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Topic: BUSTED! Re: Myth-busting pentagons by compass & straight edge.
Replies: 9   Last Post: Aug 23, 2013 11:11 PM

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GS Chandy

Posts: 7,608
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05
Re: BUSTED! Re: Myth-busting pentagons by compass & straight edge.
Posted: Aug 21, 2013 4:47 AM
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[Message re-sent]
+++++
Wayne Bishop (WB) posted Aug 19, 2013 9:35 AM:
>
> The "myth" has always been in your brain not in the
> mathematics. In
> 6th grade it was obvious to me that constructing a
> regular hexagon
> with a compass was a triviality (and understood the
> trivial
> underlying mathematics) but had already observed
> that, if I really
> wanted a regular hexagon, not a cute construction
> were the final arc
> never hits the starting point, I should use a
> protractor. It's the
> proof that matters and, even more so, that the
> regular septagon can
> NOT be so constructed, not some Easter Bunny in your
> head. Why
> persist in embarrassing yourself?
>
> Wayne
>

I observe that the construction of a regular hexagon with compasses and straight-edge (to draw the sides) was never a problem at all (in the 6th grade or even earlier, as I recall). The final arc, more often than not, did hit the starting point or got so very close that the error didn't matter at all. I tried it some time ago, and it worked just fine. Whether during our schooldays, or now, a protractor is needed only to reconfirm the angles - the 'protractor construction' is in any case only approximate - it will NEVER be 100% accurate. Your example (or lesson, if that was what you were planning to help us draw from it) fails.

Jonathan Crabtree (JC) has been facing some some problems with the pentagon, as have I. I suspect the reason is that we're both making some minor errors at various steps, which add up cumulatively in the rather more complex steps of the construction. These minor errors add up and one gets a fifth side that is quite significantly different in many cases. By the way, my architect friend came back to me with a pretty well-drawn (and adequately accurate) pentagon - he did confess that he'd had some trouble getting there. (And yes, I can confirm that one does get a sense of 'going crazy' when one sees errors in something that one KNOWS should fetch you a right result).

I found a couple of interesting links to 'approximate heptagon construction' (earlier wrongly written as 'pentagon'): one is the regular story at Wikipedia: http://en.wikipedia.org/wiki/Heptagon

The other is http://hptgn.tripod.com/heptagon.htm.

I read through both of the write-ups, but did not try to do any of the constructions.

I really don't see where JC has been "embarrassing himself" as you claim he has been. The Easter Bunny and the septagon (heptagon??) are a different matter, of course.

By the way, I might observe that mathematicians as eminent as John Horton Conway (not to mention Euclid, of course) apparently spent considerable time and effort with these 'compass-and-straight-edge constructions' without feeling embarrassed about the matter AT ALL! So I don't understand why JC (or I, for that matter) should feel "embarrassed" about sharing our thoughts on difficulties faced in 'getting a pentagon right'.

In my view, things that really should embarrass are weird slogans like:

- -- "BLOW UP THE SCHOOLS OF EDUCATION!" (whether it belongs to Reid Lyon or to you; whether meant literally or metaphorically or delusionally);

- -- "PUT THE EDUCATION MAFIA IN JAIL!";

- -- "Children must be PUSHED to learn math (or anything else). ENCOURAGEMENT not the issue".

IMHO, these should REALLY be embarrassing!

Why persist in embarrassing yourselves?? !!!

GSC
("Still Shoveling! Not PUSHING!")

> At 04:16 PM 8/18/2013, Jonathan Crabtree wrote
> >"Although they make use of the visible forms and
> reason about them,
> >they are not thinking of these, but of the ideals
> which they
> >resemble, not the figures which they draw..."
> >
> >So in conclusion, the myth is busted and my

> relationship with Euclid
> >is now merely Platonic.
> >
> >Jonathan Crabtree
> >Mathematics Researcher
> >Melbourne Australia
> >(Currently over at http://bit.ly/Vrrx9Q)



Message was edited by: GS Chandy



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