Can an n-dimensional object fit entirely in n-1 dimensions?

[Einstein's Cat]

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

 

[fresh_42]

Your questions become vague when you use the terms "exist entirely" and "fit in". Those have to be clearly defined to answer the questions.

could an object of N dimensions exist entirely in N-1 dimensions?

No.

could an infinitely flat object have 3 degrees of freedom and also be able to fit entirely in 2D space?

You can paint out a square by a line.
see post #10: https://www.physicsforums.com/threads/summing-a-function-value-over-an-interval.853431/#post-5352033

 

[WWGD]

Maybe , OP, you want to know if an n-dimensional object can be embedded in (n-1)-dimensions? Or are there other types of properties of the object that you want to preserve? I think you can say no for n-spheres (I think a corollary of Borsuk-Ulam theorem) and for $\mathbb R^n$, but I don't know of a more general result. But I think the answer ultimately depends on what (types of) intrinsic properties of the object you want to preserve in the lower dimensions: topology, geometry, etc.? Interesting question, though.
In one sense of dimension, the answer is no: if you see the dimension n of an object as the minimal number of coordinates of a point needed to uniquely identify each point in the space, then the answer would be (is) no.