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[math-fun] Headlights (was Re: Miss those IBM/DEC blinking lights?)
by Keith F. Lynch 14 Apr '18

14 Apr '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > I'm surprised that auto LED headlights don't already broadcast their > vehicle VIN# (maybe they do?). I don't know, but many of them do rapidly flash on and off. So do LED taillights. > Apparently, one of the countries in the Middle East is now requiring > some sort of RFID IFF for automobiles. Sure makes the CIA drone > program a lot more accurate... RFID doesn't work over long distances. I'd worry more about RFID tags in US passports, which make it possible to build a IED that will detonate only when a US citizen is in range. > I seem to recall that someone at MIT had developed an LED display > for bicycle spokes that was time-multiplexed in such a way as to > provide a full circular bit-mapped display -- kinda like those WWII > radar displays. Synchronized clock displays, and the like, on things like spinning fan blades can be found in any novelty shop. But I think that would be a bad idea on bicycle wheels, since cyclists don't want motorists to be the least bit confused as to what they are seeing. Speaking of which, traditional headlights are a very bad way to illuminate your path, as they make it difficult for other road users to see anything except your headlights. Car headlights are designed to only light what's directly in front or below, so as to not shine directly in the eyes of motorists, cyclists, or pedestrians. And maybe that works well on perfectly flat terrain, but it doesn't work well anywhere else. Gyroscopically stabilizing them wouldn't help, since other road users follow the terrain. Decades ago, Edwin Land proposed that all headlights and all windshields be diagonally polarized. Pedestrians and cyclists could wear diagonally polarized glasses. You'd see the reflected light from your own headlights, but you wouldn't be dazzled by oncoming headlights as you are now. Since polarizers aren't perfect, you could still *see* the oncoming headlights, but they'd no longer be blindingly bright. The disadvantage is that all lights would have to be twice as bright to compensate. Anyhow, it never caught on, and is unlikely to at this late date. Today's technology gives more options. I suggest that car headlights use something like Google's face-detection algorithm to keep bright light out of everyone's eyes. Some light would be sent toward eyes, but not enough to keep people from seeing anything else. Bike headlights, at least in brightly lit urban and suburban areas, are mostly for being seen rather than for seeing. The same algorithm could be used in reverse for them, aiming all their much dimmer light toward eyes. That would greatly increase the battery life, assuming the computer chips don't use more power than the lights. The light could also be directed toward anything dangerous that the cyclist might not have noticed, such as a fallen branch across the road.

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[math-fun] Paper-cutters
by James Propp 14 Apr '18

14 Apr '18

Why do paper-cutters have curved blades? Is the usual design intended to optimize something? Jim Propp

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Re: [math-fun] More information begets less?
by Dan Asimov 13 Apr '18

13 Apr '18

----- > At first, "turn of the century" was unambiguous, but now even the > augmented phrase "turn of the 20th century" is ambiguous. According to the Wikipedia article on it, the British English version of the augmented phrase is still unambiguous. ----- Interesting! (I have little knowledge of other languages.) —Dan

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[math-fun] More information begets less?
by Dan Asimov 13 Apr '18

13 Apr '18

I don't know from information theory, but consider this: (As a post-W.W.II baby boomer) I had always heard the phrase "turn of the century" used unambiguously to mean around the time that 1899 became 1900. But as someone pointed out to me, nowadays the same phrase "turn of the century" is rarely mentioned without *which century* being added ... evidently because *without* the added info, the bare phrase "turn of the century" could mean the recent one or the earlier one. However, the called-for information is not necessarily the "which century" that people tend to assume is what's missing. For most people, the phrase "turn of the 20th century" is about as likely to mean 1899 —> 1900 as 1999 —> 2000 (or for pedants, add one). At first, "turn of the century" was unambiguous, but now even the augmented phrase "turn of the 20th century" is ambiguous. Clearly, more information is less. —Dan

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Re: [math-fun] "Random" cards
by Henry Baker 13 Apr '18

13 Apr '18

I claim that "1" is the *least-picked*/*least-random?* number. Three Dog Night even had a popular song about this phenomenon: "One is the Loneliest Number" At 06:06 PM 4/12/2018, James Propp wrote: >Cf. the observation that when people are asked to pick a number between 1 >and 20, the most common answer is 17.

