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

In summary: One can, however, embed a space filling curve in ##\mathbb R^{n+1} ## for any positive integer n.In summary, the conversation discusses the possibility of an object of N dimensions existing entirely in N-1 dimensions and whether an infinitely flat object can have 3 degrees of freedom and fit in 2D space. The experts mention that the questions become vague without clear definitions and that the answer may depend on the intrinsic properties of the object, such as topology or geometry. They also mention that an n-dimensional object cannot be embedded in n-1 dimensions in one sense of dimension, but it is possible to embed a space filling curve in a higher dimension.
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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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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.
 
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Related to Can an n-dimensional object fit entirely in n-1 dimensions?

1. What is a 3D object in 2D space?

A 3D object in 2D space refers to a three-dimensional object that is represented or displayed on a two-dimensional surface, such as a computer screen or a piece of paper. This representation is usually done using techniques like perspective drawing, where parallel lines in 3D space appear to converge in 2D space, giving the illusion of depth.

2. How are 3D objects represented in 2D space?

There are various techniques used to represent 3D objects in 2D space, such as orthographic projection, isometric projection, and perspective projection. These techniques involve transforming the coordinates of the vertices of a 3D object onto a 2D plane, while also accounting for factors like depth and distance.

3. What are the limitations of representing 3D objects in 2D space?

One of the main limitations is the loss of depth perception. In 2D space, objects appear flat and often lack the sense of depth that we experience in the real world. Additionally, some complex 3D objects may be difficult to accurately represent in 2D space, resulting in distortions or inaccuracies in the final image.

4. How is 3D modeling used in 2D space?

3D modeling involves creating digital representations of 3D objects, which can then be displayed in 2D space using various techniques. These models are often used in fields like architecture, engineering, and animation to create realistic and accurate representations of objects in a two-dimensional format.

5. What are some applications of 3D objects in 2D space?

3D objects in 2D space have a wide range of applications, including computer graphics, video games, virtual reality, and product design. They are also used in scientific research and simulations, as well as in medical imaging and visualization, allowing for a better understanding of complex structures and processes in the human body.

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