h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Commutative and Non-commutative Spaces - A1003294
Author: Toy Box - U208464

For the moment, I yet have to do the following changes:

I'll add another example (involving three variables); I wouldn't dare to go into how to define points and lines and whatever in non-commutative spaces, though, as this involves quite abstract ring theory. It has been suggested that I add a few basic calculations; what kind of calculations should I include?

The description of how to make a torus from a plane should become more step-by-step.

Besides, I would like to include a few pictures: one to illustrate the construction I just mentioned, another one to make the two 'circular' coordinates more visual: could someone tell me how to proceed?

As for the Klein bottle, I'll change the current link as soon as possible. A few pictures may replace a link; or -better still- I may write an entry about it.

Further suggestions welcome .

I have included calculations and changed the link. Other classical examples may be a little bit too complicated actually to be described here, as they involve a more considerable set of defining relations. As two-dimensional examples go, these are the most important anyway.

This is good stuff. I'm working my way through it and will give you a detailed response later.

Big Points:

1. It needs an introduction to say that it is mathematics we are talking about.

2. You use the phrase 'I claim that' a few times. Don't. This is not your personal opinion, so you don't need to claim anything.

3. The section on Klein bottles only serves to confuse things. I think it would be better left out completely or made into a footnote.

4. The section entitled Non-Commutative spaces is mainly about Commutative spaces. Split it up into two sections: 'Commutativity' and 'Non-commutatite Spaces'.

5. At some stage you seem to slip from xy meaning 'measure x, then measure y' to meaning 'multiply x by y'. You should make this clear that you are doing this.

6. Where I come from, doughnuts like flattened spheres are just as common as toroidal ones. You might like to say 'ring doughnut' of '(ring) doughnut'.

Small points (grammar, spelling etc.)

billiards ball --> billiard ball
ont two lines --> on two lines
like a doughnut --> like the surface of a (ring) doughnut
with a little habit --> with a little practice
consiting --> consisting
pairwise different --> all different from each other
shere --> sphere
coffe --> coffee
henceforth --> hence

Done, more or less. I'm not quite satisfied with my explanation as to how to translate 'measure x first, then y' into 'multiply x by y on the right' (the ring of coordinates bit seems a bit too technical to me).

The use of 'I claim' is just a bad habit of mine, meaning that I am to state a result the proof of which will follow eventually (possibly after having drawn some consequences first). Sorry about that.

Concerning billiard(s): do you say 'to play billiard' or 'to play billiards'?

The game is called billiards and you say to play billiards but for some reason the ball is called a billiard ball.

That's what confused me .

OK, Toy Box,

I have no idea what commutative means. I've only read the first bit so far, but even if the explanation is further down it the entry, it would be good to say what it means early on!

Hmm. I've now read to the end and I still don't understand. Do you have to be a mathematician to understand the entry, or should I come at it fresh and all will be revealed?

I don't know. Apparently, it helps to be a mathematician to understand the entry . The problem is, if I have to start going into the detail of coordinate rings, how you can identify numbers and operators and all that stuff, the entry will start becoming far too technical for my purpose.

What would you think if I left out the whole non-commutative part and just focused on the first part, namely introducing plane, sphere and torus and proceeding to explain how to show these are not homeomorphic?

This way I could actually put the Klein bottle back into this entry, and possibly write another one in which I would deal with more technical stuff, like rings of coordinates, algebraic and non-commutative varieties, etc.

There would certainly be enough in that for an entry.

So what do you think of just keeping the first part (stopping just before "Why Coordinates are Useful" and adding the Klein bottle)? How do you suggest I rename the entry?

The Topology of Two-Dimensional Spaces

Thanks!

I'll remove this one from PR and put the modified version back when it's ready.

There. Does it look better? It's still endless .

I was happy with this until the final paragraph, which I had to read 3 times before I understood what you meant. (Assuming I do now.)

Perhaps breaking it into 2 paragraphs would help - Proof would be ----, Failure would mean----

I developed somewhat the last paragraph, all the more since I seem to remember that some people do not feel at home with reductio ab absurdum (my brothers don't find it convincing at all, for instance).

Much better.

Good PR thread this - well done all! Especially Toybox.

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