March 2011 - mathfun@mailman.xmission.com

Re: [math-fun] perpendicular planes
by Dan Asimov 27 Mar '11

27 Mar '11

If two lines are given parametrically by {u0 + su} and {v0 + tv} for (s,t) in R^2, vectors u0, v0, and unit vectors u, v, then the distance between the lines is just (setting w0 = u0-v0) |w0 * u x v|, or am I missing something? --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] perpendicular planes
by Dan Asimov 27 Mar '11

27 Mar '11

The one thing I don't agree with in this paper is the claim that because the geodesic distance between two k-planes P, Q in R^n (in the Grassmannian manifold of k-planes in R^n) is not differentiable at some points (it's given by d(P,Q) = sqrt( theta_1^2 + . . . + theta_n^2) ) it is therefore flawed. All the most basic widely accepted distances -- geodesic distances on any smooth Riemannian manifold, including the Euclidean distance on R^n -- e.g., D(x,y) := |x-y| on R -- have this "flaw". --Dan << In case you haven't read the paper below, you should give a look: "Packing lines, planes ..." by Conway, Hardin and Sloane. http://www2.research.att.com/~njas/doc/grass.pdf >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Perpendicular planes
by Fred lunnon 27 Mar '11

27 Mar '11

If the lines are parallel, u x v (Gibbs vector product) vanishes ... Also, from an algorithmic point of view, putting your lines into this (non-canonical) form in the first place is not entirely trivial --- several extra branches are required. I'd guess that this wasn't what Henry wanted either! WFL On 3/27/11, Dan Asimov <dasimov(a)earthlink.net> wrote: > If two lines are given parametrically by {u0 + su} and {v0 + tv} > for (s,t) in R^2, vectors u0, v0, and unit vectors u, v, > then the distance between the lines is just (setting w0 = u0-v0) > > |w0 * u x v|, > > or am I missing something? > > --Dan >

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Re: [math-fun] how do you structure causality?
by Henry Baker 27 Mar '11

27 Mar '11

Some modern physicists have already started to give up on "causality", as a result of EPR experiments showing that "spooky action at a distance" does indeed happen. So "causality" may indeed be an illusion that results from our (vastly imperfect) "memory" system. This is essentially a version of the anthropic principle: we see what our minds & memories allow us to see. http://en.wikipedia.org/wiki/Anthropic_principle Szilard/Maxwell/Bennett found that the baseball umpire's rule is true: there aren't any balls or strikes until the umpire calls them. Translation: there is a large energy cost to nailing down a memory; the cost may not show up, however, until you _clear_ the memory to zeros. http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theo… At 12:00 PM 3/26/2011, Bill Gosper wrote: >In freshman Humanities, we had to read Hume's > > 1. *An Enquiry Concerning Human >Understanding*<http://www.infidels.org/library/historical/david_hume/human_understanding.h…> > >**wherein he concluded causality was essentially illusory. Or so I gathered. >In his gedanken mechanics, the terminology was so muddled and >in conflict with modern terminology that I boggled. >--rwg

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Re: [math-fun] perpendicular planes
by Henry Baker 27 Mar '11

27 Mar '11

Perhaps closely related is the problem of the shortest distance between two non-parallel ("skew") lines in 3 (or more) dimensions. I googled this & found this wikipedia entry: http://en.wikipedia.org/wiki/Skew_lines I'm not happy about the ugliness of these expressions. Yes, the "triple product" is a bit more satisfying, but I seem to recall seeing an expression somewhere that was much more satisfying -- something along the lines of the problem of the "distance of a point to a line", where the distance is obtained by plugging the point into the standard form of the line in terms of direction cosines: directed distance from point (x0,y0) to sin(theta)*x+cos(theta)*y+C is sin(theta)*x0+cos(theta)*y0+C. http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point… I checked about 10 pages of Google's results, but couldn't find anything better than the wikipedia-type calculation for skew lines. At 11:29 AM 3/19/2011, Fred lunnon wrote: >Given non-parallel lines L,M in Euclidean 3-space, there exist unique planes >S,U meeting in L, and T,V in M, with both S,T perpendicular to both U,V. > >An algebraic proof of this turned out to be surprisingly nontrivial --- >is there a more intuitive synthetic demonstration? > >[The appropriate generalisation of this this apparently tedious little lemma >provides one crucial link in the classification of isometries for a quadratic >inner-product space. It may well be a (special case of) some well-known theorem >concerning bases of vector subspaces, though I didn't recognise it as such.] > >Fred Lunnon

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Re: [math-fun] how do you structure causality?
by Bill Gosper 26 Mar '11

26 Mar '11

In freshman Humanities, we had to read Hume's 1. *An Enquiry Concerning Human Understanding*<http://www.infidels.org/library/historical/david_hume/human_understanding.h…> **wherein he concluded causality was essentially illusory. Or so I gathered. In his gedanken mechanics, the terminology was so muddled and in conflict with modern terminology that I boggled. --rwg

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[math-fun] Art Robinson affair
by Eugene Salamin 26 Mar '11

26 Mar '11

My apologies for having posted a lengthy, mostly political, message. Had I been more attentive to my own email, I could have simply provided a web site that provides complete, up-to-date information. Here it is, and I will say no more on this forum about the matter. http://www.oregonstateoutrage.com/ -- Gene

