- mathfun@mailman.xmission.com - mailman.xmission.com

Re: [math-fun] Everyone's a winner!
by Dan Asimov 24 Jul '18

24 Jul '18

----- You're a physicists? How many? :-) ... ... Similarly, Randall Monroe pointed out that, per unit mass, a firefly is thousands of times brighter than the sun. See https://what-if.xkcd.com/151/ ----- Maybe you mean Randall Munroe? :-) —Dan

3 2

Re: [math-fun] math-fun Digest, Vol 185, Issue 49
by Stuart Anderson 24 Jul '18

24 Jul '18

Just to be clear, the first paragraph of my post was a quote from Kevin Lynch (beginning with "Also, given that official weather... And ending "... Even if there's no climate change. ") in his post 'Everyone's a winner " My apologies for not editing the subject line and making the quote apparent. So I was commenting on his quote, not asking a question. On Wed, Jul 25, 2018, 09:31 <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: Everyone's a winner! (Dan Asimov) > 2. Re: Everyone's a winner! (James Propp) > 3. Re: math-fun Digest, Vol 185, Issue 48 (Stuart Anderson) > 4. Re: math-fun Digest, Vol 185, Issue 48 (Dan Asimov) > 5. Re: 3D printers & musical instruments (Hilarie Orman) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Tue, 24 Jul 2018 11:07:46 -0700 (GMT-07:00) > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Everyone's a winner! > Message-ID: > <1439810851.8502.1532455666695(a)wamui-jasmine.atl.sa.earthlink.net> > Content-Type: text/plain; charset=UTF-8 > > ----- > You're a physicists? How many? :-) > > ... > ... > > Similarly, Randall Monroe pointed out that, per unit mass, > a firefly is thousands of times brighter than the sun. See > https://what-if.xkcd.com/151/ > ----- > > Maybe you mean Randall Munroe? :-) > > ?Dan > > > > ------------------------------ > > Message: 2 > Date: Tue, 24 Jul 2018 15:15:42 -0400 > From: James Propp <jamespropp(a)gmail.com> > To: Dan Asimov <dasimov(a)earthlink.net>, math-fun > <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Everyone's a winner! > Message-ID: > <CA+G9J-fM-G46625f5+LMJkzuZPVrvawh+dfrG= > TE+F_DjiU0Fw(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > I don't know if everyone's a winner, but it seems everyone's a comedian. > :-) > > Jim > > On Tue, Jul 24, 2018 at 2:07 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > > > ----- > > You're a physicists? How many? :-) > > > > ... > > ... > > > > Similarly, Randall Monroe pointed out that, per unit mass, > > a firefly is thousands of times brighter than the sun. See > > https://what-if.xkcd.com/151/ > > ----- > > > > Maybe you mean Randall Munroe? :-) > > > > ?Dan > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > Message: 3 > Date: Wed, 25 Jul 2018 07:32:16 +1000 > From: Stuart Anderson <stuart.errol.anderson(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] math-fun Digest, Vol 185, Issue 48 > Message-ID: > <CA+3-r9PiXBQN1fx= > 8qagPXeDomt4oDzoEreh_S6EgAfTQRi8Xw(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > Also, given that official weather records have only been kept for about 150 > years in the US, as there are more than twice that many days in a year it's > unusual if at least one day each year *doesn't* set a record for highest or > lowest temperature, or most rainfall or snowfall, for that date, even if > there's no climate change. > > If each of the years were statistically independent with no el Nino climate > correlation or climate forcing wouldn't the expected number of record > setting events be equal to the sum of the number of terms of the harmonic > series? > For 150 years that would be about 5.6 and as time goes on records would be > set less often. > > > > > > > > > > ------------------------------ > > Message: 4 > Date: Tue, 24 Jul 2018 16:01:01 -0700 (GMT-07:00) > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] math-fun Digest, Vol 185, Issue 48 > Message-ID: > < > 1123546516.18500.1532473261302(a)wamui-kristoff.atl.sa.earthlink.net> > Content-Type: text/plain; charset=UTF-8 > > As I try to think about this, I'm getting confused between records for > *years* > (of what?) and records for *specific days* in a year, which are also > mentioned. > > Aha ? I now see that I'm *also* confused about "expected number of > record-setting > events"). Since records