September 2014 - mathfun@mailman.xmission.com

[math-fun] bit commitment blues?
by Warren D Smith 16 Sep '14

16 Sep '14

http://www.anagram.com/jcrap/Volume_2/bc_blues.html truly, a masterpiece.

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[math-fun] Apraphulian rope&pulley XOR ?
by Henry Baker 16 Sep '14

16 Sep '14

I spent a few moments with this paper (thanks for posting!) today & noticed that it showed only NOT, OR, AND, RS flipflop and "multiplex" gates. Is there an elegant XOR in this logic system? Also, it would be interesting to work out an _invertible_ rope&pulley logic system, much closer to (perhaps isomorphic to?) the billiard ball model. Pulleys must be very, very old, as ships & ropes are very old. We know that the Greeks&Romans already had "odometers" in which the number of turns of a wheel would "count" a number of balls, or at least wind a certain amount of rope. Given that we now know they had sophisticated gear trains, it is conceivable that they also had sophisticated rope & pulley mechanisms.

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[math-fun] Turning the Tables
by Gareth McCaughan 15 Sep '14

15 Sep '14

There's a famous (and easy, once you've seen the idea) theorem: If you have a rectangular table with one leg in each corner, and an arbitrary (smooth) uneven floor, you can always find a way of placing the table so that all four feet touch the floor. If instead your table is triangular, with a leg in each corner and one somewhere in the middle, then it's easy to make a floor for which no position puts all the feet on the floor. (E.g., height = x^2+y^2.) What happens for arbitrary convex quadrilaterals? (Note: I have elided a few details above; e.g., if your floor is finite and smaller than the table then there's nowhere to put it :-).) -- g

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[math-fun] Alan Adler, of Aerobie fame,
by Bill Gosper 14 Sep '14

14 Sep '14

finds Euler's product over primes for zeta(s) somewhat magical, and wonders if we funsters know other really neat sums or products involving primes. Hence yesterday's "Minor twist". Otherwise, all I could remember was our Valentine's Day 2011 discussion of "Euler's crazy pi product". --rwg

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[math-fun] string salesman Lunnon
by Warren D Smith 14 Sep '14

14 Sep '14

Your machine asymptotically does NOT exhibit constant time "lookup" to solve an A-B shortest path problem, that is where you are confused. The mass you are pulling on and the friction both grow with N... So anyhow as I tried to demonstrate this machine really is worse than conventional algorithms...

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Re: [math-fun] "Steampunk" mathematics?
by Henry Baker 13 Sep '14

13 Sep '14

Yes, there are a whole bunch of "ant colony"-type algorithms that have been studied, trying to figure how how large numbers of parallel computers and/or robots can solve certain tasks. At 12:29 PM 9/12/2014, Victor Miller wrote: >Does this qualify: http://arxiv.org/abs/1106.0423 ? > >Victor

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Re: [math-fun] "Steampunk" mathematics?
by Henry Baker 13 Sep '14

13 Sep '14

Whatever happened with Ralph Merkle's mechanical nanocomputers? http://archive.wired.com/science/discoveries/news/2001/11/48278 At 12:04 PM 9/11/2014, Mike Stay wrote: >There's this wooden ternary calculator from a contemporary of Babbage, >who seems to have been denied funding because Babbage's efforts didn't >pan out. >http://www.mortati.com/glusker/fowler/ > >------------ >TEA lasers are surprisingly low-tech. They're not directly related to >computation, but I'm sure it would make certain things easier. >http://www.sparkbangbuzz.com/tealaser/tealaser7.htm > >Quote: > >I used to tell people "There is no such thing a true home made laser. >There is always a requirement for exotic parts that can only be >obtained from a laser manufacturer, and - or there is the requirement >to perform exotic high vacuum, glass blowing and gas mixing processes. >This would defeat much of the satisfaction of building your own >laser." > >When I read about TEA lasers recently though, that all changed. Here >is a laser that is built from aluminum foil, a dielectric and some >pieces of aluminum. It is amazing to think of a laser project where a >simple 4 to 6 KV DC power supply is the most elaborate component. > >------------ >Scientific American had an article about analog computation in its >June 1985 issue: >http://www.softouch.on.ca/kb/data/Scan-130202-0003.pdf > >------------ >Adi Shamir's TWINKLE factoring device is amusingly analog. >http://web.archive.org/web/20010615145445/http://porsche.ls.fi.upm.es/Mater… > >------------- >The best steampunk comics: > >The Thrilling Adventures of Babbage and Lovelace! >http://sydneypadua.com/2dgoggles/stories/ > >Girl Genius >http://www.girlgeniusonline.com/comic.php?date=20021104 > >---------- >An excellent site for all things steampunk: >http://brassgoggles.co.uk/blog/ >http://brassgoggles.co.uk/forum/ > >On Thu, Sep 11, 2014 at 11:39 AM, Dave Dyer <ddyer(a)real-me.net> wrote: >> >> I nominate Turing's analog computer to find zeta function zeros, described >> in the Hodges biography. >> >> Also, obviously, the concept of physically realizing Turing machines >> using punched tape and mechanical reader/writer mechanisms.

