October 2011 - mathfun@mailman.xmission.com

[math-fun] Elementary triangle puzzle
by Dan Asimov 17 Nov '11

17 Nov '11

Given an arbitrary triangle T in R^2, characterize p in R^2 such that the sum of distances S(p) := d(p, L_1) + d(p, L_2) + d(p, L_3) is minimized, where the L_j are the affine lines containing the sides of T. --Dan "Things are seldom what they seem." --W.S. Gilbert

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[math-fun] Question about amicable numbers
by David Wilson 08 Nov '11

08 Nov '11

Is there an upper bound to the ratio of amicable numbers? To the ratio of maximum and minimum of an amicable chain?

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[math-fun] Tetrahedral altitudes
by Fred lunnon 03 Nov '11

03 Nov '11

Richard's paper mentions mentions that the 4 altitude lines of a tetrahedron, through vertices and perpendicular to faces, do not necessarily concur in a common point. A long time ago I came across a charmingly antiquated tome: Nathan Altshiller-Court "Modern pure solid geometry" (1935) which I seem to remember stated the theorem that the altitudes always lie on a quadric; though his notation --- undefined --- for this property was something confusingly obscure. There were several other theorems of a similar nature. Can anybody confirm this? Has any further work in this direction taken place more recently? WFL On 10/29/11, Richard Guy <rkg(a)cpsc.ucalgary.ca> wrote: > ... I've just submitted a paper, ``Five-point > circles, the 76-point sphere, and the Pavillet tetrahedron'' > to the Monthly. The theme is that triangle geometry is not > dead. I would attach a copy if attachments were allowed; I > will entertain individual requests. ...

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Re: [math-fun] Elementary triangle puzzle
by Dan Asimov 30 Oct '11

30 Oct '11

Yes, that's the last case: any point p on the base of an isosceles triangle with apex < pi/3 equally minimizes the sum of distances from p to the sides of the triangle. I like Fred's coinage of inceeding. I would also like to coin words for each connected component of non-equilateral isosceles triangles: Let the elongated equilaterals be called "sharp" and the squashed ones be called "flat". --Dan Fred wrote: << For isosceles triangles T with apical angle inceeding pi/3 --- exasperated at having no contrapositive for "exceeding", I hereby coin one --- any point along the base serves as centre p. For collinear T (with area zero) any point on the line serves as p. >> "Things are seldom what they seem." --W.S. Gilbert

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[math-fun] the sextic for Cantrell's 5 circle packing
by Bill Gosper 30 Oct '11

30 Oct '11

maximizing sum(radii), http://www2.stetson.edu/~efriedma/cirRcir/5.gif, namely Root[-87131 + 32779 #1 + 18567 #1^2 + 10931 #1^3 - 14350 #1^4 + 2538 #1^5 + 162 #1^6 &, 2], is unsolvable in radicals, according to my (Ssexy) solver. --rwg

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Re: [math-fun] Tetrahedral altitudes
by Dan Asimov 30 Oct '11

30 Oct '11

Another way to see what Joshua seemed to recall, besides by the theorem that Fred quoted elsewhere, is via the ----- Lemma: Given a nondegenerate tetrahedron T, two of its altitudes (to, say, apices A and B) are intersecting <=> the edge AB of T, and its opposite edge, are perpendicular. (Here "perpendicular" means the two skew lines containing these two edges -- after orthogonal projection to any plane parallel to both of them -- are perpendicular. Or equivalently the isometry group of the union of these skew lines is of size 8.) Proof: Left to the reader. ----- Since the perpendicularity condition is symmetrical with respect to the two edges involved, it must imply that the other two altitudes intersect each other as well. --Dan Joshua wrote: << Fred wrote: << Richard's paper mentions mentions that the 4 altitude lines of a tetrahedron, through vertices and perpendicular to faces, do not necessarily concur in a common point. >> I seem to recall that if one pair of altitudes is intersecting, then the other pair is as well. Does anyone know an easy proof or counterexample? >> >> "Things are seldom what they seem." --W.S. Gilbert

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Re: [math-fun] math-fun Digest, Vol 104, Issue 16 re Drilling
by Stuart Anderson 30 Oct '11

30 Oct '11

If learning involves wiring changes in the brain perhaps the question is what is the best learning environment for facilitating brain changes? http://www.smh.com.au/tv/show/brain-fitness/brain-fitness-for-kids-20111028… Stuart > > Kerry> > > I'm a bit of two minds about drilling. I get how over-drilling can > kill any creative spark a student may have for the beauty of math. > Having taught it at the undergraduate level (which too often means, > "at the middle-school/high-school level"), I've had to endure the > scowls and sneers of those who didn't care a lick about math but had > to do their calculations anyway, because that was part of the > syllabus. However, I can also see how it certain facility with > numbers and arithmetic operations is necessary to even begin to deal > with math at anything more than a pure computational level. I had a > buddy who also taught and was quite proficient in martial arts. In > his class, he approached arithmetic drills as he would repetitive > practice/warm-up in the martial arts context--do something relatively > mundane so often that it becomes automatic, then move on to more > sophisticated stuff. > > If there is an answer, it's probably the same one that it's always > been--adjust the material with the level and goals of the student in > mind. > > Kerry > -- lkmitch(a)gmail.com > <http://gosper.org/webmail/src/compose.php?send_to=lkmitch%40gmail.com>www.kerrymitchellart.com > >