began? Okay, now it makes sense. > > I would need to know *exactly what* type of record-setting events, and also > *exactly what knowledge* of these events is assumed when calculating the > probability. > > E.g., is this asking what the expected number of records will be, before > record-keeping began and assuming no knowledge of new records after they > are made? Looks that way. > > ?Dan > > ((( > Many real-world records, like fastest marathon ( > https://en.wikipedia.org/wiki/Marathon_world_record_progression#Men) > show periods > of steady, almost differentiable growth over the years, but also have > plenty of what > look like discontinuities. > ))) > > ?Dan > > Stuart Anderson wrote: > ---- > Also, given that official weather records have only been kept for about 150 > years in the US, as there are more than twice that many days in a year it's > unusual if at least one day each year *doesn't* set a record for highest or > lowest temperature, or most rainfall or snowfall, for that date, even if > there's no climate change. > > If each of the years were statistically independent with no el Nino climate > correlation or climate forcing wouldn't the expected number of record > setting events be equal to the sum of the number of terms of the harmonic > series? > > For 150 years that would be about 5.6 and as time goes on records would be > set less often. > ----- > > > > ------------------------------ > > Message: 5 > Date: Tue, 24 Jul 2018 17:30:55 -0600 > From: "Hilarie Orman" <ho(a)alum.mit.edu> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] 3D printers & musical instruments > Message-ID: <201807242330.w6ONUtpq015735(a)rumpleteazer.rhmr.com> > > There are 3D printed violins, but they are indeed too heavy. There > are electric violins with lovely 3D printed pieces, and there is some > hope for customizing such instruments to the anatomy of the artist, > and that might be worth the cost of the instrument (for jazz or > country musicians). > > Let me note (a 441 A) in passing that the mystique of a Strad is > largely due to marketing hype from a couple of hundred years ago. > Yes, they are generally very good instruments, but they have been > modified from their original construction to meet modern demands (and > to repair them from sometimes awful injuries!). Newly minted > instruments from a master luthier can sound equally good. > > In considering design alternatives based on "aesthetics", it is > important to consider the artist as part of that. The instrument must > respond to control by the artist, it must be predictable, it must have > a range of dimensions and an intuitive interface with tactile > feedback. I think that this might be achieved with nanomaterials and > active control. I've no idea how to do this, nor if it amounts to > "reinvent wood." > > Hilarie > > On Mon, 16 Jul 2018 at 09:22:19 -0400 Richard Howard contended: > > > Airplanes don't have feathers. > > Given 3D printing and full acoustic simulation, why would you make a > > "violin". > > That design came from another set of constraints (properties of wood, > > available configuration tools, etc.). > > New instrument would be based on other constraints, which could use > > principles of aesthetics. > > > --R > > > On Mon, Jul 16, 2018 at 3:34 AM, Simon Plouffe <simon.plouffe(a)gmail.com > > > > wrote: > > > > Hello, > > > > > > they made one experiment once to make a violin > > > from scratch using a design based on acoustic and > > > physics only, they actually succeeded in making one, > > > the sound was good, even comparable to a Stradivarius, > > > it worked, the only thing is that the violin was ugly. > > > The article was on Scientific American a long time ago. > > > > > > I think that 3D printing is about the same, we probably > > > could print a violin, it could also sound pretty good, > > > but it won't replace the look and feel and lightness of > > > a wooden real model. Does anybody ever saw the film > > > : The Red Violin ? > > > > > > Best regards, > > > Simon Plouffe > > > _______________________________________________ > > > math-fun mailing list > > > math-fun(a)mailman.xmission.com > > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 185, Issue 49 > ***************************************** >

