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

Entry: An Introduction to Metric Spaces - A248159
Author: Archangel Dr Justin, BSc (Hons), BF, Patron Saint of Paper-Cuts, Ace, Artist, Scout, Sub {1+0-8+40+9=42}, Harrasser of Italics - U108409

One for the mathematically minded...

Definitely. I started skimming after the second section. I did try to follow your arguments. Small problem reading some of the formulae - see below.

Typo note:
three-dimensionsal > three-dimensional
'Manhatten metric' > 'Manhattan metric' ?

The only other quite puzzling thing I believe I did understand is those squares in the first set of formulae, and four out of the eight other sections, do they indicate special characters you’ve used that aren’t in the font? Clearly if you can't see any squares this may be a drawback in identifying them.


Pimms

From <./>GuideML-Characters</.>:

'Please note that most browsers can't display all these characters (though one day one assumes they will).'

The symbols all appear properly in my version of Netscape, but not in IE. I'll try and re-write it without using those symbols.

I'll also fix those typos...

Typos fixed, and those nasty characters replaced with text

Looks good to me (still don't pretend to be able to follow all of it easily, but I think that is inherent with the subject)

I noted in the examples of metric spaces the example that moves between a diamond and a square at infinity. This struck a chord as I am trying to prepare a simple Entry on superellipses (A1029845, not ready for PR yet), where the Lame curve equation is equivalent to the Manhattan metric (I think). You could mention that d2 gives you an actual circle rather than a simple closed curve.


Pimms

>>> 'You could mention that d2 gives you an actual circle rather than a simple closed curve'

Did you look at the diagram? That'll be obvious once you see it.

Forgot to refresh my memory of the diagrams when I reread the Entry

Pimms

Ummm... where's everyone gone?

I didn't have any further suggestions to improve this.

How did you create your diagrams? I was wondering if it would be possible for me to create a very similar one for the superellipse entry I mentioned earlier in this conversation (but obviously including a superellipse in it).

Pimms

I did all the diagrams in CorelDraw 10, then exported them as jpegs.

Tum te tum te tum...

Bumping, Dr J?

I've re-read it. I got further

Still hard going. In your initial unpacking of the formulae I would benefit from some sentence about how to read "d: A×A &#8594; bold R", even if only a footnote.

As an aside: I wondered if it was possible to have a distance function in a set with only one point? You do say one point is enough. I suppose it would be trivial, with zero distances.

You mention much later - it might be worth bringing it in earlier - that bold R is the real numbers, and I have assumed bold B is the set of points defined by a Ball?

My main issue is that not enough of the terms used have been explained enough, and I think IMHO I would be classed as above average in mathematical general knowledge. As it stands it might be *too* esoteric for the edited guide.

When I can follow it, I'll recommend it, ? Of course I may be out of step, and in that case it will someone else who pushes it forward for inclusion.

Pimms

Bumping, me?

I've added a footnote explaining how to read the function.

I've added an explanation about single-point metric spaces.

Bold R explained in footnote (at the end of the 'functions' footnote)

Bold B (I think) is already explained at the start of the section on 'open balls' - or do you find this unclear? If so, how...?

Ok... which terms (in particular) do you think need more explaining?



J

Yes that footnote helped me a lot .
Some of the unpacking of definitions might be assisted by being even more detailed. I can now get to the bounded sets before being derailed, when it comes to "a in A and K in R" - simplistic examples would assist my visualization. Could you say of the sentence
'A subset S of a metric space M = {A, d} is said to be bounded if there exist a in A and K in R such that d(x, a) &#8804; K for all x in S.'

"S is a bounded set of points if there is a point 'a' in the set of points A such that all the points in the subset S (any one of which you can call 'x') are no further from 'a' (using the distance function 'd') than a distance K"
(sorry I can't put the relevant italics in) The reason K must be in R is probably obvious, but for some reason I stumbled when trying to understand it.

Pimms

Your Guide Entry has just been picked from Peer Review by one of our Scouts, and is now heading off into the Editorial Process, which ends with publication in the Edited Guide. We've therefore moved this Review Conversation out of Peer Review and to the entry itself.

If you'd like to know what happens now, check out the page on 'What Happens after your Entry has been Recommended?' at EditedGuide-Process. We hope this explains everything.

Thanks for contributing to the Edited Guide!

Well done

Pimms

Yay!

Thanks Pimms