# Does there exist a constructible (by unmarked straightedge and compass) angle that cannot be quintsected?

I know that for example an angle of 20 degrees cannot be quintsected because an angle of 4 degree cannot be constructed (I'm thinking in terms of (unmarked) straightedge and compass. But an angle of 20 degrees cannot be constructed (as above) and I would be interested to see an example of a constructible angle that cannot be quintsected, assuming one exists.

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What about $60^\circ$? –  Berci Feb 4 '13 at 21:58
@Berci quintsection of $60^\circ$ is constructible because $12^\circ=72^\circ-60^\circ$ and $72^\circ$ and $60^\circ$ is constructible. –  tetori Feb 4 '13 at 22:17

The $72$ degree angle is constructible but the $\dfrac{72}{5}$ degree angle cannot be.

This follows from the nice theorem that a regular polygon of $n\ge 3$ sides can be constructed by straight edge and compass if and only if $n$ is of the shape $$2^e p_1p_2\cdots p_k,$$ where the $p_i$ are distinct Fermat primes.

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You is too quick for me, dude. –  Will Jagy Feb 4 '13 at 22:04
Thank you Andre'--yes that answers my question. But how about if we were to require that the resulting angle that can be constructed (after quintsecting) is a whole number degree angle? –  Elliot Benjamin Feb 4 '13 at 22:32
There is no integer $k$ such that the $k$-degree angle is not constructible, while the $5k$ degree angle is. This is basically because $180$ is not divisible by $5^2$. We get a whole degree example for trisection because $3^2$ divides $180$. –  André Nicolas Feb 4 '13 at 22:56
Thanks again Andre' for your quick response. I realized this soon after I asked the question, though the way I think of it is that an angle is constructible if and only it is a multiple of 3, and unlike trisecting, quintsecting doesn't give us the multiple of 3 that we would need. Can you tell me something about the case if we use a marked ruler--such as what is the criteria for being able to construct angles? And how about using both marked ruler and compass? And what about criteria for quintsecting angles (whole number and general) for marked ruler, and marked ruler and compass? Thanks! –  Elliot Benjamin Feb 5 '13 at 2:14
Sorry, I remember the marked ruler stuff for trisecting, but not for quinquisecting. There is a nice algebraic characterization of the numbers constructible by marked straightedge and compass. I forget the name of the book that does these things nicely, it is yellow. –  André Nicolas Feb 5 '13 at 2:22