1 0

[math-fun] 3D printers & musical instruments
by Henry Baker 24 Jul '18

24 Jul '18

I was at a symphony concert last night & I started thinking about "high tech" musical instruments (or the lack thereof). It occurred to me that a lot of existing instruments could be partially or wholly made using 3D printers. But then it also occurred to me that whole new classes of instruments could be made using 3D printers that could never be fabricated in any other way. For example, many instruments have resonant chambers of varying sizes (i.e., frequencies) and other characteristics. But stereo speaker cabinets also have resonant chambers of varying sizes. By using a 3D printer, it should be possible to make instruments (including speaker housings) with many, many different resonant frequencies and characteristics. While 3D printing may not be cheap enough for large volume :-) production, it is cheap enough for home experimentation. I'm curious if anyone has married *fractal* patterns and musical instruments & audio speakers with 3D printers? Also, I don't think that the final word has been said about antennae for the electromagnetic spectrum. There may be sophisticated antennae which can only be fabricated using 3D printing, and which have much more interesting properties than the simply antennae we learned as EE undergrads.

6 6

Re: [math-fun] math-fun Digest, Vol 185, Issue 48
by Dan Asimov 24 Jul '18

24 Jul '18

As I try to think about this, I'm getting confused between records for *years* (of what?) and records for *specific days* in a year, which are also mentioned. Aha — I now see that I'm *also* confused about "expected number of record-setting events"). Since records began? Okay, now it makes sense. I would need to know *exactly what* type of record-setting events, and also *exactly what knowledge* of these events is assumed when calculating the probability. E.g., is this asking what the expected number of records will be, before record-keeping began and assuming no knowledge of new records after they are made? Looks that way. —Dan ((( Many real-world records, like fastest marathon (https://en.wikipedia.org/wiki/Marathon_world_record_progression#Men) show periods of steady, almost differentiable growth over the years, but also have plenty of what look like discontinuities. ))) —Dan Stuart Anderson wrote: ---- Also, given that official weather records have only been kept for about 150 years in the US, as there are more than twice that many days in a year it's unusual if at least one day each year *doesn't* set a record for highest or lowest temperature, or most rainfall or snowfall, for that date, even if there's no climate change. If each of the years were statistically independent with no el Nino climate correlation or climate forcing wouldn't the expected number of record setting events be equal to the sum of the number of terms of the harmonic series? For 150 years that would be about 5.6 and as time goes on records would be set less often. -----

1 0

Re: [math-fun] math-fun Digest, Vol 185, Issue 48
by Stuart Anderson 24 Jul '18

24 Jul '18

Also, given that official weather records have only been kept for about 150 years in the US, as there are more than twice that many days in a year it's unusual if at least one day each year *doesn't* set a record for highest or lowest temperature, or most rainfall or snowfall, for that date, even if there's no climate change. If each of the years were statistically independent with no el Nino climate correlation or climate forcing wouldn't the expected number of record setting events be equal to the sum of the number of terms of the harmonic series? For 150 years that would be about 5.6 and as time goes on records would be set less often. > >

