axis of gravity, z as the axis of sight, and x as the axis that's perpendicular to both of those.

When it comes to 2D, I generally dismiss the z direction, as I don't imagine being in the plane, rather looking at the plane...
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Postby pat » Thu Mar 29, 2007 10:44 pm

Keiji wrote:Eh, I always use y as the axis of gravity, z as the axis of sight, and x as the axis that's perpendicular to both of those.

When it comes to 2D, I generally dismiss the z direction, as I don't imagine being in the plane, rather looking at the plane...


Scratches head... if you dismiss z, and z is your line of sight, then you're not looking anywhere? I mind not having a line of sight, and so use x in all dimensions.
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Postby Keiji » Fri Mar 30, 2007 7:02 am

No, I mean that on a sheet of paper for instance, the axes would be x and y, without z.
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Postby daemonflower » Fri Mar 30, 2007 1:08 pm

pat wrote:Care to describe your "elegant solution to create a n+1-dimensional Go board from an n-dimensional one"?
Basically it's the old way to construct an n-dimensional hypercube: Take an n-1 dimensional hypercube of size a, then move it a units along the nth axis. You can do the same with a grid with 19 (or, more generally, m) crossing points.

The point I really like about my idea is that I use generic programming to define an n-dimensional grid and project it to n-1-dimensional space (and so on, until you've reached 3 dimensions). I'd post a simple code example for a hypercube here, but I have discovered that the code I already have is too complex to boil it down to a few lines... at least it would take me more than a few minutes to do it. I'll post it in the "Programming" forum when I find the time.
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