# Conic-Constructible Regular Polygons

Antreas P. Hatzipolakis xpolakis at otenet.gr
Wed Dec 15 21:20:44 CET 1999

```John Conway wrote:

>On Tue, 14 Dec 1999, Antreas P. Hatzipolakis wrote:
>>
>> Does anyone know which is the sequence of the R. polygons which are
>> conic-constructible? (since right now I have not handy the Math. Int.
>> volumes to check)
>
>   I think it's obvious that the answer is those for which n has the form
>2^a.3^b.p.q.r....   ,  where  p,q,r,...  are distinct primes of the
>form  2^x.3^y + 1.   Yes, that IS obvious.

And I think that what is obvious for JHC is not obvious for everyone else! :-)

There is a theorem (by Vieta) which says:

(conic-constructibility) <=> (marked ruler constructibility: trisect an angle)

Now, for the (marked ruler constructibility) the formula is the one given
by Conway above.

So, the Conics sequence is: 7,9,13,14,18,19,21,26,27,28,35,36,37,38,39,
42,45,52,54,56,57,63,65,70,72,73,74,76,78,81,84,90,91,95,97,...........

<q>
I am sorry, but the terms
7,9,13,14,18,19,21,26,27,28,35,36,37,38,39
do not match anything in the table.
</q>

Note that the primes of the form 2^x * 3^y + 1 are called Pierpoint Primes
(A005109)

Note also that the "Greek Sequence" (2^a * 3^b * 5^c; where:
a = 0,1,2,3..., b,c in {0,1}, excluding the terms 1,2; that is:
(a,b,c) =/= (0,0,0), (1,0,0)):
3,4,5,6,8,10,12,15,16,20,24,30,32,40,48,60,64,80,96,120,144,160,.....

<q>
I am sorry, but the terms
3,4,5,6,8,10,12,15,16,20,24,30,32,40,48,60,64,80,96,120,144,160
do not match anything in the table.
</q>

Bibliography:
George E. Martin: Geometric Constructions.
New York etc: Springer, 1997, p. 140

Antreas

```