constructible angles with integer values in degrees

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

∘

or

36

∘

).
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

x

2

,

x

4

,

x

8

,

…

,

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,

2

⁢

x

,

3

⁢

x

,

4

⁢

x

,

…

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

2

⁢

x

. Repeat the procedure and you get

3

⁢

x

,

4

⁢

x

,

…

.

Now, a theorem.

Theorem 1

The angle measuring

3

∘

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

∘

and

60

∘

.

By first lemma we can construct then

72

∘

,

72

∘

2

=

36

∘

,

36

∘

2

=

18

∘

,

18

∘

2

=

9

∘

,

9

∘

2

=

4.5

∘

=

4

∘

⁢

30

′

and also we can construct

60

∘

,

60

∘

2

=

30

∘

,

30

∘

2

=

15

∘

,

15

∘

2

=

7.5

∘

=

7

∘

⁢

30

′

But if we can construct

4

∘

⁢

30

′

and

7

∘

⁢

30

′

we can then construct their

difference

, which is exactly

3

∘

.

Alternative (J. Pahikkala): Since

72

∘

and

60

∘

can be constructed,

12

∘

=

72

∘

-

60

∘

can be also constructed. Bisecting

12

∘

gives

6

∘

and bisecting again shows that

3

∘

can be constructed.

Theorem 2

We can construct any angle measuring an integer multiple of

3

∘

.

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

∘

.

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

∘

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

3

⁢

m

-

t

⁢

n

=

1

(

n

or

m

could be

negative

).

By the second lemma, we can construct both

3

⁢

m

∘

and

t

⁢

n

∘

, so we can construct their sum (or difference), which would prove

1

∘

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

∘

cannot be constructed with ruler and compass, this contradicts what we just showed, and therefore only angles being an integer multiple of

3

∘

can be constructed.

Q.E.D.

For a more general proof for other real values besides integers, see the

theorem on constructible angles

.

Title

constructible angles with integer values in degrees

Canonical name

ConstructibleAnglesWithIntegerValuesInDegrees

Date of creation

2013-03-22 14:16:36

Last modified on

2013-03-22 14:16:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11S20

Classification

msc 11R32

Classification

msc 51M15

Classification

msc 13B05

Synonym

constructible angle

Related topic

ExactTrigonometryTables

Related topic

TheoremOnConstructibleAngles

Related topic

ClassicalProblemsOfConstructibility

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