1 0

Re: [math-fun] Everyone's a winner!
by Keith F. Lynch 23 Jul '18

23 Jul '18

Brent Meeker <meekerdb(a)verizon.net> wrote: > Keith F. Lynch wrote: >> I like to collect unexpected numbers. For instance how would you >> guess the heat generated by the sun, per unit time per unit mass, >> compares to the metabolic heat generated by a person? Check it >> out for yourself, as you probably wouldn't believe my answer. > Being a physicists I already knew the answer. :-) You're a physicists? How many? :-) Ironically, given that I'm not a visual thinker, I discovered this surprising fact by plotting mass against power output of a wide variety of things on a log-log chart. I've learned a lot by doing the same with other pairs of physical parameters. Similarly, Randall Monroe pointed out that, per unit mass, a firefly is thousands of times brighter than the sun. See https://what-if.xkcd.com/151/ I discovered that our galaxy has about the same area density, i.e. kilograms per square meter, as a potato chip. In other words a potato chip of the same diameter as our galaxy would have about the same mass. I wonder if anyone else has noticed this -- or gone on to calculate how long it would take for a person to eat such a chip. Of course it would either collapse under its own gravity or fly apart, depending on how fast it was rotating. It could be stable if it consisted of many concentric rings, each with its own rotation rate. That led to the question of how large a solid object could exist in space. Lets replace the chip with a steel cable in a loop, spinning at just such a rate as to counteract gravity. Assuming it's not near anything, I think it could be of cosmological size. I'm not sure what would finally make it fail. Probably frame dragging due to the Hubble expansion. Or maybe whatever mysterious effect is causing the expansion to accelerate. Or maybe differential pressure from the cosmic microwave background, whose rest frame is different in different places due to the expansion. > Ten miles is plenty of distance for a big difference in rain fall. > Didn't they nominate it a thousand year flood in Ellicott City based > on measured rainfall?? Was it the same as in Baltimore? Yes, that was the point in my argument. Sorry if I wasn't clear. >> The United States could be divided into about 38,000 squares ten miles >> on a side. So it can be expected that each year about 38 of them have >> a thousand-year flood. > But in a lot of them a thousand year "flood" might be 6 inches of > rain instead of the mean 2 inches. So however statistically unusual > it was, it wouldn't get reported beyond the local newspaper and > nobody would call it a flood. Right, which is why newspapers aren't full of such stories. Also, given that official weather records have only been kept for about 150 years in the US, as there are more than twice that many days in a year it's unusual if at least one day each year *doesn't* set a record for highest or lowest temperature, or most rainfall or snowfall, for that date, even if there's no climate change. For instance each of the past 150 July 24ths has the same odds of being the hottest (or coldest, or rainiest, etc.) July 24th in your city in that period. But the newspaper calls it the hottest *ever*, as if time began in 1870. Getting back to the density of galaxies, there are at least three objects in Earth's sky that are about a half a degree across: The sun, the moon, and the Andromeda galaxy. It's easy to show that the volume of an object of given angular size scales with cube of distance, and that the tidal effect of an object scales with the inverse cube of distance, so the two cancel out. As such the tidal effect of spherical objects of the same angular size is proportional to their densities. And indeed the ratio of the moon's to the sun's tidal effect on Earth is about equal to the ratio of their densities. Andromeda must also have a tidal effect on Earth, but since its density is so low, the effect in unnoticeably small. (Also, Andromeda isn't spherical.) This doesn't seem to be widely known, otherwise visitors to my home could answer when I point to my autographed copy of Misner, Thorne, and Wheeler's _Gravitation_, and ask them whether its tidal effect on them exceeds that of the moon. (Okay, that book isn't spherical either, but if it's so close that it would be more than half a degree wide no matter which way it was rotated....) Does anyone else here have any unexpected numbers or little-known scaling laws to share? Thanks. It's only barely mathematical, but I'm also interested in unexpected facts about chronology. For instance the first tyrannosaurs lived closer to the present than to the Jurassic. Cleopatra lived closer to the present than to when the pyramids were built. When the pyramids were built, there were still living mammoths. The Berlin Wall has been down longer than it was up. JFK has been dead longer than he was alive. The movie Gone With the Wind was released closer to the Civil War than to the present. (Plenty of Civil War veterans watched it. But an adult actress in it is still alive today. Ironically, the character she played died in the movie.) When Star Wars was released, Charlie Chaplin was still alive and France was still using the guillotine. Harvard University was founded before calculus was invented. Fax machines were invented before Texas, California, and Florida were states. The TV show The Simpsons has been broadcast for more than a third of the time since TV was invented. There's a TV interview of a witness of the Lincoln assassination. There are photos of veterans of the Revolutionary War. People who weren't born until after the 9/11 attacks will be old enough to vote in the next presidential election. The first manned moon landing was closer to Lindbergh's flight than to the present, and Neil Armstrong got a congratulatory telegram from Lindbergh. Every ex-president born in the past hundred years is still alive. I've been using email for a quarter of the time since the Civil War.

