August 2010 - mathfun@mailman.xmission.com

[math-fun] Isodiametric problem
by victor miller 16 Sep '10

16 Sep '10

A few days ago, at the AMS meeting, I saw a neat talk by Michael Mossinghoff about "isodiametic" problems -- e.g. Fix the diameter of a conv ex n-gon and ask for the ones with maximum perimeter. He reduced the problem to finding univariate polynomials whose non-zero coefficients alternate between +/- 1 and are divisible by a particular cyclotomic polynomial.. He was able to calculate all such up to about 500. Each such polynomial is compactly encoded by th length of the gaps between the non-zero terms. For small n almost all were periodic. He showed how a certain family of periodic constrctions give solutions, but as n got bigger the "sporadic" ones swamped them. He conjectued that these are eventually all of them. He had lots of pictures. The paper is not available yet. Victor

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[math-fun] Ever seen a spacefilled pentagon?
by Bill Gosper 10 Sep '10

10 Sep '10

Yesterday I told tutor Julian an idea for spacefilling a pentagon, and he said I could just conformally map Peano's filled square onto whatever, but then he got into it and helped make www.tweedledum.com/rwg/pentagonfill.pdf . It self-contacts, but the median curve doesn't. Note the density variations, which become unbounded in the limit. I don't have a fix, but I can turn a slight variant of this one into one of those exact continuous maps of the rational unit interval onto some dense set of algebraic points. This would permit sampling at various frequencies and phases, which has proven interesting in previous cases (www.tweedledum.com/rwg/sampeano.htm, www.tweedledum.com/rwg/sam306090.htm ). --rwg

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Re: [math-fun] left vs. right
by rcs＠xmission.com 01 Sep '10

01 Sep '10

Forwarded from SBG ... ------- Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg(a)roadrunner.com> Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer. Steve Gray On 8/30/2010 11:44 AM, James Buddenhagen wrote: > Interesting. I always assumed that people who had trouble with maps, > cardinal directions, etc. were not math people. Perhaps it is because I was > around maps since a child that I have always puzzled that some people have > trouble with them. When heavy fog or overcast I get lost much easier than > on clear sunny days, but I do not consciously think about the sun or > shadows. > > Whereas visualizing geometric solids seems easy for me, this I assume is > learned. Nonetheless, I am almost always confused by illustrations of how > to insert cards in ATM machines, and have even resorted to first trying the > opposite of what I think it says, my wife never has this problem. Also, she > finds ambiguous directions that others give about how to get somewhere > unambiguous, while I am weighing the odds of what interpretation to give. I > guess people are just different. >

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Re: [math-fun] left vs. right, from SBG
by rcs＠xmission.com 31 Aug '10

31 Aug '10

Steve Gray sez ... ------ Date: Mon, 30 Aug 2010 17:24:53 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg(a)roadrunner.com> To: math-fun <math-fun(a)mailman.xmission.com> 2.1 unnamed [text/html] 1.77 KB Download On 8/30/2010 12:25 PM, Andy Latto wrote: > On Mon, Aug 30, 2010 at 3:00 PM, Stephen B. Gray > <stevebg(a)roadrunner.com> wrote: > >> I lived on the East Coast for 36 years. When I moved to L.A. I realized that >> my sense of direction depended on which way the ocean was. It was actually >> quite difficult to adjust to the ocean being on the west. Then I visited >> Orlando, where the ocean is in both directions. Hopeless. >> > When I first moved to Cambridge, the primary landmarks I referred > everything to were Mass Ave and the Charles River, two parallel lines. > At least in the Harvard Square area, my usual haunts. The first my > travels took me to MIT, I was completely disoriented; my two parallel > lines met there, at right angles! > > Some people may claim that the Charles curves. I prefer to think of > Cambridge as a surface of positive curvature, balanced by the negative > curvature of Boston on the other side of the river. > > You don't believe Boston has negative curvature? Try driving around > the common, with its 5 right-angled-corners! > > Andy > > If the positive curvature includes the wide part of the Charles River, a boat in the middle would drift downhill to the side. That's why the marinas there are not in the middle of the river.

