Visualising Topology: How Important is it to Get the Visualisation Clear?

[Math Amateur]

I am reading myself into a basic understanding go topology with a view to algebraic topology.

I try to get a visual picture and intuitive feel for what I am learning .. but wonder if I am worrying too much about gaining this type of understanding early on ...

For example I am at the moment struggling to visualise the projective plane as constructed from a sphere as indicated in the book Surface Topology by Firby and Gardiner … see page 36 of their book ….

View attachment 2198Should I really worry about visualising this construction … or would a vague intuition do … then work from the analysis and algebra ...

How important is it to get the visualisation clear? What do members think? Or do we just work with the formal algebra/analysis ...

BTW does anyone know of any text or online resources that helps one visualise the above construction ...

I hope I can progress with only vague impressions of the constructions … do others have difficulties with visualising the constructions of topology?

Peter

 

[Math Amateur]

I am reading myself into a basic understanding go topology with a view to algebraic topology.

I try to get a visual picture and intuitive feel for what I am learning .. but wonder if I am worrying too much about gaining this type of understanding early on ...

For example I am at the moment struggling to visualise the projective plane as constructed from a sphere as indicated in the book Surface Topology by Firby and Gardiner … see page 36 of their book ….

View attachment 2198Should I really worry about visualising this construction … or would a vague intuition do … then work from the analysis and algebra ...

How important is it to get the visualisation clear? What do members think? Or do we just work with the formal algebra/analysis ...

BTW does anyone know of any text or online resources that helps one visualise the above construction ...

I hope I can progress with only vague impressions of the constructions … do others have difficulties with visualising the constructions of topology?

Peter

I am re-posting the above post as there have been some problems - which Jameson and others have now corrected … but i need to re-paste the image which is not showing … and I cannot edit the above ….

So my above post should read:

==================================================================

I am reading myself into a basic understanding go topology with a view to algebraic topology.

I try to get a visual picture and intuitive feel for what I am learning .. but wonder if I am worrying too much about gaining this type of understanding early on ...

For example I am at the moment struggling to visualise the projective plane as constructed from a sphere as indicated in the book Surface Topology by Firby and Gardiner … see page 36 of their book ….https://www.physicsforums.com/attachments/2210

Should I really worry about visualising this construction … or would a vague intuition do … then work from the analysis and algebra ...

How important is it to get the visualisation clear? What do members think? Or do we just work with the formal algebra/analysis ...

BTW does anyone know of any text or online resources that helps one visualise the above construction ...

I hope I can progress with only vague impressions of the constructions … do others have difficulties with visualising the constructions of topology? Do problems with visualising spaces and constructions heavily impede understanding of the mathematics of topology?

What do MHB members think?

Peter

==================================================================

EDIT : I am beginning to think that I am trying to see too much in this construction and hence worrying over what is possibly a simple construction … but maybe hard to visualise as the construction progresses step by step …

Maybe someone knows of a video or animated graphic that shows progress 'point by point' ...

 

[Deveno]

There are a couple of ways to do this, which depend on "how faithful" you want to be to the mathematics.

In older video games, screens used to have a feature called "wrap-around": for example if a moving object exited the top of the screen, it would re-enter at the bottom. In this context, the projective plane can be viewed as "wrap-around with mirror image": if an object moves off the top of the screen moving right, it reappears at the bottom moving LEFT.

Another way to look at it is this way: Imagine a 2-D drawing with perspective-all the parallel lines share a common "vanishing point" (which is kind of like "the point at infinity").

The way I like to view it personally, though is THIS way: take a sphere (like a basketball) and slice it open. Now flip one edge of the cut upside-down (an actual basketball is probably not "stretchy" enough to do this), and start to sew the edges back together (you'll have to start in the middle, because the "ends" would just tear). The basketball will assume a strange "puckered" shape (much like most mobius bands you make out of paper have a "kink" in them), and you can see that you can "mostly" do it, but you are getting some "impossible" holes to close at the ends. The only way we can "finish" (in three dimensions, at least) would be to allow the basketball to go though itself (self-intersection).

 

[Math Amateur]

There are a couple of ways to do this, which depend on "how faithful" you want to be to the mathematics.

In older video games, screens used to have a feature called "wrap-around": for example if a moving object exited the top of the screen, it would re-enter at the bottom. In this context, the projective plane can be viewed as "wrap-around with mirror image": if an object moves off the top of the screen moving right, it reappears at the bottom moving LEFT.

Another way to look at it is this way: Imagine a 2-D drawing with perspective-all the parallel lines share a common "vanishing point" (which is kind of like "the point at infinity").

The way I like to view it personally, though is THIS way: take a sphere (like a basketball) and slice it open. Now flip one edge of the cut upside-down (an actual basketball is probably not "stretchy" enough to do this), and start to sew the edges back together (you'll have to start in the middle, because the "ends" would just tear). The basketball will assume a strange "puckered" shape (much like most mobius bands you make out of paper have a "kink" in them), and you can see that you can "mostly" do it, but you are getting some "impossible" holes to close at the ends. The only way we can "finish" (in three dimensions, at least) would be to allow the basketball to go though itself (self-intersection).

Thanks Deveno ... Helpful and interesting post ...

Just wondering about what (exactly) you mean by "faithful to the mathematics"

Peter

 

[Deveno]

Mathematics typically characterizes an object by properties it possesses, not "objects in and of themselves".

When we "visualize" a mathematical object, we make a comparison to real, physical objects that typically possesses "extraneous" properties (such as the color of the basketball, or the make and model of the monitor on which a video game is displayed). Often, the comparisons are to "drawings" that are definitely NOT "the objects themselves".

So I was just alluding the the gap you yourself noticed between the "abstract" presentations of $\Bbb{RP}^2$, and how we PICTURE it.