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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
>>
>>
>>
>>
>>
>





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