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[math-fun] How can I get Mathematica to solve this?
by Dan Asimov 31 Aug '10

31 Aug '10

I want Mathematica to solve this equation (*) Integral from z to Q of 1 / K*(z-a)*(z-b) dz == t for Q, in terms of z, t, K, a, b, where all 5 of these are complex with a unequal to b. It's a theorem that even if the quadratic polynomial denominator is replaced more generally by *any* function V(z) analytic for all z in C, and |t| < eps, where eps = eps(z,V), then there is a solution Q = Q(z,t,K,a,b), and this is unique. (As long as the unspecified path of integration does not contain a point z_0 where V(z_0) = 0 -- which we assume.) In fact for the case of the quadratic polynomial, Q = Q(z,t,K,a,b) is defined for all (z,t) in CxC. In fact, the solution is a linear fractional transformation of z, in K, a, b, t: (a e^(-Lt) - b e^(Lt)) z + ab (e^(Lt) - e^(-Lt)) (**) Q(z,t,K,a,b) == ----------------------------------------------------- (e^(-Lt) - e^(Lt)) z + (a e^(Lt) - b e^(-Lt)) where L := K / 2(a-b). But no way can I get Mathematica to come up with anything remotely as simple, no less (**). The commands I used are just: << In[1]:= f[u_] := K*(u-a)*(u-b) In[2]:= FullSimplify[Solve[Integrate[f[u],{u,z,Q}] == t, Q]] >> but I suspect, or at least hope, there are smarter ways to get Mathematica to simplify its answer. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] sums of reciprocal multinomials
by Dan Asimov 31 Aug '10

31 Aug '10

Fred wrote: << I illustrate with the tetranomial case --- the generalisation to any dimension is obvious: \sum_{0<=j<oo} p!q!r!j!/(p+q+r+j)! = p!q!r!/(p+q+r-1)(p+q+r-1)! Does anyone know of a reference for these and similar results . . . >> Hmm, dividing both sides, one gets: \sum_(0<=j<oo) j!/(k+j)! = 1/((k-1)(k-1)!) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] left vs. right
by James Propp 31 Aug '10

31 Aug '10

hvm writes: >I still have difficulty with left and right; for the longest time >I had to stop and think about which hand I wrote with. Or ... . While we're discussing such things, I wonder if other math-funsters have my peculiar sort of trouble with using maps: If I'm heading north, navigating is easy; if I'm heading east or west, I do the appropriate 90 degree rotation; but if I'm heading south, I can't do a 180 degree rotation, so I *pretend* I'm going north (travelling the reverse of the route I actually want to take), figure out what I need to do, and then reverse the instructions (so that left turns becomes right turns and vice versa). Jim Propp

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Re: [math-fun] 5-ray spoiler
by Fred lunnon 31 Aug '10

31 Aug '10

That's quite nice --- though SBG did inadvertantly hint at the answer by asking for an explanation! Here's my stab at the impossibility: given some arrangement of rays R1,...,Rn meeting at mutually obtuse angles in the origin, let F1,...,Fn denote their perpendicular planes through the orgin. Evidently R2,...,Rn lie on the far side of F1 from R1; similarly R3,...,Rn lie in the acute slice between F1 and F2; and R4,...,Rn lie in the acute trihedron formed by F1,F2,F3. Therefore n <= 4. Fred Lunnon On 8/30/10, rcs(a)xmission.com <rcs(a)xmission.com> wrote: > Forwarded from SBG ... > > ------- > > Date: Mon, 30 Aug 2010 12:08:08 -0700 > Subject: Re: [math-fun] left vs. right > From: "Stephen B. Gray" <stevebg(a)roadrunner.com> > > Here's an exercise in 3D visualization. Given a point P, is it possible to > construct FIVE rays coming out from P such that every ray makes an obtuse > angle (>90 degrees) with every other one? (That's 10 angles that must be > obtuse.) Explain your answer. > > Steve Gray

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Re: [math-fun] left vs. right
by Dan Asimov 31 Aug '10

31 Aug '10

Here's a possibly harder 3D visualization challenge: Draw a picture of a (topological) Moebius band in 3-space whose boundary is a perfect circle. (This can be done by continuously deforming the usual version.) --Dan -------------- Stephen wrote: << Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] left vs. right
by Henry Baker 30 Aug '10

30 Aug '10

True story. When I was a freshman at MIT in 1966, we rowed several of the 8-oared crew shells through the locks & into Boston Harbor. We were doing fine until a huge ship went by, sending out an enormous wake. Our coach yelled at us to turn our boats "parallel to the wave", but all of us being engineers/mathematicians/physicists just out of 8.01 & 18.02, we dutifully turned the main axis of our boats parallel to the _wave vector_. One of the boats was then lifted by its bow & stern -- leaving the center unsupported -- and broke in half, leaving 8 MIT oarsmen & 1 coxswain in the water! Since only engineers/mathematicians/physicists would even consider the wave vector, rather than the crest of the wave, as the axis to align to, we were undone by knowing "too much". At 11:44 AM 8/30/2010, James Buddenhagen wrote: >Interesting. I always assumed that people who had trouble with maps, >cardinal directions, etc. were not math people. Perhaps it is because I was >around maps since a child that I have always puzzled that some people have >trouble with them. When heavy fog or overcast I get lost much easier than >on clear sunny days, but I do not consciously think about the sun or >shadows.

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