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[math-fun] eta(e^(-80 pi))
by Bill Gosper 26 Mar '11

26 Mar '11

A week or so of exhaustive LatticeReducing (FindIntegerNullVector cr*ps out around dimension 50) "factorizes" the big surd in eta(e^(-80 pi)) (just south of Monterey) into binomial surds: DedekindEta[ 40 I] == ((3/(5 (1 + 2 Sqrt[2/5])))^(7/8) Sqrt[ 7/((1 + 3/Sqrt[2]) (1 + Sqrt[2]/5^( 1/4)))] (((1 + 3 Sqrt[2/5]) (1 + 1/5^(3/4)) (1 + 2/5^( 3/4)) (1 + 3/5^(3/4)) (1 + Sqrt[2]/5^(3/4)))/((1 + ( 2 2^(1/4))/Sqrt[5]) (3 + 5^(1/4)) (1 + 5^(1/4)/Sqrt[ 2]) (1 + 2^(3/4) 5^(1/4)) (1 + 2^(1/4) Sqrt[5]) (4 + 2^(3/4) Sqrt[5]) (1 + 5^(3/4)) (2 + 5^(3/4)) (1 + 5^(3/4)/ Sqrt[2]) (2 + 10^(1/4))))^( 1/4) (((1 + (3 Sqrt[5/2])/2) (1 + 8/5^(3/4)) (1 + 2^(1/4)/Sqrt[ 5]) (4 + 5^(1/4)) (4 + Sqrt[2] 5^(1/4)) (4 + Sqrt[5]) (4 + 2^(1/4) Sqrt[5]))/((1 + 2 (2/5)^(1/4)) (1 + 4/Sqrt[ 5]) (1 + (2 2^(3/4))/Sqrt[5]) (1 + 4/5^(1/4)) (1 + ( 2 Sqrt[2])/5^(1/4)) (1 + (2 Sqrt[5])/3) (1 + Sqrt[5]/2^( 1/4)) (3 + 5^(3/4)) (1 + 1/3 2^(1/4) 5^(3/4)) (5 + 3 10^(1/4)) (4 + 5 10^(1/4))))^(1/8) Gamma[1/4])/(4 (1 + 1/Sqrt[2])^( 5/4) ((1 + (2/5)^(1/4)) (2 + Sqrt[5]) (3 + Sqrt[5]))^( 5/8) ((1 + (2 5^(1/4))/3) (1 + 5^(1/4)) (1 + Sqrt[5]) (1 + 10^( 1/4)))^(3/8) ((1 + Sqrt[2/5]) (1 + 1/2^(1/4)) \[Pi])^(3/4)) For what algebraic numbers are such factorizations possible? (Massively nonunique, in this case.) (Apropos Allan's roadogeny (pathogenesis?), I could swear seeing a pointer here to a website featuring a cellular type (celluloid?) algorithm for synthesizing diagrams strongly resembling maps of towns.) --rwg

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[math-fun] Please - no more political messages on the Math-Fun list.
by Schroeppel, Richard 25 Mar '11

25 Mar '11

Please -- no more political messages. If you have something borderline, let me see it first. Rich rcs(a)xmission.com

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[math-fun] how do you structure causality?
by Gary Antonick 25 Mar '11

25 Mar '11

Feels like you guys are the village elders and I'm the kid who's gotten into some trouble with something simple. Am hoping for basic logic advice on structuring "reasons" — perhaps something from AI or rhetoric. I post puzzles online periodically and several days ago ran a Gardner classic: Why Are Manhole Covers Round? Readers have suggested tons of reasons: won't fall into the hole, etc. I'm now trying to structure the reasons and even potentially reject the question and the idea that there can be a "reason" in the logical sense. Any suggestions would be great. Cheers, Gary *Structuring Reasons* *1. rejecting the question* First of all, I think the question "Why Are Manhole Covers Round?" is itself invalid because it's begging the question. Am wondering if anyone familiar with logic might confirm this. I was thinking of using this analogy. Why are racecar tires rubber? Why not wood? Romans used wood. Someone might say wood is inflexible and too slippery. Wait. If wood were the only option for tires then the sport of racecar driving as we know it wouldn't exist. *2. parallel and serial causes* Second, am hoping someone might suggest a method for structuring reasons where there's not a necessary and sufficient condition. I'll use the racecar example again: Racecar x crashes in race y. Why did racecar x crash? It was going too fast. OK. If it had been going slower it would not have crashed? Well, no. It had also been struck from the side.. and the track was slippery. These seem to be in parallel.. that is, they have similar status. There's always one more. Hydra heads. Then there's a list of reasons that seem to be in series. 1. cars were invented 2. car racing developed 3. car racing conditions/rules give rise to/allow crashes 4. racecar x created 5. racecar x is in good condition 6. racecar x participates in race y Take out one of these and the car doesn't crash. But they aren't really "reasons." *3. rejecting causality: emergent results* Then there's the final outcome: racecar x [crashes, not crashes], which has me wonder if "reasons" can only be assigned to conditions and not to outcomes. That is, the track was wet because it rained. No rain, no wet.

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