Re: DWB: Trisecting an Angle

From: shedar (nobody_at_nonesuch.com)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 01:34:46 GMT

"shedar" <nobody@nonesuch.com> wrote in message
news:DKpad.708560$gE.181960@pd7tw3no...
> "Ryan Reich" <ryanr@uchicago.edu> wrote in message
> news:2su6ldF1n0l6oU2@uni-berlin.de...
> > On Sunday 10 October 2004 19:37, David Bandel wrote:
> >
> > > For years, 'trisecting an angle' has been synonymous with "crank,"
> > > "crackpot," and any other variety of derogatory terms. The situation
> > > here is a problem mathematics has been face with since mathematicians
> > > have taken it upon themselves to judge others before judging their
> > > work.
> > >
> > > What follows is a perfectly accurate method of trisecting an angle
> > > using only a straitedge and a compass. Will it be viewed as correct? I
> > > suspect not. But not for any logical fallacy in itself. Merely because
> > > mathematicians are blinded by the jargon they've been shackled with by
> > > their colleagues and teachers.
> > >
> > > You are given an angle.
> > >
> > > Construct a circle around the vertex of the angle. The center of the
> > > circle is point A, and the points where the rays of the angle
> > > intersect the perimeter of the circle are B and C. Hence angle BAC
> > > becomes the angle of interest.
> > >
> > > Continue the line AC in the direction opposite of ray AC.
> > >
> > > Draw a line from point B such that it strikes the circle at one point
> > > E on it's way out and hits the ray CA at point D and such that line
> > > segment DE is equal to the radius of the circle.
> >
> > This sentence is where you depart from the straightedge-compass rules.
> > There is no way to make this line BE without using a ruler, since it
> relies
> > on two as-yet-unconstructed lengths being simultaneously constructed to
be
> > equal. Archimedes had this construction, actually (it's discussed in
> > Dummit and Foote's _Abstract_Algebra_, p. 515 in the second edition).
> >
> > > Angle EDA is x.
> > > Triangle DAE is isosceles, therefore angle DAE is x and angle AED is
> > > 180 - 2x.
> > > Triangle BEA is therefore 2x. It is also isosceles so EBA is 2x and
> > > EAB is 180 - 4x.
> > > This leave angle BAC as 3x.
> > >
> > > BAC is the angle we started with. Angle EDA is one third of it's
> > > angular measure.
> > >
> > > The angle is trisected.
> > >
> > > The proof is fairly rigorous as far as construction-based proofs are
> > > concerned.
> >
> > It is, in fact, totally rigorous...but it uses an axiom that isn't one
of
> > the straightedge-and-compass rules, so it does not "trisect the angle"
in
> > the Greek sense.
> >
> > > Are we dealing with a paradox? Is it a contradiction when a seemingly
> > > impossible problem is solved? I'm afraid that many posters here seem
> > > to think so. Mathematics as it exists today is shrouded in hypocricy
> > > and bigotry against the young and new. Those with fresh ideas are
> > > shunned from the mathematical community like sooth sayers with the
> > > plague. When difficult problems are solved, mathematics marches
> > > forwards regardless of the reactions of the few fools who try to hold
> > > it back.
> >
> > Spare the bitterness, please. It's okay to be wrong, less so to be
> grouchy.
> >
> > --
> > Ryan Reich
> > ryanr@uchicago.edu
>
> You have the patience of a saint, Ryan. Look at the number of ridiculous
> rants he's swamped this NG with.
>
> It is okay for a student to be wrong if the student is willing to learn
from
> his mistakes. But a distinctive trademark of a "crackpot" is to PURPOSELY
> muddle the precise constraints of what is and is not a "legal"
construction
> so that he might be able to confuse and lull the laymen into incorrectly
> thinking that he is smarter than Archimedes or Appollonius--not to even
> mention the fact that the technique he described has already been reported
> by Archimedes over two millennia ago (but unlike the "crackpots",
Archimedes
> understood perfectly the "illegality" of the construction according to the
> constraints of classical Greek geometry).
>
> I can only imagine the precocious and prodigal Apollonius--who, on more
than
> one occasion, was highly critical of some of Archimedes'
ingenuity--laughing
> himself to death upon seeing the "illegal" construction, or laughing with
> the rest of the geometry "court" as they kick the "crackpot" out of their
> place of gathering.
>
> Shedar

> I can only imagine the precocious and prodigal Apollonius ...

Should have used my dictionary. But I meant "prodigious" (the adjective for
"prodigy"), not "prodigal" (Apollonius of Perga was a child prodigy). The
dictionary also says the word "prodigious" is now *archaic* (that's only
befitting for Apollonius too as he was ancient).

Shedar

Quantcast