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That'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
by psykotic 15y ago
That'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.
- btilly 15y agoNo, the rules specify "unmarked straightedge". And marking the straightedge is not allowed. If you were allowed to mark the straightedge then you could put 2 random marks on it. With 2 random marks, you can trisect any angle. The location of the marks doesn't matter, just their existence.
- Someone 15y agoThanks for the pointer. It led me easily to http://planetmath.org/encyclopedia/TrisectionOfAngle.html http://planetmath.org/encyclopedia/TrisectionOfAngle.html. Step 3 in that construction is not one I had thought of as allowed. It also, to me, isn't evident to me how one can proof that that Neusis construction is possible to place the ruler in that way without resorting to analytical means. I guess somebody must have thought about that. Are you aware of more rigid geometric proofs?