On: Geometric proof that angle trisection by straightedge and compass is impossible

[strmpnk]

At least with a finite number of arcs and lines... true of anything I guess.

Very interesting as it might be for lovers of flat paper, I wonder if this is true of all topologies. I'd guess not but I'm not mathematician.

[lmkg → strmpnk]

Well, the first problem is defining "line" and "circle" in your new topology, and this in itself tends to complicated not just the answer, but the question. For example, on the surface of a sphere, "lines" are a special case of circles, and all circles have two possible center points with two radii, which are distinct iff the circle is not a line. Once you've defined circles and lines, constructable sets looks very different. For example, you can always construct an absolute measure of distance, but the concept of parallel line segments doesn't exist, although you can have circles that are parallel to lines.

So can you trisect an angle in this slightly exotic topology? There's a good chance that you can, actually. There's a construction in the plane for trisecting a segment. If that construction is still valid on the sphere, you can take your angle and re-construct the legs with length equal to 1/4 of the sphere's circumference. Now, the line segment connecting the endpoints of these legs is an arc of a circle whose center is your angle. Trisecting this segment, and connecting the trisection back to the angle, will trisect the angle.

[lloeki → lmkg]

> all circles have two possible center points with two radii

Reminded me of those p-adic sets where given the |a-b| metric:

- a circle radius is equal to its diameter

- all points inside the circle are a centre of that circle

- all triangles are isosceles

- appending a segment to itself may actually make it shorter

[xyzzyz → strmpnk]

The notion of "angle" does not really make much sense in topology. However, one could ask this question in context of inner product spaces where one can define an angle in a natural way, but all real 2-dimensional inner product spaces are essentially all alike, so the answer is negative as well.

[sp332 → strmpnk]

At least with a finite number of arcs and lines... true of anything I guess.

Actually, there are a lot of geometrical constructions that are possible with a finite number of operations with just a straightedge (not a ruler) and compass. Exactly bisecting an angle is easy. Trisecting is impossible.

[phoobahr → sp332]

See here I am a little confused. I never really understood why one couldn't do this:

construct an equilateral triangle atop the angle to be trisected bisect any side, construct a new triangle with said 30deg angle giving a line segment 1/sqrt(3) of the original eq tri bisect the new 60deg angle, constructing a new 30/60/90 tri

voila, 1/[sqrt(3)sqrt(3)] == 1/3. You have trisected a line with a compass and unmarked straightedge

[sp332 → phoobahr]

If I'm starting with a 40 degree angle, how do I construct an equilateral triangle on it?

[phoobahr → sp332]

Given angle(PQR) begin as you would when bisecting the angle: http://www.mathopenref.com/constbisectangle.html.

Your arc intersections with |PQ| and |RQ| (call them p & q) give you an isosceles triangle pQr.

Construct an equilateral triangle on |pr| similar to: http://www.mathopenref.com/constequilateral.html

[ColinWright → sp332]

Square roots are easy, cube roots are impossible.

[psykotic → strmpnk]

This classical result holds in some generality, e.g. you also cannot trisect every angle with straightedge and compass in the hyperbolic plane (an exercise in Hartshorne's book Companion to Euclid).

[Sniffnoy → strmpnk]

You need rather more than just a topology before you can start talking about lines and circles!