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RE: Angle Trisection
Posted:
May 28, 2002 1:04 PM
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>
> Here are few examples of angles that can be triseced, but this is not
> obvious:
>
> 1. Pi/7 can be trisected since Pi/21 = Pi/3 - 2*Pi/7
Dear Sergei,
Thank you for the interesting comments on angles that can be trisected.
There is, however, a problem with the statement I isolated above, and
the argument is not correct. To see why, let me give a more transparent
example.
It is like saying, for example, that an angle of 10 degrees can be
constructed because 10 = 30 - 2.10.
On the other hand we know that an angle of 10 degrees cannot be
constructed (If it could, so would its double 20, so 60 could be
trisected. But it is well known that it cannot).
To return to the above, it only says: IF (this is a big "if" here) you
could construct Pi/7, then you could trisect it. The problem is that you
cannot construct Pi/7. If you could, then doubling it would give you the
central angle of a regular septagon. But it is well known that the
regular septagon cannot be constucted using ruler and compass.
(Incidently, for the benefit of those who like some history of maths,
this particular regular polygon was constructed by Archimedes using a
type of neusis construction. The original Greek text is lost, but there
is an Arabic translation of it. I highly recommend that you find a text
that has an English translation of it. "The Works of Archimedes" by T.
Heath is one such.)
All the best,
Michael Lambrou.
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