Re: Magic number 23
- To: mathgroup at smc.vnet.net
- Subject: [mg41465] Re: [mg41445] Magic number 23
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 21 May 2003 08:01:01 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
On Tuesday, May 20, 2003, at 10:41 pm, Ingolf Dahl wrote:
> I did not specify exactly what I meant by "exact value", and of course
> one
> could consider Sin[Pi/23] to be an "exact value" by itself. But what I
> meant
> was an expression involving integers and radicals. If we allow Root
> expressions, the question about the number of exact values becomes
> almost
> trivial.
There is really no question of "the number of exact values". All such
such trigonometric expressions can be expressed in terms of radicals
because they can be expressed in terms of roots of unity and these can
be expressed in terms of radicals, by rather elementary Galois theory.
But finding these expressions is a different matter. I am not an
expert, but i suspect it's probably hard.
>
> Maybe the operation ToRadicals should be redefined to also convert
> expressions of this type.
>
> I find it a bit surprising that Mathematica does not give radical
> expressions for Sin[Pi/8] and Cos[Pi/8]. These were given in my school
> math
> tables.
You need to direct it a bit.
In[32]:=
ToRadicals[RootReduce[FullSimplify[TrigToExp[Cos[Pi/8]]]]]
Out[32]=
Sqrt[1/2 + 1/(2*Sqrt[2])]
>
> One (recreational) use of these expressions are to find different ways
> to
> express Pi. If we know an exact expression of Tan[Pi/12], for
> instance, we
> might use this value as x, and use the power expansion of ArcTan[x] to
> obtain a series with the sum Pi/12. And if we use Tan[Pi/2^20] in the
> same
> way, we should obtain a commplicated expression, but a very rapid
> convergence.
>
> But, still, what is so special with Sin[Pi/23]? What is FullSimplify
> trying
> to do with it? And why do some other primes take so long to evaluate?
You should look at the corresponding cyclotomic polynomials. Some of
them are much more complicated than others.
In[36]:=
Table[Length[Cyclotomic[i,x]],{i,1,23}]
Out[36]=
{2,2,3,2,5,3,7,2,3,5,11,3,13,7,7,2,17,3,19,5,9,11,23}
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
> FullSimplify does not succeed to do anything with these expressions,
> it just
> is wasting time.
> On a second run of the evaluations with the same Kernel, FullSimplify
> has
> learnt the lesson, and the evaluations are immediate.
>
> Ingolf Dahl
> Sweden
>
>
>> -----Original Message-----
>> From: Andrzej Kozlowski [mailto:akoz at mimuw.edu.pl]
To: mathgroup at smc.vnet.net
>> Sent: Tuesday, May 20, 2003 12:54
>> To: Ingolf Dahl
>> Cc: mathgroup at smc.vnet.net
>> Subject: [mg41465] Re: [mg41445] Magic number 23
>>
>>
>> Actually, I was to impatient and in fact the second RootReduce did
>> arrive at an answer:
>>
>>
>> Root[1 - 12*#1 - 60*#1^2 + 280*#1^3 + 560*#1^4 - 1792*#1^5 -
>> 1792*#1^6 + 4608*#1^7 + 2304*#1^8 - 5120*#1^9 -
>> 1024*#1^10 + 2048*#1^11 & , 11]
>>
>> which, I think, is a pretty satisfactory exact expression for
>> Cos[Pi/23] even if it is not in radicals.
>>
>> Note that if you apply ToRadicals to the above Mathematica will return
>> you the input instantly, which suggests it does not even attempt to
>> radical expressions for root objects of such high degree.
