- #1

- 1

- 1

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- #1

- 1

- 1

- #2

Mentor

- 37,034

- 9,132

No, it's not possible.The question is this; could an object of N dimensions exist entirely in N-1 dimensions?

duyix said:In other words, could an infinitely flat object have 3 degrees of freedom and also be able to fit entirely in 2D space? [\quote]

If by "infinitely flat object" you mean "a plane" it's already a two-dimensional object that can be determined by two nonparallel direction vectors. I.e., two degrees of freedom.

Last edited:

- #3

Science Advisor

Gold Member

- 6,376

- 8,714

There are different definitions of the term Dimension. One of them is that of number of data points needed to fully describe every point in the n-th dimensional object. And that number is precisely n.

There are results to the effect that ##\mathbb R^{n+k} ; k >0 ##; k a positive Integer, cannot be embedded in ##\mathbb R^n ##. There are similar results for n-spheres ## S^n ##. that cannot be embedded in ## \mathbb R^n ## or lower IIRC, the main result is that of Borsuk -Ulam.

Edit: A 1-dimensional object embedded in n-space is describable as ##(f_1(x), f_2(x),...,f_n(x))##.

An m-dimensional object in k-space is describable as ## (f_1(x_1,..., x_m), f_2(x_1,x_2,..,x_m),,..,f_k(x_1,x_2,..,x_m) )##

There are results to the effect that ##\mathbb R^{n+k} ; k >0 ##; k a positive Integer, cannot be embedded in ##\mathbb R^n ##. There are similar results for n-spheres ## S^n ##. that cannot be embedded in ## \mathbb R^n ## or lower IIRC, the main result is that of Borsuk -Ulam.

Edit: A 1-dimensional object embedded in n-space is describable as ##(f_1(x), f_2(x),...,f_n(x))##.

An m-dimensional object in k-space is describable as ## (f_1(x_1,..., x_m), f_2(x_1,x_2,..,x_m),,..,f_k(x_1,x_2,..,x_m) )##

Last edited:

Share:

- Replies
- 6

- Views
- 676

- Replies
- 3

- Views
- 1K

- Replies
- 7

- Views
- 803

- Replies
- 2

- Views
- 1K

- Replies
- 0

- Views
- 354

- Replies
- 3

- Views
- 485

- Replies
- 2

- Views
- 829

- Replies
- 3

- Views
- 560

- Replies
- 13

- Views
- 756

- Replies
- 3

- Views
- 748