Re: Re: Re: Magic number 23
- To: mathgroup at smc.vnet.net
- Subject: [mg41505] Re: [mg41488] Re: [mg41466] Re: Magic number 23
- From: David Terr <dterr at wolfram.com>
- Date: Fri, 23 May 2003 03:27:34 -0400 (EDT)
- Organization: WRI
- References: <B5322966-8C57-11D7-9A0F-00039311C1CC@mimuw.edu.pl>
- Sender: owner-wri-mathgroup at wolfram.com
Andrzej Kozlowski wrote: > They certainly can all be expressed in radicals, since they can be > expressed in terms of roots of unity, which are of course radicals. > Actually, even if you do not want to considers this kind of radicals > (roots of 1) you can obtain radical expression that look more like > what was meant in the original posting. > Perhaps you meant "real radicals", but even then it is not true that > 23 is in any sense the smallest, since Cos[Pi/7] can't be expressed > in terms of real radicals but you can see its representation by > applying FunctionExpand. Sorry, my mistake. I did mean radicals by the way, not real radicals. Perhaps I was confused because 23 is the smallest integer n such that the ring of integers of the number field Q(E^(2 Pi I/23)) is not a unique factorization domain. David > Actually, in the Mathematica Guidebooks Michael Trott seems to claim > that Mathematica will eventually return the answer to > FunctionEpand[Cos[Pi/23]] although I have not heard of anyone else who > has had the patience to wait do long. > > Andrzej Kozlowski > Yokohama, Japan > http://www.mimuw.edu.pl/~akoz/ > http://platon.c.u-tokyo.ac.jp/andrzej/ > > > > > On Thursday, May 22, 2003, at 07:55 pm, David Terr wrote: > >> Dave Langers wrote: >> >>>> Mathematica knows the exact values of the trigonometric functions >>>> for some >>>> special angles. I was curious how many such values there are. >>>> >>> >>> Take a look at: >>> http://mathworld.wolfram.com/TrigonometricAngles.html >>> >>> It doesn't explain what might be special about sin(pi/23), except that >>> it cannot be written as a simple exact value. >>> >>> BTW: This is interestingly enough related to constructions with compass >>> and straightedge: >>> http://mathworld.wolfram.com/ConstructibleNumber.html >>> http://mathworld.wolfram.com/ConstructiblePolygon.html >>> >>> Greetings, >>> Dave >>> >>> >> 23 is magic in the sense it's the smallest positive number n such that >> sin(pi/n) is not solvable with radicals. To get radical expressions for >> smaller values of n, use FunctionExpand. >> >> David >> >> >> >> >> >