>>
>>
>> On Tuesday, May 20, 2003, at 07:37 pm, Andrzej Kozlowski wrote:
>>
>>> The meaning of "exact values" is not entirely clear. For example,
>>> the
>>> following can be considered an "exact value":
>>>
>>>
>>> TrigToExp[Cos[Pi/23]]
>>>
>>>
>>> 1/2/E^((I*Pi)/23) + (1/2)*E^((I*Pi)/23)
>>>
>>> and so can:
>>>
>>>
>>> FullSimplify /@ TrigToExp[Cos[Pi/23]]
>>>
>>>
>>> (1/2)*(-1)^(1/23) - (1/2)*(-1)^(22/23)
>>>
>>> and
>>>
>>> (RootReduce[FullSimplify[#1]] & ) /@ TrigToExp[Cos[Pi/23]]
>>>
>>> Root[1 - 2*#1 + 4*#1^2 - 8*#1^3 + 16*#1^4 - 32*#1^5 +
>>> 64*#1^6 - 128*#1^7 + 256*#1^8 - 512*#1^9 + 1024*#1^10 -
>>> 2048*#1^11 + 4096*#1^12 - 8192*#1^13 + 16384*#1^14 -
>>> 32768*#1^15 + 65536*#1^16 - 131072*#1^17 +
>>> 262144*#1^18 - 524288*#1^19 + 1048576*#1^20 -
>>> 2097152*#1^21 + 4194304*#1^22 & , 21] +
>>> Root[1 - 2*#1 + 4*#1^2 - 8*#1^3 + 16*#1^4 - 32*#1^5 +
>>> 64*#1^6 - 128*#1^7 + 256*#1^8 - 512*#1^9 + 1024*#1^10 -
>>> 2048*#1^11 + 4096*#1^12 - 8192*#1^13 + 16384*#1^14 -
>>> 32768*#1^15 + 65536*#1^16 - 131072*#1^17 +
>>> 262144*#1^18 - 524288*#1^19 + 1048576*#1^20 -
>>> 2097152*#1^21 + 4194304*#1^22 & , 22]
>>>
>>> Attempting to RootReduce this last expression seems indeed to take
>>> for
>>> ever (but I have little patience and a fairly slow computer)
>>>
>>> But presumably what you are are talking about is a "radical
>>> expression". Since the Galois group of cyclotomic extension over the
>>> rationals is solvable, we know that that all Cos[2Pi/n] and
>>> Sin[2Pi/n]
>>> can be expressed in terms of radicals, but I doubt that there is a
>>> practical algorithm for writing it out. In any case even if it exists
>>> implementing it in Mathematica would be of no practical value.
>>>
>>> Andrzej Kozlowski
>>> Yokohama, Japan
>>> http://www.mimuw.edu.pl/~akoz/
>>> http://platon.c.u-tokyo.ac.jp/andrzej/
>>>
>>>
>>> On Tuesday, May 20, 2003, at 04:24 pm, Ingolf Dahl wrote:
>>>
>>>> Hello Mathgroup!
>>>> Mathematica knows the exact values of the trigonometric functions
>>>> for
>>>> some
>>>> special angles. I was curious how many such values there are. In
>>>> principle,
>>>> there should be an infinite number of such angles available, if I
>>>> have got
>>>> it correctly: at least all angles which can be written as Pi times
>>>> an
>>>> integer fraction, where the denominator can be written as a product
>>>> of
>>>> powers of two and three. Also at least one factor five can be
>>>> included in
>>>> the denominator. I have not investigated further. The trigonometric
>>>> expressions might get very complicated, of course. Mathematica knows
>>>> about
>>>> the denominators 2, 3, 4, 5, 6, 10 and 12.
>>>> In the attempt to investigate further, I asked Mathematica to
>>>> perform
>>>> the
>>>> following operation:
>>>>
>>>> Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}]
>>>>
>>>> The first run of this command gives very varying times, from 0.
>>>> Second for
>>>> i=2 to 2.3 Second for i=19. If we change the limits of table,
>>>> Mathematica
>>>> get completely stuck at i=23 (?!?!). For i=29, it takes 119.73
>>>> Seconds,
>>>> while i=36 requires 0.06 Second.
>>>>
>>>> What is the magic of i=23?
>>>>
>>>> I think that this might be an interesting feature, not a bug, so
>>>> therefore I
>>>> send it to Mathgroup. To handle the case that this really is a bug,
>>>> I
>>>> also
>>>> send it to the Wolfram support.
>>>>
>>>> I use Mathematica 4.2.0.0 on a fast Windows Me machine.
>>>>
>>>> Ingolf Dahl
>>>> Sweden
>>>>
>>>>
>>>>
>>>>
>>>
>>>
>> Andrzej Kozlowski
>> Yokohama, Japan
>> http://www.mimuw.edu.pl/~akoz/
>> http://platon.c.u-tokyo.ac.jp/andrzej/
>>
>
>
>