# 3D object in 2D space

1. Jan 31, 2016

### 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

2. Jan 31, 2016

### Staff: Mentor

3. Jan 31, 2016

### 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.

Last edited: Jan 31, 2016