Re: Trisecting an arbitrary angle

On 19 Jul 2005 12:34:29 -0700, "Proginoskes"
<proginoskes@xxxxxxxxxxxxx> wrote:

>
>quasi wrote:
>> On Tue, 19 Jul 2005 12:37:14 +0200, "Jutta Gut"
>> <gut.jutta.gerhard@xxxxxxxxx> wrote:
>> >
>> >If I understand correctly, you have shown that in an triangle with
>> >the sides a^3 , a*(b^2-a^2) , b*(b^2-2*a^2) one angle is three times
>> >another one.
>> >
>> >The more interesting question would be: given an angle, how to
>> >construct a triangle with the sides a^3 , a*(b^2-a^2) , b*(b^2-2*a^2)
>> >and the given angle?
>> >
>> >Jutta
>>
>> Contructing the triangle effectively trisects the given angle which in
>> general has been shown to not be possible using straight edge and
>> compass. There are only countably many angles for which trisection by
>> straight edge and compass is possible.
>
>The OP isn't requiring that an arbitrary angle be trisected.
>
>Maybe, during the construction process, the smaller angle will be
>tripled, which _is_ allowed using straightedge and compass.
>
> --- Christopher Heckman

Tripling an angle is a rather trivial construction (just concatenate 3
copies of the angle).

The formulas discussed parametrize triangles with the property that
one angle is a triple of another. They don't really lead to an
algorithm for trosecting an arbitrary angle with straight edge and
compass, and as we all know, such an algorithm is not possible. Of
course, some angles can be constructibly trisected and in fact all
such angles can be trisected by standard methods simply constructing
the length cos(theta/3) and using that to contruct theta/3. Thus,
these paramerization formulas give no new trisectable angles -- we
have them all already.

Also, there is nothing in the parametrization formulas that even uses
an angle as input. Rather it goes the other way -- the inputs are
parameters, the ouput is a triangle (with associated lengths and
angles).

That doesn't mean the formulas are uninteresting, just that they are
not useful as a tool for trisecting angles.

quasi
.

Relevant Pages

  • The Philosophical Physicist, Statement No. 3
    ... that of geometry is discussed. ... geometer's angle related. ... And the means of analogy between mathematics ... And when the triangle appears nonstandard ...
    (sci.physics)
  • Re: Hawking and distance of stars
    ... >>> stretch the top up the triangle as long as you want to. ... >>> compass or a triangle and try to determine the angle. ... The instruments used by astronomers are able to resolve to 1 milil arc ... it says the measurements were made. ...
    (talk.origins)
  • Re: Towards a Formula for Primes
    ... triangle, its perimeter matches that of the triangle. ... So we take the coprime pair 3:2 and get the coprime triple 8:15:17 ... E a point on line appears a two equal triangle angle vertex. ... For ca>Ec as line segment magnitudes segment Ec defines the unit ...
    (sci.math)
  • Re: Sumito
    ... Do you really think you can learn everything without a teacher? ... The Triangle - have significantly and abruptly improved my game. ... "Angle it out about 45 degrees. ... that, but every time they do a leg takedown, they are doing at least ...
    (rec.martial-arts)
  • Re: trigonometric functions
    ... What's shown in the picture is how one extends the trig functions to ... Note that the angle shown is ... by the right triangle in terms of point P? ... This then determines the sign of the trig function within each quadrant ...
    (sci.math)