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Geometric proof that angle trisection by straightedge and compass is impossible
- samstave 15y agoThis should be a vid on Khan Academy.
- Estragon 15y agoI don't know about Khan Academy, but a video of the rotations would definitely be cool.
- strmpnk 15y agoAt 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.
- xyzzyz 15y agoThe 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.
- lmkg 15y agoWell, 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 15y ago> 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
- sp332 15y agoAt 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.
- ColinWright 15y agoSquare roots are easy, cube roots are impossible.
- phoobahr 15y agoSee 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 15y agoIf I'm starting with a 40 degree angle, how do I construct an equilateral triangle on it?
- phoobahr 15y agoGiven angle(PQR) begin as you would when bisecting the angle: http://www.mathopenref.com/constbisectangle.html 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 http://www.mathopenref.com/constequilateral.html
- psykotic 15y agoThis 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 15y agoYou need rather more than just a topology before you can start talking about lines and circles!
- btilly 15y agoAny time I see this claim I want to bring up the fact that you can do it if the straightedge is marked in 2 places. See http://www.geom.uiuc.edu/docs/forum/angtri/ http://www.geom.uiuc.edu/docs/forum/angtri/ for a random link that explains the construction.
- monochromatic 15y agoThis is exactly the type of thing that has always made me think compass and straightedge constructions were even sillier than a lot of very obviously contrived mathematics. /math major
- psykotic 15y agoThat's an odd attitude for a math major to have. When geometers in ancient times thought mechanical curves were lesser second-rank objects, that was obviously a counterproductive prejudice that for a long time retarded the development of mathematics. But if you ask me, there is nothing "obviously contrived" about studying a configuration of lines and circles, their points of intersection, the lines and circles directly constructible from those intersections, and so on, ad infinitum. It strikes me as marvelously natural and beautiful and anything but contrived. That this corresponds to studying quadratic field extensions is a confirmation of that, if there were ever any serious doubt.
- monochromatic 15y agoWhat I was saying is that I've always thought it was a silly restriction to say that you can't put a couple of arbitrary marks on your straightedge. It's just a matter of taste, I guess. The connection between these constructions and field theory is interesting, but I derive no pleasure from compass and straightedge proofs.
- Someone 15y agoBut you _can_ put _arbitrary_ marks on your straightedge. I haven't seen proofs needing or even using them, but you are free to place a new point at an arbitrary distance from a point A. You can then use the compass to transfer that distance to any line segment. What you cannot do is put specific points on it, say three equidistant ones, or two at a distance of PI. If you allowed such constructs, the 'game' would become extraordinarily dull.