jeffd@ficc.uu.net (jeff daiell) (07/03/89)
I've been following the discussion of trinary computers,
which brought to mind another question: has anyone ever
succeeded in trisecting an angle?
Jeff Daiell
--
"Thomas Jefferson still survives!"
-- John Adamshollombe@ttidca.TTI.COM (The Polymath) (07/06/89)
In article <4859@ficc.uu.net> jeffd@ficc.uu.net (jeff daiell) writes: } }I've been following the discussion of trinary computers, }which brought to mind another question: has anyone ever }succeeded in trisecting an angle? The ancient Egyptians worked out a method called the "conchoid" (pronounced "KONKoyd"). However, it isn't considered clean or elegant enough to qualify. Apart from that, no, it's never been done. (I recall hearing of a proof that it's impossible, but that may just be my aging neurons mis-firing. (-: ). -- The Polymath (aka: Jerry Hollombe, hollombe@ttidca.tti.com) Illegitimati Nil Citicorp(+)TTI Carborundum 3100 Ocean Park Blvd. (213) 452-9191, x2483 Santa Monica, CA 90405 {csun|philabs|psivax}!ttidca!hollombe
mjm@eleazar.dartmouth.edu (Michael McClennen) (07/07/89)
In article <4859@ficc.uu.net> jeffd@ficc.uu.net (jeff daiell) writes: >I've been following the discussion of trinary computers, >which brought to mind another question: has anyone ever >succeeded in trisecting an angle? To (hopefully) tie up a trivial but interesting thread, the answer is: Sure. The trisection of 60 degrees, for instance, is 20 degrees. Oh, you meant the general case! Well, how closely can you measure the angle? If you know the exact angle you wish to trisect, simply find a pocket calculator and divide by three. If not, well, your answer will be as accurate as the protractor you use to measure it... Oh, you meant using only a straightedge and compass! Well, certain angles can certainly be trisected by this method (60 degrees, for instance), but there are many angles that cannot. The proof that I have seen for this assertion involves field theory (a branch of abstract algebra), and will probably be found in any standard introduction to the subject, for it is a basic exercise. All rumours of a general method for trisecting an angle using only an unmarked straightedge and compass are false! Note, however, that if one allows oneself to make marks on the straightedge, the problem becomes solvable, and is indeed not that difficult. Michael McClennen mjm@dartmouth.edu
mm@mdbs.UUCP (Michael MacKenzie) (07/08/89)
In article <4859@ficc.uu.net>, jeffd@ficc.uu.net (jeff daiell) writes: > > I've been following the discussion of trinary computers, > which brought to mind another question: has anyone ever > succeeded in trisecting an angle? > Jeff Daiell > -- > "Thomas Jefferson still survives!" > -- John Adams I recall a method that uses a spiral of Achimedes. Unfortunatly you can't draw a Archimedean spiral with a compass & straight edge. There exists a proof about the impossibility of trisecting an arbitrary angle with compass & straight edge. This doesn't seem to stop anyone who needs to cut a piece of pie into 3 parts.
strgh@warwick.ac.uk (J E H Shaw) (07/10/89)
In article <14266@dartvax.Dartmouth.EDU> mjm@dartmouth.edu (Michael McClennen) writes: >... using only a straightedge and compass! Well, certain angles can >certainly be trisected by this method (60 degrees, for instance), but there >are many angles that cannot. ... > Nitpick: note that 60 degrees is one of the (many!) angles that certainly cannot be trisected using compass and straight-edge; see, for example, `Topics in Algebra' by I.N.Herstein.
Horne-Scott@cs.yale.edu (Scott Horne) (07/10/89)
In article <4704@ttidca.TTI.COM>, hollombe@ttidca (The Polymath) writes: > In article <4859@ficc.uu.net> jeffd@ficc.uu.net (jeff daiell) writes: > } > }I've been following the discussion of trinary computers, > }which brought to mind another question: has anyone ever > }succeeded in trisecting an angle? > > The ancient Egyptians worked out a method called the "conchoid" > (pronounced "KONKoyd"). However, it isn't considered clean or elegant > enough to qualify. Apart from that, no, it's never been done. (I recall > hearing of a proof that it's impossible, but that may just be my aging > neurons mis-firing. (-: ). It's impossible by Euclidean methods. So only a few of your neurons misfired. :-) --Scott Scott Horne Hacker-in-Chief, Yale CS Dept Facility horne@cs.Yale.edu ...!{harvard,cmcl2,decvax}!yale!horne Home: 203 789-0877 SnailMail: Box 7196 Yale Station, New Haven, CT 06520 Work: 203 432-6428 Summer residence: 175 Dwight St, New Haven, CT Dare I speak for the amorphous gallimaufry of intellectual thought called Yale?