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[math-fun] “Random” cards
by James Propp 13 Apr '18

13 Apr '18

If you ask people to name a random card in a standard deck, what regularities govern their answers? (Is this something someone has looked at?) Cf. the observation that when people are asked to pick a number between 1 and 20, the most common answer is 17. Jim Propp

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[math-fun] Radial disk-dissection
by James Propp 13 Apr '18

13 Apr '18

If you dissect a unit disk radially into a large number of equal wedges, it’s well known that you can reassemble them to form a shape that in the limit converges to a 1-by-pi rectangle. What other limiting shapes can we form in the limit from patterns that join the wedges edge-to-edge? I’m guessing that you get a class of curvy pseudoquadrilaterals ABCD where AB is a unit segment, CD is a unit segment, and the curves BC and AD satisfy some sort of curvature-compatibility condition. Jim Propp

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Re: [math-fun] "Random" cards
by Henry Baker 13 Apr '18

13 Apr '18

How big a % of the population has any clue what a "random" card means? It only took mathematicians a few *hundred* years to figure this out, and perhaps they still aren't done yet. I'm not blaming the mathematicians; these are very, very subtle and tricky notions, so it doesn't surprise me that Ivy Leaguers -- particularly the ones in Washington, DC -- don't really understand these things either. Perhaps the minimum wage folk who purchase lottery tickets have a better idea that most about what "random" means. At 06:06 PM 4/12/2018, James Propp wrote: >If you ask people to name a random card in a standard deck, what >regularities govern their answers?

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[math-fun] The Moor of Venice
by françois mendzina essomba2 12 Apr '18

12 Apr '18

Did you read Othello my dear ?? a recurrence for functions a(n+1)=2^(n+3/2)*sqrt(1-sqrt(1-2^(-2*n-2)*a(n)^2)) ; for : a(0)=sqrt(2-2/sqrt(1+x^2)) ; n -------- infinity ; a(n) ------- atan(x) ; for : a(0)=sqrt(2-2*x) ; n -------- infinity ; a(n) ------- acos(x) ; for : a(0)=sqrt(2-2*sqrt(1-x^2)) ; n -------- infinity ; a(n) ------- asin(x) ; for : a(0)=sqrt(2-2*x/sqrt(1+x^2)) ; n -------- infinity ; a(n) ------- acot(x) ; The results are in absolute values FME...

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[math-fun] Generalized stable fummin problem
by Dan Asimov 12 Apr '18

12 Apr '18

I found Jim Propp's idea of generalizing a problem to irrational numbers very interesting. So can this be done for the fummin problem? Hmm. Suppose we have N unit-circumference circles that we want to distribute evenly among K recipients. Then the "(N,K) fummin problem" is to find the smallest number f(N,K) of pieces that can be rearranged to give each recipient a total length of N/K. The *stable* fummin problem tries to find the limit s(N,K) = lim (1/m)f(mN,mK), m—>oo if it exists. (Question: Does s(N,K) <> f(N,K) for some (N,K) ?) Clearly s(N,K) depends only on the ratio N/K so by abuse of notation we define s(N/K) = s(N,K). Let alpha be a positive real number. Then the "generalized stable fummin problem" is to find the limit g(alpha) = lim s(N_j / K_j), j—>oo where N_j / K_j is a sequence of rational numbers converging to alpha, *if* the limit exists and is independent of the sequence N_j / K_j approaching alpha. Question: Is g(alpha) defined for any (all?) irrational numbers alpha > 0 ??? —Dan

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