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3D object in 2D space

  1. Jan 31, 2016 #1
    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
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  3. Jan 31, 2016 #2


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  4. Jan 31, 2016 #3


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    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
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