The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics » geometry.puzzles

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Angle Trisection
Replies: 3   Last Post: Jan 13, 2005 4:53 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Earle Jones

Posts: 92
Registered: 12/6/04
Angle Trisection
Posted: May 25, 2002 11:15 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

I found this on sci.math and thought this group might be interested:
(Original posted by Sergei Markelov

It is well-known, that some angles (for instance, Pi/3) cannot be
trisected using the ruler and compass. However, some angles still can be
triseced. But which angles can be, and which cannot be trisected? I have
found no answer in literature.

My ideas are:

1. If cos(Alpha) is transcendental, then construction is impossible.
2. If Alpha/Pi is rational, Alpha=p/q*Pi, then Alpha cannot be trisected
if q=3k and can be trisected if q=3k+1 or q=3k+2 (see example with Pi/7
3. If cos(Alpha) is algebraic, but Alpha/Pi is irrational - I have
where Alpha can be triseced, and where Alpha cannot be trisected
(artan(11/2) can be trisected, arctan(1/2) cannot, see below).

I have the proof of (1), have some ideas about how to prove (2), and no
ideas about how to determine, whether the angle can be trisected in the
case (3).

Here are few examples of angles that can be triseced, but this is not

1. Pi/7 can be trisected since Pi/21 = Pi/3 - 2*Pi/7
2. arctan(11/2) can be trisected since arctan(11/2) / 3 = arctan(1/2)
3. arctan((1+3*2^(1/3))/5) can be trisected since
arctan((1+3*2^(1/3))/5) = 3*arctan(2^(1/3)-1)

Both these formulas can be proved using

tan(3*arctan(x)) = (3*x-x^3)/(1-3*x^2)

However, arctan(1/2) cannot be trisected, because minimal polynom for
tan(arctan(1/2)/3) is 2*x^3-3*x^2-6*x+1 and it is irreducible over Q.

Both arctan(1/2) and arctan(11/2) are incommensurably with Pi (I have
ideas, how to prove this), but first cannot be trisected, and second can

Any suggestions?

Thank you!

Sergei Markelov

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.