Partitioning high-dimensional space

[jblade]

Hello to all,
I know (as almost anyone) how to divide an square in let's say 4 equal parts, a cube in 8 or more, i.e. sub-squares and sub-cubes respectively. But a high dimensional space? . ..I mean where dimensions are >>3, for example 100 or 1000?. In fact I don't know if the space is "hyper-rectangle" or "hypersphere". Probably this second part (to know or determine the shape of the hyperspace) is another question...

I apologise if this is not the correct forum, then I'll apreciate if you can suggest another forum to post this/these questions

thanks

 

[HallsofIvy]

The only way I can make sense of your question is to assume you are asking about partitioning an n-dimensional "parallelapiped".

If you divide each edge into n parts, then you divide the entire figure into:
dim 2: n² parts
dim 3: n³ parts
dim 4: n⁴ parts
.
.
.
dim 1000: n¹⁰⁰⁰ parts.

Get the idea?

You have to be dividing into parallelopipeds (your "hyper-rectangles"). Spheres and "hyper-spheres" won't partition space.

 

[jblade]

First than all thanks for helping to clarify this subject.

Let's see, if I have a dim=1000 space where each dimension is 100 units length, is it possible to divide it in 4 (or any arbitrary number of) independent and equally sized parts? and if so, how can I reference each of these "parts".

I mean, is it correct to say

part 1: (0,25) in all 1000 dimmensions
part 2: (25,50) in all 1000 dimmensions
part 3: (50,75) in all 1000 dimmensions
part 4: (75,100) in all 1000 dimmensions

just as it would happens in 2 and 3 d cubes.

 

[nworm]

May be it will help.

Let we have n-dimensional "parallelepiped".

part 1: $$0< x_1< 1, 0< x_2< 1,... 0< x_{n-1}< 1, 0< x_n< 1$$,

part 2: $$0< x_1< 1, 0< x_2< 1,... 0< x_{n-1}< 1, 1< x_n< 2$$,

part 3: $$0< x_1< 1, 0< x_2< 1,... 1< x_{n-1}< 2, 0< x_n< 1$$,

part 4: $$0< x_1< 1, 0< x_2< 1,... 1< x_{n-1}< 2, 1< x_n< 2$$,

part 5: $$0< x_1< 1, 0< x_2< 1,... 1< x_{n-2}< 2, 0< x_{n-1}< 1, 0< x_n< 1$$,
...

part $$n^{999}$$: $$0< x_1< 0, 999< x_2< 1000,... 999< x_{n-1}< 1000, 999< x_n< 1000$$,
...

part $$n^{1000}$$ -1: $$999< x_1< 1000, 999< x_2< 1000,... 999< x_{n-1}< 1000, 998< x_n< 999$$

part $$n^{1000}$$: $$999< x_1< 1000, 999< x_2< 1000,... 999< x_{n-1}< 1000, 999< x_n< 1000$$


You can replace "<" by "<=".

 

[HallsofIvy]

First than all thanks for helping to clarify this subject.

Let's see, if I have a dim=1000 space where each dimension is 100 units length, is it possible to divide it in 4 (or any arbitrary number of) independent and equally sized parts? and if so, how can I reference each of these "parts".

I mean, is it correct to say

part 1: (0,25) in all 1000 dimmensions
part 2: (25,50) in all 1000 dimmensions
part 3: (50,75) in all 1000 dimmensions
part 4: (75,100) in all 1000 dimmensions

just as it would happens in 2 and 3 d cubes.

It doesn't make sense to say "each dimension is 100 units length". I assume you meant to say "there is an edge of the parallelopiped in the direction of each axis having length 100 units.
Now, when you say " divide it in 4 (or any arbitrary number of) independent and equally sized parts?" do you mean divide the parallelopiped into 4 equal parts? That's how I would interpret it but your "part 1, part 2, part 3, part 4" seems to be dividing each edge into 4 parts: that divides the parallelopiped into 4¹⁰⁰⁰ equal parts.

In particular I am puzzled by your "just as it would happens in 2 and 3 d cubes.". It is true that, since 2²= 4, you can divide a square into 4 equal parts by dividing each side into 2 equal parts. Since n³= 4 does not have any integer solution, there is no way to divide a 3-d cube into 4 identical parts by dividing each edge into equal parts. You might well be able to do it by dividing different edges into different numbers of (non-equal) parts. The same is true for 10000 dimensional space.

 

[jblade]

Thanks again,
Effectively a "hyper-figure" can be divided in equally sized parts by using 2^n parts, where n is the dimensionality.
So a 2d square can be divided into 2^4 equally sized parts, a 3d cube in 2^3 that is 8 little equally sized cubes. But is is partially true, as a 2d square can be divided into just 2 equally sized rectangles (or subspaces or sub areas), and a 3d squared space or cube can be divided into 2 big subcubes and also in 4 3d rectangles o (prisms?).
Example:
For a cube with one vertice in the origin, the exteme coordinates for each edge are in (100, 100, 100) for x, y,z axis, it can be divided into 4 equally sized parts. To help me explain I made an illustration. I know how to handle each one by its coordinates, ie. I know the region in the "delimited space" they occupy.

So I wonder if for higher dimensions it is still possible to make something as illustrated and hence to be able to handle each part individually and how

I apologise if the vocabulary is not technically acceptable, I should tell I am not expert in this subject. But as I want to learn and dissipate ignorance I ask , and appreciate your time to answer.

 

[jblade]

I admit that in the example each edge is not equally divided, otherwise it would be imposible to divide a 3d into 4 parts as HallsofIvy said.
But any way it is possible to divide the 3d space into arbitrary number of regions.

 

[nworm]

For example you want to divide n-dimensional "parallelepiped" in s parts.

part 1: $$0<x<\frac{1000}{s},0<y<1000,0<z<1000,...,0<t<1000$$
part 2: $$\frac{1000}{s}<x<\frac{2000}{s},0<y<1000,0<z<1000,...,0<t<1000$$
...
part s: $$1000-\frac{1000}{s}<x<1000,0<y<1000,0<z<1000,...,0<t<1000$$

If you want then you can divide x,y dimensions or y,z dimensions (your figure).