posted
I've got to know how you bisect an angle without being able to duplicate a length.
Also, how doing a construction with a fixed compass is "wrong." I've never seen a "collapsing" compass.
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posted
The rules we had that a compass could only measure lengths from a fixed point, or a fixed point that could be gotten to by swinging the compass around.
quote:In many commonly accepted constructions (e.g., congruent angles), the compass radius is set by the distance between two points, and then the compass is centered on some third point, elsewhere on the drawing. Under the strictest traditional rules (thank the Greeks again), this was not allowed. The radius was established only by the distance from the center point to some second fixed point. It was assumed that the moment the compass point was lifted, the instrument collapsed, making it impossible to retain the radius.
The constructions in this site ignore the collapsing compass rule. Given two points for the radius and a center point in the same plane, however distant, it is possible to draw the arc, even under the old rules. The construction for transferring the radius is elementary, but rather tedious. There is no clear purpose for the rule other than torturing students. Modern geometry classes can accomplish that with fifty-pound hall passes.
The bold portion is true, and it follows from it that any construction that can be done with a non-collapsing compass can be done with a collapsing one. So that answers your first question.
Were I teaching students this, I would make them do some transfer the radius constructions at the beginning of the year, but later let them lift the compass. But them, I'm a programmer and very comfortable with named subroutines.
quote:I've got to know how you bisect an angle without being able to duplicate a length.
"Duplicating" a length is a bit unclear. The construction I will outline for bisecting an angle does use a compass to measure, in a way, but does not "transplant" that measurement, if I'm making sense.
Put the compass point on the vertex of the angle and open it to any given radius, shorter than the legs of the angle. Make marks on the angle that are equidistant from the vertex with the pencil end of the compass. Put the point of the compass on one of the marks just made, and the pencil end on the other. Draw an arc in the interior of the angle. Now repeat that step, reversing the point and the pencil end and draw another arc. Your arcs should intersect at two points. A ray (drawn with the straightedge) connecting the vertex to those two points will be the bisector of the angle.
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Yeah, I'm comfortable with non-collapsing compasses, but I prefer unmarked straightedges (which can be purchased, but student IDs do the job for free).
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posted
BTW, WRT Mr. Carlin... he has become very anti-Catholic... he used to be witty but some of his more recent routines ("I used to be Catholic... until I reached the age of REASON [emphasis his]") have become mere vitriol and I found them highly offensive... which is sad because I continue to find him a funny and interesting performer (but I also liked "Jersey Girl" so say what you will about my taste).
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To add on to the straight-edge versus ruler thing: a long time ago, it was common to teach kids in high school geometry class to do constructions.
Constructions aren't de rigeur in HS geometry anymore? They sure are at the high school I attended and where I teach (the same textbook is used in both places)!
Do they not teach proofs either?!
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posted
It seems that proofs are predominantly the domain of honors classes. Another alternative that has become common is to accept Sketchpad demonstrations as "proofs."
I love Sketchpad. It's a great application. But a demonstration is not a proof.
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posted
So I've been e-mailing back and forth with this guys, he's been toeing the line, but just barely the whole time; last time he opened up some anti-Mormon quotes, to which I said I would not respond. Well with my deadline breathing down his neck he's opened the whole can on me and I have to say, I'm not really happy about the whole thing.
Every stereotype every created about calling Mormons to repentance and getting them to amend their ways has just been e-mailed to me in a rather vitriolic letter, proclaiming hell-fire, proclaiming me to be a terrible person. Anyone remember that thread I posted about feeling bad, bad-mouthing someone in class? Long time ago, but that came up too (he was in the class).
Wow, I haven't vented on Hatrack in a while. Sigh. Hmm, I guess I'm upset, I guess I should get over that huh?
posted
Hmm, it's weird, I never thought this stuff would effect me, and it does tend to be very difficult to get me angry, but I kind of was! I kind of still am! Was it that he was hitting at something so important to me? Am I less stable, with the stress of leaving two, wonderful parents who can't understand why I'm leaving? I don't know, but I'm waiting on writing back until I can be a little more civil.
posted
Well, I got out my "Euclidean and Non-Euclidean Geometry" book, and indeed it says that Euclid assumed that a compass should be collapsible. I guess my teacher never assigned that one.
But it doesn't show how to "transfer the radius." I'm not sure I really understand what that means. (a picture would help. Hah!)