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Re: [math-fun] Elementary triangle puzzle
by Dan Asimov 30 Oct '11

30 Oct '11

That is almost completely correct, but there is still one case to consider. --Dan Richard wrote: << Is the following the answer? If the triangle is equilateral, then all points inside the triangle serve (and all points outside if you assign signs to the distances). Otherwise the point is at the vertex with the largest angle. R. I wrote: << Given an arbitrary triangle T in R^2, characterize p in R^2 such that the sum of distances S(p) := d(p, L_1) + d(p, L_2) + d(p, L_3) is minimized, where the L_j are the affine lines containing the sides of T. >> >> "Things are seldom what they seem." --W.S. Gilbert

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Re: [math-fun] grade school/jr high arithmetic
by Bill Gosper 29 Oct '11

29 Oct '11

Kerry> I'm a bit of two minds about drilling. I get how over-drilling can kill any creative spark a student may have for the beauty of math. Having taught it at the undergraduate level (which too often means, "at the middle-school/high-school level"), I've had to endure the scowls and sneers of those who didn't care a lick about math but had to do their calculations anyway, because that was part of the syllabus. However, I can also see how it certain facility with numbers and arithmetic operations is necessary to even begin to deal with math at anything more than a pure computational level. I had a buddy who also taught and was quite proficient in martial arts. In his class, he approached arithmetic drills as he would repetitive practice/warm-up in the martial arts context--do something relatively mundane so often that it becomes automatic, then move on to more sophisticated stuff. If there is an answer, it's probably the same one that it's always been--adjust the material with the level and goals of the student in mind. Kerry -- lkmitch(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=lkmitch%40gmail.com>www.kerrymitchellart.com --------------- Indeed, everybody's different. I think I benefited from drilling. Why wasn't I bored? Maybe I could fell my skills improving. Maybe I sensed a challenge in successive examples becoming trickier or otherwise harder. Maybe there was a pattern to the answers suggesting a little theorem/general formula. The latter seems like a cool way to motivate lots of concealed drilling: Give sequences of special cases of little identities whose pattern the student should eventually guess. --rwg And where there is no pattern, make the last few examples extra credit, with steeply rising difficulty. And yes, Richard, may I have a copy of your triangles paper, too?

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Re: [math-fun] grade school/jr high arithmetic
by Henry Baker 29 Oct '11

29 Oct '11

The only rule students need to know about "mixed fractions" is how to interpret them: "9 1/2 weeks" means "(9+(1/2)) weeks"; "-8 3/4 inches" means "-(8+(3/4)) inches", and how to produce them: "+-(n/d)" becomes "+-(floor(n/d)+(mod(n,d)/d))". Thus, if you see a fraction "n/d", you type n/d into your calculator, followed by "+/-" if there is a "-" sign in front. If you get a mixed fraction "i n/d", you type the following sequence into your calculator: n/d+i, followed by "+/-" if there is a "-" sign in front. Everything else is a massive waste of time, until _after_ the student has learned enough algebra to start learning a little number theory. There's an appropriate time to learn about putting fractions into "common denominators", but it isn't until considerably later. At the appropriate time, what would have been months of torture, then becomes one or two lessons, and is then immediately obvious. Here's a bad analogy: How many high school (American) football players would there be if each one had to be able to kick a 20 yard field goal prior to playing his first game? While kicking field goals is an important & valuable capability for a football team, hardly anyone can do it, and eventually someone specializes in this task. At 04:28 PM 10/28/2011, Thane Plambeck wrote: >People write mixed fractions all the time: eg 9 1/2 weeks or whatever. > >I've had a couple of my own kids run this gauntlet at school and I just told >them to immediately convert them to what I've taken to calling "fractions >for grown ups" as a way of disparaging them and motivating my kids to not >attempt to manipulate them in math homework, but to say that no one on Earth >actually uses them is overstating the case in my opinion. > >On Fri, Oct 28, 2011 at 3:20 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> I've been going through the Khan Academy math exercises to see what math >> kids are learning. >> >> www.khanacademy.org >> >> While I am very impressed with the Khan Academy & its exercises, I am >> appalled by the time & effort in teaching fractions -- especially "mixed >> fractions". >> >> I realize that this is not the Khan Academy's fault -- KA is merely >> providing exercises to teach for the tests that the kids will have to pass. >> >> But who on Earth actually uses this stuff & why do we torture kids this >> way? >> >> No wonder people hate math. >> >> --- >> >> I think that the ability to do rudimentary programming is far more valuable >> to every kid than the ability to handle "mixed fractions". >> >> In fact, the ability to sum/difference numerical fractions at all is simply >> not terribly relevant. Ditto for greatest common divisor and least common >> multiple. (Truth time: when was the last time you used "least common >> multiple", except on a test?) Yes, there is a time for "number theory" & >> gcd's, but it should come considerably later, in conjunction with algebra & >> groups, rather than with basic arithmetic. Yes, there is a place for >> summing fractions: when simplifying algebraic expressions, but not in basic >> arithmetic. >> >> --- >> >> The hours in every kid's life are valuable, and the hours in every >> teacher's day are even more valuable. Why do we want to waste these hours >> teaching them skills that bore them out of their minds, and that they will >> never use? >> >> If there is an educational crisis, it would appear to be caused by clueless >> adults. > >-- >Thane Plambeck >tplambeck(a)gmail.com >http://counterwave.com/

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