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[math-fun] "Steampunk" mathematics?
by Henry Baker 12 Sep '14

12 Sep '14

I've just become aware of the concept of "Steampunk", which is an alternate universe in which Babbage-style steam computers actually worked in the 19th Century & thereby changed the world. (Yes, I know; I'm Rip Van Winkling myself; no snarky remarks!) Steampunk today means an exceedingly retro anachronistic concept -- e.g., a spring-wound software-defined digital radio -- so it doesn't actually require steam, per se. https://en.wikipedia.org/wiki/Steampunk Prior to the discovery of the Antikythera Mechanism, any mention of Archimedes in conjunction with geared wheels might have initially seemed "steampunk", but modern X-ray & gamma-ray images and recreations have proved it to be a remarkably sophisticated analog computer. https://en.wikipedia.org/wiki/Antikythera_mechanism Probably one of the coolest "steampunk" exercises of the 20th Century was economist William Phillips's 1949 "MONIC" water-based analog computer which represented the macro economy of the UK (called by one wag "hydraulic Keynesianism"). (Phillips could easily have been the inspiration for the TV series MacGyver -- parodied by SNL's "Gruber" -- as Phillips built all sorts of stuff while a Japanese POW -- including a radio in his shoe.) https://en.wikipedia.org/wiki/William_Phillips_%28economist%29 https://en.wikipedia.org/wiki/MONIAC I would propose that "steampunk mathematics" would be math that Cayley or Hamilton would not only understand, but might have written themselves. No Bourbaki, no axioms of choice, no Goedel (or at least Goedel only in an analog computer world). No Hilbert (especially no Hilbert's Tenth Problem), no quantum theory, very little topology (except early knot theory, thanks to Cayley, etc.). Unfortunately, my emphasis on 19th Century technology leaves out things like relay computers & Strowger switches, which I think would have been understood by, and really appealed to, the Victorians, so long as they don't include vacuum tubes or germanium diodes. I think that a lot of computer graphics technology might have appealed to Cayley & Hamilton -- particularly the use of Hamilton's quaternions for "in-betweening". It is conceivable that _fractal_ graphics, such as some of Gosper's cool pix, would have been understood by Cayley, at least. Cayley was perhaps the first to have notice the chaotic behavior of Newton's method for finding polynomial roots in the complex plane. Computer algebra systems would have been very appealing to Cayley & Hamilton, as the 19th Century mathematicians all seem to have waded through prodigious amounts of symbolic calculations. I'd be interested in any other proposals for "steampunk" mathematics.

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Re: [math-fun] "Steampunk" mathematics?
by Bill Gosper 12 Sep '14

12 Sep '14

HGB>I guess the Galton box qualifies as a mechanical calculator for a Gaussian: https://en.wikipedia.org/wiki/Bean_machine Also, Fredkin's billiard ball computer: https://en.wikipedia.org/wiki/Billiard-ball_computer And then there's those wonderful 1960's mechanical desktop "calculators"; I recall seeing one in the 1950's multiply two large numbers; they were amazing to watch! <HGB Even from a distance. The Friden (and I think, Marchant) machines had a column of 10 togglable keys for each of 10 digit positions, and a 20 digit accumulator carriage that chugged back and forth. The Friden I used in 1961 to Monte-Carlo submarine kill probabilities(!) could also take the 10 digit square root of the 20 digits in the accumulator. Unless the carriage was too close to an immovable object, whereupon you enjoyed the screech of stripping gears, and a multihundred dollar, multiday repair job. HGB> Some of IBM's early punch card machines were deliciously electromechanical, < and LOUD (and plugboard programmable) HGB> with grease everywhere. Also, some of their disk drives and printers I used in the 1960's had _hydraulic_ mechanisms; the IBM CE's looked more like car mechanics than computer engineers! And UNIVAC CEs were more like shipyard workers than car mechanics. UNIVAC tape drives were frightening. And the 1206 CPU, built like a bank vault, though solid state, required such deafeningly ferocious cooling that you could Bernoulli-levitate a football in the vertical exhaust. --rwg Also, selected Stanford AI alumni have coffee tables made from Librascope disk platters. At 11:39 AM 9/11/2014, Dave Dyer wrote: I nominate Turing's analog computer to find zeta function zeros, described in the Hodges biography. Also, obviously, the concept of physically realizing Turing machines using punched tape and mechanical reader/writer mechanisms. <

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[math-fun] all shortest paths & the "string machine"
by Warren D Smith 12 Sep '14

12 Sep '14

In a V-vertex E-edge graph... What string-machine salesman Lunnon forgot to say, is that enormous edge-length integers, N bits long, would require an exponentially large budget to purchase the string, plus the thing would gravitationally collapse into a black hole. If the edge lengths were selected from a BOUNDED set of integers only, though, then Lunnon's "string machine" for all shortest paths would work... it takes O(E+V) steps to build it, then to solve the A-B shortest path problem requires effort (force, energy expenditure versus friction, also time if you have bounded power source) of order O(E+V) to pull on that much weight. Meanwhile the conventional computer has a linear time algorithm O(V+E) for single source, all destinations, shortest paths (assuming bounded nonnegative integer edge lengths), then constant time lookups... so, sorry, Lunnon+string has not really beat conventional computers. A nicer try was made by Ken Steiglitz who once invented a machine made of rigid parts to solve NP-complete problems instantly. (Some part can move, or not, if the SAT problem has solution, or not.) Really, though, it won't work, albeit the physical reasons for the failure are a bit subtle and were not understood by KS himself.

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