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

In summary, it is not possible for an object of N dimensions to exist entirely in N-1 dimensions. This is because an object's dimension is defined by the number of data points needed to fully describe it, and a lower-dimensional space would not have enough data points to fully describe an object of higher dimensions. Additionally, there are mathematical results that show that certain objects cannot be embedded in lower-dimensional spaces.
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duyix
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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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duyix said:
The question is this; could an object of N dimensions exist entirely in N-1 dimensions?
No, it's not possible.

duyix said:
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.
 
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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) )##
 
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FAQ: Can an Object with N Dimensions Exist in N-1 Dimensions?

What is a 3D object in 2D space?

A 3D object in 2D space refers to a three-dimensional object that is represented or projected onto a two-dimensional surface, such as a piece of paper or a computer screen. This is often done in order to visualize or analyze the object in a simpler and more easily understandable way.

How is a 3D object represented in 2D space?

A 3D object can be represented in 2D space through various methods, such as using drawings, photographs, or computer-generated images. These representations may use different techniques, such as perspective, shading, or isometric projection, to convey the three-dimensional characteristics of the object.

What are the limitations of representing a 3D object in 2D space?

One limitation of representing a 3D object in 2D space is that it can result in a loss of depth and dimensionality. This can make it difficult to accurately perceive the true size, shape, and orientation of the object. Additionally, certain features or details of the object may not be visible or easily discernible in a 2D representation.

How can a 3D object in 2D space be useful in scientific research?

Representing a 3D object in 2D space can be useful in scientific research as it allows for easier analysis and comparison of different objects. It also enables scientists to study and manipulate the object in a controlled and simplified manner, which can aid in understanding its properties and behavior.

Can a 3D object in 2D space accurately represent a real-life object?

While a 3D object in 2D space may not fully capture all the complexities and nuances of a real-life object, it can still provide a useful and accurate representation for scientific purposes. With careful consideration and use of appropriate techniques, a 2D representation can effectively convey important information about the 3D object and its characteristics.

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