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Talk:Compass and straightedge (view source)
Revision as of 00:54, July 13, 2019
, 00:54, July 13, 2019→"Limp" compass: reply
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 [[joke]]s 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. --[[User:Dataclarifier|Dataclarifier]] ([[User talk:Dataclarifier|talk]]) 17:02, 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 [[joke]]s 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. --[[User:Dataclarifier|Dataclarifier]] ([[User talk: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.
:[[User:SamHB|SamHB]] ([[User talk:SamHB|talk]]) 20:54, 12 July 2019 (EDT)