请输入您要查询的字词:

 

单词 ConstructibleAnglesWithIntegerValuesInDegrees
释义

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

x2,x4,x8,,x2k

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 multipleMathworldPlanetmath of x, that is, 2x,3x,4x, can be constructed

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


Now, a theorem.

Theorem 1

The angle measuring 3 can be constructed.

It is well known that both regular pentagon and equilateral triangleMathworldPlanetmath can be built with ruler and compass. That allows us to construct angles measuring 72 and 60.

By first lemma we can construct then

72,722=36,362=18,182=9,92=4.5=4 30

and also we can construct

60,602=30,302=15,152=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 coprimeMathworldPlanetmath, that is, gcd(3,t)=1.

But then, by Euclid’s algorithm we can find integers m,n so that3m-tn=1 (n or m could be negative).

By the second lemma, we can construct both 3m and tn, 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.

Titleconstructible angles with integer values in degrees
Canonical nameConstructibleAnglesWithIntegerValuesInDegrees
Date of creation2013-03-22 14:16:36
Last modified on2013-03-22 14:16:36
OwnerPrimeFan (13766)
Last modified byPrimeFan (13766)
Numerical id9
AuthorPrimeFan (13766)
Entry typeTheorem
Classificationmsc 11S20
Classificationmsc 11R32
Classificationmsc 51M15
Classificationmsc 13B05
Synonymconstructible angle
Related topicExactTrigonometryTables
Related topicTheoremOnConstructibleAngles
Related topicClassicalProblemsOfConstructibility
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/3 0:48:07