However, I thought this was kind of interesting:
"The Danish mathematician G. Mohr and the Italian Mathematician L. Mascheroni dicovered independently that all Euclidean constructions of points can be made with a compass alone. A line, of course, cannot be drawn with a compass, but it can be determined by constructing two points lying on it. In this sense, the straightedge is unnecessary.
On the other hand, The German J. Steiner and the Frenchman J.V.Poncelet showed that all Euclidean constructions can be carried out with a straightedge alone if we are first given a single circle and its center."
So if you really want to get picky (or sadistic) you can say that constructions should be done with a straightedge or a compass, but not both, or you're doing it wrong.
I shouldn't be up at 2:13 am.
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posted
Hobbes, I'm very sorry to hear that. I hope you can find it in you to respond with calm but strong words.
Since he's such a devout Christian, maybe you could ask him where in Christ's ministry does he find the example of Jesus specifically telling anyone that they are going to hell? Maybe remind him that God is our judge and you feel right with God.
Anyway, I don't really have any advice on how to "fix" this situation, but it sure does seem like a sad way to end a friendship.
posted
rivka, we did proofs in my geometry class if that makes you feel ANY better. actually...geometry was the only math subject that I was really good at.
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Icky, it depresses sometimes how right you always are. Thanks.
Ela, I think my response was based on my misunderstanind of what you were saying. I'm sorry for misunderstanding.
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quote:quid...was that a televangelist or a televised church service from a recognized denomination? Sounds weird either way, but I'd tend more to believe that some whack-job televangelist would spew that nonsense than I would believe that a program sponsored by a particular denomination would. Although I'm not really "up" on religious broadcasting these days.
As I recall, it was a religious program that aired either five or seven days a week in southern Manitoba (could have been more areas than that, but that's all I know) back in the early 1980s. The programming was most likely for the Mennonite audience (thousands and thousands of Mennonites in southern Manitoba), although it possibly could have been Baptist (there are many similar beliefs, and when a Mennonite either can't find a Mennonite church or decides to change religions, it's frequently to a Baptist church they go.) The host had a guest speaker on that day, and he was not a preacher (reverend, whatever the title is) but someone higher up in the food chain of that church (like I said, I believe it was Mennonite. I mean, my grandparents were watching it, and they're firmly Mennonite.) - so possibly a representative from the Mennonite Central Committee (which oversees all churches and doctrine within the MCC congregations) or Mennonite Brethren (same thing).
As for the preacher at the local church, it happened many times that a sermon was given on the evils of Mormons. Perhaps 3-6 times from the time I was a child until I was sixteen. (I used to visit the grandparents most summers.) And that was the local preacher, not a substitute as far as I'm aware.
And as I think about it, I wouldn't put my grandparents past mentioning to the reverend that their Mormon grandchild was visiting and would be coming to church, so it's entirely possible that it was geared specifically towards us Mormons who would be in the congregation that week. (My grandparents also had a LOT of anti-Mormon literature at home - dozens of books, pamphlets, and the like.)
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posted
Well, yeah. My grandparents were not, shall we say, the least bit open-minded. My entire childhood, they'd talk about being saved, and they'd ask all the grandchildren who hadn't already told them they'd been saved if they've been saved yet. They just didn't understand that, as far as I'm concerned, being saved isn't a matter of saying "I believe in Jesus Christ and I accept him as my Savior." So as far as they were concerned, I was going to burn in hell.
It didn't matter that some of the "saved" grandchildren did drugs, were alcoholic, had children out of wedlock, broke the law, or whatever. They were automatically better than me, simply because they uttered those magical words, and my grandparents made sure I knew it, just like they made sure that I knew their opinions on my religious beliefs.
Eh. Whatever.
Now I'm married to a Muslim, and I wonder what my grandmother thinks. It must have caused her a great deal of anxiety and stress, to think a grandchild of hers could stray so far as to marry so far away from Christianity. Although my grandmother has mellowed quite a bit (my grandfather is now dead), she still holds on to some of this stuff. And yes, I mock her, a little, and I'm intolerant of her intolerance, and yet, I still love her. She's much better now than she used to be.
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Constructions aren't de rigeur in HS geometry anymore? They sure are at the high school I attended and where I teach (the same textbook is used in both places)!![[Roll Eyes]](rolleyes.gif)
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She's much better now than she used to be.
