Talk:Compass and straightedge
Perhaps my deletion of this article was hasty, but it didn't seem well-written and I thought it contained deliberately inserted errors.
I'll give it another week or two, if Sam or anyone wants to fix it up.
- Greetings, Ed. Your comment about making it understandable to people who completely skipped plane geometry certainly raises the bar in terms of the amount of work I have to do. I accept the challenge, but it will take a while. By those standards, you're absolutely right in that it isn't well written. But at least it no longer contains deliberately inserted errors. BTW, the article goes off in more directions than I had remembered. But I accept the challenge. (Also, didn't you say somewhere that you skipped plane geometry? That raises the bar even further.) SamHB 15:56, 13 November 2010 (EST)
- I could, but I don't have upload rights. The "construct a square root" problem is very nice. SamHB (talk) 20:57, 12 July 2019 (EDT)
You did not establish that the ancients had "no adjustable compass" that can "remember" the radius of a circle or be held in a stable position of adjustment by the hand. In the "Rules of the Game" you direct the player to adjust the compass and implicitly assume that it cannot be held steady by the hand, so that it collapses: "you can place the point of the compass on one of them [one of two points], adjust it so that the pencil touches the other, and draw a circle." If the compass can be so held without altering the span of distance between point and pencil (stylus) so that it can describe a perfect circle with unalterable stability during the drawing, then after drawing the circle it can "remember" the radius by the stabilizing hand of the operator, and draw another circle of identical diameter. If the ancients did use adjustable compasses, and certainly were able to hold them steady, then the restriction that a compass is so "limp" that it cannot be firmly adjusted, and is not to be held stable by the hand, has introduced a difficulty the ancients did not have, and is a "cheat", deliberately designed to guarantee obstructing solution of the puzzle by intentionally making it impossible to solve (a mathematical form of the straw man fallacy and Reductio Ad Absurdum). Also, I really must reiterate together in agreement with you the fact that such puzzles were and are indeed forms of mathematical amusement, but such jokes of fun do not ipso facto make such puzzles and exercises a joke, even if they are presented as one, or a mockery of serious mathematics making it into a humorless joke. While mathematical geometry is serious and no joke, it should also be fun. --Dataclarifier (talk) 17:02, 12 July 2019 (EDT)
- I didn't say whether it was within the technology of the time for the ancient Greeks to have a compass that can "remember". But the lack of a compass that can "remember" is simply one of the rules of the game. That's the way it was taught to me in 8th grade, and the way all mathematicians understand it. I suggest that you read the Dunham book that I mentioned in the "squaring the circle" talk page. He explains it all very clearly.
- I've seen a diagram of a special device, a sort of modified carpenter's square, that can precisely trisect an angle. But it's out of the rules of the game.
- The lack of memory in a compass leads to the theorem that the only constructible lengths are those that are algebraic, with a minimal polynomial that has a degree that is a power of two. This is a profound theorem, which wasn't proven until the 19th century.
- This has nothing to do with fallacies, "straw man" or otherwise.