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Re: Re: Re: Magic number 23

  • To: mathgroup at smc.vnet.net
  • Subject: [mg41502] Re: [mg41488] Re: [mg41466] Re: Magic number 23
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 23 May 2003 03:25:36 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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. 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
>
>
>
>
>



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