What kinds of surfaces exist in Maths?

[geordief]

I have been a little confused by this recently

When are surfaces 2D and when they higher dimensional?

I understand hypersurfaces can be of any number of dimensions so long as they have one less dimension than the space they are embedded in.

 

But are there also 2D surfaces embedded in these higher dimensional spaces?

And how does this apply to 3D+1 space-time?

 

 

[studiot]

42 minutes ago, geordief said:

I have been a little confused by this recently

When are surfaces 2D and when they higher dimensional?

I understand hypersurfaces can be of any number of dimensions so long as they have one less dimension than the space they are embedded in.

 

But are there also 2D surfaces embedded in these higher dimensional spaces?

And how does this apply to 3D+1 space-time?

 

 

 

A surface is always two dimensional.

 

It can exist (be embedded in) 3D,  4D, any number of D, but the surface itself is always 2D.

 

 

 

 

[geordief]

1 hour ago, studiot said:

 

A surface is always two dimensional.

 

It can exist (be embedded in) 3D,  4D, any number of D, but the surface itself is always 2D.

But a hypersurface is as I described?

If they are , are they solely mathematical curiosities or are there practical applications?  (hypersurfaces ,;not surfaces that is)

Edited January 8, 2018 by geordief

[Strange]

A surface is always 2 dimensional. There are higher dimensional equivalents. For example, the surface of a sphere (a 2-sphere or 2D surface) has a 3D equivalent, but it is not a surface: https://en.wikipedia.org/wiki/3-sphere

[studiot]

6 hours ago, geordief said:

I understand hypersurfaces can be of any number of dimensions so long as they have one less dimension than the space they are embedded in

Yes this is true for hypersurface dimensions greater than 2.

A surface divides 3 dimensional space into two disjoint parts.
That is you cannot get from one part to the other without crossing the surface.

A one dimensional object (manifold or line) cannot divide 3 dimensional space in this way.

A hypersurface divides n+1 space into two parts with similar restrictions.
An n-1 dimensional manifold cannot divide n+1 space in this way.

Further a surface has two sides.
Some objects have only one side such as  Mobius strip.

 

 

 

[geordief]

2 hours ago, studiot said:

Yes this is true for hypersurface dimensions greater than 2.

A surface divides 3 dimensional space into two disjoint parts.
That is you cannot get from one part to the other without crossing the surface.

A one dimensional object (manifold or line) cannot divide 3 dimensional space in this way.

A hypersurface divides n+1 space into two parts with similar restrictions.
An n-1 dimensional manifold cannot divide n+1 space in this way.

Further a surface has two sides.
Some objects have only one side such as  Mobius strip.

Can the Mobius  strip be curved like the surface of the sphere? Does it have to be stationary?

Motion seems to be built into the description as after one (or two ?) revolutions you return to the "same side" (isn't that it?)

Edited January 9, 2018 by geordief

[Strange]

12 hours ago, geordief said:

Can the Mobius  strip be curved like the surface of the sphere?

In four dimensions you can create a Klein bottle, which is analogous to a sphere: https://en.wikipedia.org/wiki/Klein_bottle (it is a bit like rolling a Mobius strip uptown forma cylinder)

Quote

Motion seems to be built into the description as after one (or two ?) revolutions you return to the "same side" (isn't that it?)

I don't think that requires motion; you can just think of it as having a single side (and a single edge).

[thoughtfuhk]

19 hours ago, geordief said:

But a hypersurface is as I described?

If they are , are they solely mathematical curiosities or are there practical applications?  (hypersurfaces ,;not surfaces that is)

Hypersurfaces are extremely common in Machine learning/Deep learning:

1. We see hypersurfaces in how artificial neurons acquire activity from neighbouring neurons, by integration, summation or averaging.
2. Example 1: Weak analogy to biological brain in artificial neuron activation sum (or hypersurface): [math]Neighbouring_{activity}=W \cdot x + b[/math] (Common in old hebbian learning models)
3. Example 2: A less weaker analogy to biological brain in artificial neuron activation sum (or hypersurface): [math]Neighbouring_{activity}=W * x + b[/math] (Representing a convolution denoted by [math]*[/math], common in modern Convolutional neural networks)

Edited January 9, 2018 by thoughtfuhk