Magic number 23
- To: mathgroup at smc.vnet.net
- Subject: [mg41445] Magic number 23
- From: "Ingolf Dahl" <ingolf.dahl at telia.com>
- Date: Tue, 20 May 2003 03:24:05 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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