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# constructible angles with integer values in degrees

The aim is to characterize all constructible angles with straightedge and compass whose value is an integer number of degrees (like $60^{\circ}$ or $36^{\circ}$). From now on, every time we refer to the measurement of an angle, it is meant to be in degrees, not radians.

We need two short lemmas:

###### Lemma 1.

If an angle measuring $x$ degrees can be constructed, then angles measuring

$\frac{x}{2},\frac{x}{4},\frac{x}{8},\ldots,\frac{x}{2^{k}}$ |

can be constructed.

Notice that we are not stating all of them have integer values, only constructibility. The proof follows almost inmediately by knowing any angle can be bisected with ruler and compass.

###### Lemma 2.

If an angle measuring $x$ degrees can be constructed, then angles measuring any integer multiple of $x$, that is, $2x,3x,4x,\ldots$ can be constructed

If you can construct $x$, you can construct again an adjacent angle with the same value and you will have constructed an angle measuring $2x$. Repeat the procedure and you get $3x,4x,\ldots$.

Now, a theorem.

###### Theorem 1.

The angle measuring $3^{\circ}$ can be constructed.

It is well known that both regular pentagon and equilateral triangle can be built with ruler and compass. That allows us to construct angles measuring $72^{\circ}$ and $60^{\circ}$.

By first lemma we can construct then

$72^{\circ},\frac{72^{\circ}}{2}=36^{\circ},\frac{36^{\circ}}{2}=18^{\circ},% \frac{18^{\circ}}{2}=9^{\circ},\ \frac{9^{\circ}}{2}=4.5^{\circ}=4^{\circ}\,30% ^{{\prime}}$ |

and also we can construct

$60^{\circ},\frac{60^{\circ}}{2}=30^{\circ},\frac{30^{\circ}}{2}=15^{\circ},% \frac{15^{\circ}}{2}=7.5^{\circ}=7^{\circ}\,30^{{\prime}}$ |

But if we can construct $4^{\circ}\ 30^{{\prime}}$ and $7^{\circ}\ 30^{{\prime}}$ we can then construct their difference, which is exactly $3^{\circ}$.

Alternative (J. Pahikkala): Since $72^{\circ}$ and $60^{\circ}$ can be constructed, $12^{\circ}=72^{\circ}-60^{\circ}$ can be also constructed. Bisecting $12^{\circ}$ gives $6^{\circ}$ and bisecting again shows that $3^{\circ}$ can be constructed.

###### Theorem 2.

We can construct any angle measuring an integer multiple of $3^{\circ}$.

The proof follows directly from the second Lemma.

###### Theorem 3.

The only constructible angles measuring an integer number of degrees are precisely the multiples of $3^{\circ}$.

We are only left to prove we cannot construct any other integer value. We will work by contradiction.

Suppose we are able to construct with ruler and compass an angle measuring $t^{\circ}$ with $t$ integer and $t$ not multiple of $3$.

Since $3$ does not divide $t$ and $3$ is prime, it follows that $3$ and $t$ are coprime, that is, $\gcd(3,t)=1$.

But then, by Euclid’s algorithm we can find integers $m,n$ so that $3m-tn=1$ ($n$ or $m$ could be negative).

By the second lemma, we can construct both $3m^{\circ}$ and $tn^{\circ}$, so we can construct their sum (or difference), which would prove $1^{\circ}$ is constructible, and therefore any angle equal to an integer number of degrees could be constructed with ruler and compass.

However, the standard proof of the impossibility of trisecting an arbitrary angle goes by proving $20^{\circ}$ cannot be constructed with ruler and compass, this contradicts what we just showed, and therefore only angles being an integer multiple of $3^{\circ}$ can be constructed.

Q.E.D.

For a more general proof for other real values besides integers, see the theorem on constructible angles.

## Mathematics Subject Classification

11S20*no label found*11R32

*no label found*51M15

*no label found*13B05

*no label found*

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## Comments

## very nice, but

where is the proof that the angle of 20 degrees can not be constructed?

## Re: very nice, but

in pretty much any galois theory book

ok a nicer reply:

the proof relies on 20Âº not being constructible.

This is a corollary from the proof that

60Âº can't be trisected

which is the STANDARD proof that

the trisection of angles is impossible

so.. I'm relying in a known fact

everytime yo prove something you take some things for granted

I could add it, but it has a completely different

(and non elementary) context, so I left it out as known fact

And realize this is always done, If I had written the actual proof

would you request to add inside the proof to the galois theory results

it uses? and then the proof the the results used on those proof and ad infinitum?

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: very nice, but

Well, it really would be nice if "trisection of the angle is impossible" were an entry in PM, possibly along with doubling the cube and squaring the circle; all three could be examples somethere in Galois-theory-land.

As for needing proofs for all the Galois theory results, that's a problem for the entries on the actual results --- but it makes sense to hope that those results actually have entries.

## Re: very nice, but

Yes, that's what I meant

The style and techniques used in proving the trisection of angle don't go very well with this entry, I try to keep things in a n elementary level as possible

So yes, a "trisecting angle" entry should be added (with sysnonyms mentioning impossibility at all, it could even become a topic and then we could attach this entry and several others to it

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f