Michael O'Brien wrote :
> I already know that a regular n-sided polygon is constructible with
> straighedge and compass only if
>
> n=2^k*p_1*p_2*p_3*...
>
> where k is any nonnegative integer and the p's are distinct Fermat
primes. I
> have also seen constructions for other low-order regular polygons that
use a
> trisection device in the construction. (Conway and Guy's book "On
Numbers
> and Games" has such constructions.)
>
> My question is: Are a compass, ruler and trisection device sufficient to
> construct any n-sided regular polygon for all n>=3? If not, what is the
> smallest n for which these three items are not sufficient?
I'll relate this to the following set of construction tools :
construction by conic sections.
That is you have a tool which *draws* any conic section, so you can
construct the intersection points of any conic section.
This is not feasible with compass and straightedge only, except in
specific cases (for instance if the two conic sections share a same
focus)
The construction by conics has been studied in details [1], and the
set of all constructible numbers (=coordinates of points) is the
smallest subfield F of R for which :
if u is in F, then sqrt(u) is in F
if P(x)=0 is any *cubic* equation with coeficients in F, then all
real roots of P are in F.
That is you can solve any quadratic and cubic equations in F.
With this you can trisect any angle (because it is a cubic equation),
And trisecting an angle allows to solve any cubic equation.
Hence the construction by conics is equivallent to
compass + straightedge + trisector.
The set of all constructible regular n-gons in F is characterized by
n = 2^a * 3^b * product of p_i
with p_i distinct prime numbers in the form 1 + 2^u*3^v
This extends the constructions with compass and straightedge through
the "3" factor, and including Fermat primes (v = 0).
"Conic-primes" which are not Fermat prime (v != 0) are
7, 13, 19, 37, 73, 109, 163, 487, ...
so including the Fermat primes (3,5,17,257,65537) the first missing
primes are 11, 23, ...
And you get your counterexample :
*****************************************************************
* The regular polygon with 11 sides is not constructible with *
* compass, straightedge and trisector. *
*****************************************************************
(and Conway's book doesn't give it, just 3,5,7,9,13,17)
With a quinquisector, you might be able to do that (there are known
trisectors, I've never heard of such a tool as a quinquisector...)
The first primes p = 1 + 2^u*3^v*5^w are : 11 etc...
But 23 is still missing and requires even more weird tools
(as 23 = 1 + 2*11, an 11-sector huh ?)
ref [1] : "Théorie des corps, la règle et le compas" by JC Carréga
ISBN 2 7056 1449 4
"Field theory, the compass and straightedge"
There should be equivallent books in English.
Regards.
--
Philippe C., mail : chephip+news@[EMAIL PROTECTED]
: http://chephip.free.fr/
(recreational mathematics)
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