High School Can an Object with N Dimensions Exist in N-1 Dimensions?

[duyix]

I am concerned that this question may instead be a philosophical one although if it it mathematical, any insights would be very appreciated. The question is this; could an object of N dimensions exist entirely in N-1 dimensions? In other words, could an infinitely flat object have 3 degrees of freedom and also be able to fit entirely in 2D space? Thank you and please excuse any naivety

 

[Mark44]

The question is this; could an object of N dimensions exist entirely in N-1 dimensions?

No, it's not possible.

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.

 

[WWGD]

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 $(f_1(x_1,..., x_m), f_2(x_1,x_2,..,x_m),,..,f_k(x_1,x_2,..,x_m) )$