6
$\begingroup$

Pierpont proved that a regular $n$-gon is constructible by (singly) marked straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2}$ for $a_i \geq 0$ and $p_i = 2^{b_1} 3^{b_2} + 1 > 3$ is prime with $b_i \geq 0$.

It has been known since the time of Archimedes that a marked straightedge allows for angle trisection. Let a $q$-sector be an object which allows for angle $q$-section.

Does this result generalize to the following?

Let $q$ be a prime. A regular $n$-gon is constructible by $q$-sector, straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2} \cdots q^{a_m}$ for $a_i \geq 0$ and $p_i = 2^{b_1} 3^{b_2} \cdots q^{b_m} + 1 > q$ is prime with $b_i \geq 0$.

Update: Gleason's paper provides the complete answer for constructible $n$-gons. Here, it is shown that a regular $n$-gon is constructible by straightedge, compass and $p$-sector for each prime $p$ dividing $\varphi(n)$, the Euler totient of $n$.

Thus, I must modify my conjecture to the following:

Let $q$ be a prime. A regular $n$-gon is constructible by $\{ 3, 5, \dots, q \}$-sectors, straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2} \cdots q^{a_m}$ for $a_i \geq 0$ and $p_i = 2^{b_1} 3^{b_2} \cdots q^{b_m} + 1 > q$ is prime with $b_i \geq 0$.

One direction is certainly true by using the multiplicativity of the Euler totient function. The question is now whether the other direction also holds.

  • 3
    +1 for making me look up angle trisection with a marked ruler. For some reason I didn't know about this until today.2012-05-19

1 Answers 1

1

First of all, I'm not sure there is such a thing as a straightedge sufficiently marked to allow for, say, cutting an arbitrary angle into 5 pieces. But even if there is such a thing, I don't see how it would help you to construct, say, a regular 7-sided polygon. To construct that polygon, you need to construct solutions of an irreducible cubic, while the straightedge helps you construct solutions of an irreducible quintic. The fields have nought to do with each other.

  • 0
    @GerryMyerson you're dead on a 5-sector does not help with a regular 7-gon, but looking for a " straightedge sufficiently marked to allow for, say, cutting an arbitrary angle into 5 pieces" we have a marked straightedge and thus enough for the 7-gon construction. The edit really addresses this by using the set $\{3,5,...\}$-sectors.2017-06-29