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

"Algebra can be used to translate geometric questions into equations. This would allow one to prove difficult results which seem intractable from a purely geometric point of view. Typically, to prove that a geometric construction is not possible, you end up having to prove that some equations have no solution, a problem for the solution of which much mathematical machinery exists."

I quite like that paragraph. But I see what you mean about the last sentence, specifically 'a problem for the solution of which much mathematical machinery exists'.

I'll think of a way to rephrase it tonight then

Entry: Constructions with a Ruler and a Compass - A24020506
Author: toy box - U208464



The clumsy sentence became:

'Typically, to prove that a geometric construction is not possible, you end up having to prove that some equations have no solution. This is usually a much easier thing to prove, as there are many tricks to do so.'

with a footnote:

'For instance, you can see by drawing a graph that the equation x² + 2x + 2 = 0 has no solution. Of course there are more elaborate examples!'

How's this going? I was thought it was coming on pretty well...

>>'For instance, you can see by drawing a graph that the equation x² + 2x + 2 = 0 has no solution. Of course there are more elaborate examples

-- that footnote seems to be completely the wrong way around. You've used a geometric construction (a graph) to demonstrate that an equation has no solution, when you are using it as an example of how to use an equation to show a geometric construction is impossible.

Fair point. I must implicitly consider graphs as part of Analysis, and therefore that's not Geometry

Well, I'll remove the 2x bit. It is even easier to see that x^2 + 2 = 0 has no solution (yes, yes, no *real* solution, let's not split hair ).

How's it coming toy?

For the moment it seems to have stalled. Maybe I'll give the entry a read with a fresh eye and smooth out the rough bits during the week end. Would you have any suggestion?

I thoroughly enjoyed reading this

Here's some minor nitpicks

>>Constructions with a ruler and a compass is a classical question in...<<
This just sounds slightly odd to me - that 'Constructions' is a classical 'question'. Later you refer to 'the construction problems' which seems a better phrase. How about something like
->Constructions with a ruler and a compass presents classical problems in...<-

>>This is usually a much easier thing to prove, as there are many tricks to do so<<
But, in the footnote that follows
>>you can easily see that the equation x² + 2 = 0 has no solution<<
I do easily see that there's no real solution - so I don't see the trick. Rather than cast about for another example, you might just omit the sentence along with the footnote and put a period after '...easier thing to prove.' - and it's likely the reader will happily take your word that it is so.

>>While the first property is very easy to see, the second and third one are more difficult and rely on some astute use of Thales's theorem.<<
This is slightly problematic since you really need to explain Thale's theorem, or provide a link - but really, the sentence is not necessary, imo. If you leave it out, the reader will, again, happily accept your word that constructible numbers do indeed enjoy the remarkable properties you list.

That's all I have. Really excellent

The procedure followed ...(self-evident notions which are not defined by means of other terms, such as points and straight lines, then some axioms, ...

The bracket needs to close, presumably after 'straight lines'


One is italicised, and one is not; is this intentional?
'...as Napoleon's construction...'
'...as Pappus's construction...'

A great read, Toybox.

Thanks for the comments

I just made the suggested changes (I italicised both finally).

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.

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Coincidentally enough, the following has just been made available today:

http://arxiv.org/abs/0709.3270

Antonio Oller: Origami constructions. (A characterization of real numbers constructible by paper folding.)

One thing I could find in this Entry is how to construct a pair of parallel lines

Oh, forgot to mention that

I left out the constructions though because I didn't want to make it a list of constructions. Obviously I left one, just to illustrate what these constructions are.

My purpose in not detailing any more constructions was to avoid making a very long entry with no pictures. This could be done in conversations after the entry though.

Oh,