Re: Magic number 23
- To: mathgroup at smc.vnet.net
- Subject: [mg41461] Re: [mg41445] Magic number 23
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 21 May 2003 07:58:51 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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
>
>
>
>