Is there any supercategory for 3-space like "surfaces" is for 2-space

[ScienceNostalgia101]

One thing I notice is that many shapes in 2-space; squares, circles, etc... can all have the common word "surface" apply to them. Even non-2-space descriptions like "the Earth's surface" still refer to the kinds of things that could intersect with each other at a point, along a line, along a curve, etc... just like 2-D shapes can. It seems the word surface more generally refers to that which is either 2-dimensional or could theoretically be unfurled to FORM something 2-dimensional. (Granted, if you did that with the Earth's surface a lot of people would get hypothermia pretty quickly!) Alternatively, it seems to refer to anything which Stokes' Theorem may apply to. Are these two interpretations of the word logically equivalent?

 

Which leaves a question; is there any specific descriptor for 3-space that doesn't apply to 2-space? If I said "region" I am not sure whether or not that's a 2-space or 3-space descriptor. For instance, if I said "this region of town" or "this region of the country" I'm not sure whether that refers to the exact ground/pavement surface in that part of the country/town, or to that surface PLUS all the air above it. I in the meantime say "region for lack of a better word. Is there any super-category of words that could apply to spheres, cubes, etc... that cannot apply to surfaces?

 

While I'm at it, is there any common descriptor that a line, a path, a curve, etc... could have in common? (I don't mean function, I mean something broad enough to include curves or sequences of line segments that would fail the vertical line test, but still narrow enough to have only two directions on it; forward and backward; with no equivalent of right or left.)

Edited December 17, 2020 by ScienceNostalgia101

[studiot]

5 hours ago, ScienceNostalgia101 said:

One thing I notice is that many shapes in 2-space; squares, circles, etc... can all have the common word "surface" apply to them. Even non-2-space descriptions like "the Earth's surface" still refer to the kinds of things that could intersect with each other at a point, along a line, along a curve, etc... just like 2-D shapes can. It seems the word surface more generally refers to that which is either 2-dimensional or could theoretically be unfurled to FORM something 2-dimensional. (Granted, if you did that with the Earth's surface a lot of people would get hypothermia pretty quickly!) Alternatively, it seems to refer to anything which Stokes' Theorem may apply to. Are these two interpretations of the word logically equivalent?

 

Which leaves a question; is there any specific descriptor for 3-space that doesn't apply to 2-space? If I said "region" I am not sure whether or not that's a 2-space or 3-space descriptor. For instance, if I said "this region of town" or "this region of the country" I'm not sure whether that refers to the exact ground/pavement surface in that part of the country/town, or to that surface PLUS all the air above it. I in the meantime say "region for lack of a better word. Is there any super-category of words that could apply to spheres, cubes, etc... that cannot apply to surfaces?

 

While I'm at it, is there any common descriptor that a line, a path, a curve, etc... could have in common? (I don't mean function, I mean something broad enough to include curves or sequences of line segments that would fail the vertical line test, but still narrow enough to have only two directions on it; forward and backward; with no equivalent of right or left.)

 

Gosh you have got a lot of geometry and topology into this question.

So there is a huge amount of terminology to master.

Starting with the sequence of linear (= flat) objects    point, line , plane, hyperplane  ....... subsequent flat objects are also called hyperplanes or n dimensions or n-flats.

 

point,    zero dimensions

line,        one dimension

plane,   two dimensions

hyperplane  three dimensions, for more we do not distinguish further names just an n dimensional hyperplane.

We can follow David Hilbert and build flat or Euclidian geometries of n dimensions using these.

 

One of these objects of n dimensions divides or separates a space of (n+1) dimensions into two regions, both of (n+1) dimensions.

So a point is a 0 dimension object that divides a line into two lines

A line is a 1 dimensional object that divides a plane into two half planes

A plane is a two dimensional object that divides 3 space into two regions (region is a respectable word for a subdivision of a space of any dimension)

So the n dimensional object forms a boundary between the regions.

 

Curved objects come later and involve more concepts and terminology, depending onif your interest is geometry or topology.

 

[ahmet]

15 hours ago, studiot said:

Gosh you have got a lot of geometry and topology into this question.

I have not felt myself good in geometry (but not in basic topology) so I had not replied.

but simply,this quotation is a good reply.

and if I am not wrong, I think this user is confusing herself/himself.

(Note: I know that hyperplane did not have to be 3 dimension. because I remember from somehwere when P is a plane,then P+t is hyperplane. 

15 hours ago, studiot said:

hyperplane  three dimensions, for more we do not distinguish further names just an n dimensional hyperplane.

 

Edited December 18, 2020 by ahmet

[studiot]

1 hour ago, ahmet said:

I have not felt myself good in geometry (but not in basic topology) so I had not replied.

but simply,this quotation is a good reply.

and if I am not wrong, I think this user is confusing herself/himself.

(Note: I know that hyperplane did not have to be 3 dimension. because I remember from somehwere when P is a plane,then P+t is hyperplane. 

 

Well to continue the themed answer.

The OP asked about surfaces, and I was trying to put things into context.

I have chosen the plane and hyperplane because their geometries are linear.
The plane and hyperplane are perhaps the simplest surfaces.

We can build shapes in n dimensions considering intersections of lines, and planes.
So we have polygons in 2 dimensions, polyhedrons in three dimensions and n-polytopes in n dimensions.

Polygons are constructed from lines of (n-1) ie 1 dimension, but exist in 2 dimensions.
Polyhedrons are solid shapes with surfaces constructed from planes. Again (n-1) ie 2 dimensions, but existing is 3 dimensions.
Polytopes are shapes whose surfaces are (n-1) flats, existing in n dimensions so a polyhedron is also a 3-tope with 2-flat surfaces.

Having got this far we can generate geometry on these surfaces and hypersurfaces.
This will be the familiar Euclidian Geometry, or Cartesian Geometry if we involve coordinates.
So for instance triangles will have interior angles adding to 180 and areas  equal to 1/2 base times height, that will not vary with position on the surface.

When we come to curved surfaces equivalent triangles will not be so simple to handle.

Also we can generalise polytopes to curved closed surfaces such as spheres and ellipsoids and more and even to more complicated object we call manifolds.

[ScienceNostalgia101]

My interest is primarily in geometry... preferably geometry as generalizable as possible so as to apply to everything from physics to geography to fields involving both physics and geography.

 

So if region can apply to 2-D and 3-D regions alike, why is there no supercategory for 3-D regions the way "surface" often is used for 2-D regions?

[joigus]

In my understanding, a hypersurface is a relative concept. When you have an n-dimensional manifold (that in itself is a generalisation of a surface), any n-1 dimensional manifold embedded in it is called a hypersurface with respect to the n-dimensional manifold.

I was kind of unsure if that concept was standard or it was just a local tradition, and here's what I've found --LMGIFY:

https://en.wikipedia.org/wiki/Hypersurface

The really powerful concept is manifold, that Studiot mentions (in general, no reference to it being a surface, embedded in anything).

There are also polytopes, that Studiot mentions too, and fractals could be interesting as well, if you're interested in irregularities of geography. Studiot is building towards a classification of most all different objects of geometry.

But I think the "descriptor" that you're groping towards may possibly be the manifold and its local charts. How many variables you need to map the thing give you the dimension. Riemann intended "manifold" as a generalisation of "surface."

The subject of geometry is vast indeed.