1 0

Re: [math-fun] Everyone's a winner!
by Keith F. Lynch 22 Jul '18

22 Jul '18

Brent Meeker <meekerdb(a)verizon.net> wrote: > It may also be important to consider the range of competitors. > https://www.npr.org/sections/krulwich/2014/04/16/302943533/the-ultimate-ani… Very true. Of course animals can beat people at many tasks. For instance horses can run faster than people. (However, people can run *further* than horses.) What's surprising and disturbing about that story is that chimps are beating people at an intellectual task. We all have our strengths and weaknesses. For instance I've noticed that I seem to have no color memory. I remember colors only verbally, and only if I make a mental note of them. I'm also not a graphical thinker. I've been beaten at the pattern-matching game Jungle Speed by a small child. I've realized that I translate the patterns into words and compare those, which is slower than comparing patterns directly. I also do very poorly at the pattern-matching game Set, but I've discovered it has some interesting mathematical properties, as does Tsuro, another game I'm bad at. (I added several Tsuro- inspired sequences to OEIS.) As for what kind of contest I'd be best at, probably a trivia contest. I'd do very well in Jeopardy! if they'd get rid of the sports, potent potables, and recent popular culture categories. Especially if the questions are mostly about math, science, written science fiction, science fiction fandom, engineering, history, and geography. (I spend most of my free time reading, mostly non-fiction, and I retain what I read. I've read more words than I am seconds old.) Another kind of contest is by species rather than by individual. For instance how does the total distance walked by people compare to the total distance walked by ants? Of course nobody has kept careful track, but I'd estimate that people have walked a few hundred light years and that ants have walked a few hundred billion light years. (There are a lot more of them, and they've been around for a lot longer.) (Of course you could nitpick that ants aren't just one species, or that they crawl rather than walk.) A perhaps more interesting question is how the total distance people have walked compares to the distances we have bicycled, driven in cars, flown on airplanes, and flown in space. I like to collect unexpected numbers. For instance how would you guess the heat generated by the sun, per unit time per unit mass, compares to the metabolic heat generated by a person? Check it out for yourself, as you probably wouldn't believe my answer. Another example is that caves can't be more than a few miles deep on Earth, due to the pressure of the overlying rock, but that on a smaller planet they could be deeper, so the total lengths of caves could be much more. In the limit, caves could go all the way to the center of a small planet, and the total length of the caves on that small world could be measured in light years. (Hal Clement wrote a story about people getting lost in such a cave system.) I once took a free blood test at a health fair. They test 20 things. Anything that's in the top or bottom 5 percent of what healthy people typically score is flagged as abnormal. Some claim that it's a marketing scheme, since by chance most healthy people will score abnormal on at least one thing by that standard, and most of them will pay for followup tests. Sure enough, I scored abnormal on one thing: Cholesterol. It was flagged as too *low*. I didn't opt for any followup test. Given that that was nearly 30 years ago, and I haven't been ill for over 40 years, I'd say I made the right choice. (I'm uninsured. Friends call me the world's oldest "young invincible.") Speaking of misleading statistics, in May there was a thousand year flood in Ellicott City, Maryland, just two years after another thousand year flood in the same location. This is cited as proof of global climate change. I don't doubt global climate change, but that wasn't evidence for it. I happened to be in Baltimore that day, just ten miles from Ellicott City. There that storm was heavy, but it definitely wasn't the heaviest storm in a thousand years. More like the heaviest storm in a month. There was no storm damage in or near my hotel (unless you count a minor leak from the skylight) even though the hotel is just 14 feet above sea level, and the sea-level harbor is just across the street. The United States could be divided into about 38,000 squares ten miles on a side. So it can be expected that each year about 38 of them have a thousand-year flood. Floods, unlike volcanoes and earthquakes, don't relieve any long-term stress, so each year is an independent roll of the dice. The chances of any specific place that had a thousand-year flood having another within three years is about 1/333. So if there are 38 places that have a thousand-year flood each year, chances are about one in nine that at least one of them will have another thousand-year flood within three years (i.e. the same year, the following year, or, as in Ellicott City, two years later). So I'd expect such a repetition to happen more than once a decade somewhere in the US. The same reasoning applies independently to all other weather records, such as drought, snow, heat, cold, lightning, and wind. So what happened in Ellicott City is unfortunate, but it shouldn't be astonishing, nor is it evidence that the climate is collapsing and we are all doomed.

2 1

Re: [math-fun] Looks fishy; Smells OK.
by Bill Gosper 22 Jul '18

22 Jul '18

I probably shouldn't've rationalized the denominators. Out[971]= {(3 (1 + 2 Sqrt[2] - Sqrt[10]) (1 + 3 Sqrt[2] - Sqrt[ 10]) (1 + Sqrt[2] + 3 Sqrt[5] - Sqrt[10])^3)/((2 - Sqrt[2] + 2 Sqrt[5] - Sqrt[10]) (-1 + 3 Sqrt[5] - Sqrt[10]) (1 + 3 Sqrt[2] + 3 Sqrt[5] - Sqrt[10]) (1 - Sqrt[2] + Sqrt[10])^3), ((1 + 2 Sqrt[2]) (-1 + 3 Sqrt[2] - Sqrt[10]) (-1 + 3 Sqrt[5] - Sqrt[10])^3 (1 - Sqrt[2] + Sqrt[10]) (1 - Sqrt[5] + Sqrt[10]))/((1 + 2 Sqrt[2] - Sqrt[10])^3 (-1 + Sqrt[2] + 2 Sqrt[5] - Sqrt[10])^2 (3 - Sqrt[2] + 3 Sqrt[5] - Sqrt[ 10]) (1 + Sqrt[2] + 3 Sqrt[5] - Sqrt[10])), ((1 + 3 Sqrt[2] - Sqrt[10]) (-1 + 2 Sqrt[5] - Sqrt[10]) (-1 + 3 Sqrt[5] - Sqrt[ 10])^3 (3 + Sqrt[2] - 3 Sqrt[5] + Sqrt[10]))/((1 + 2 Sqrt[2] - Sqrt[10])^3 (-1 + 3 Sqrt[2] - Sqrt[10]) (1 + 2 Sqrt[5] - Sqrt[ 10]) (3 + Sqrt[2] + 3 Sqrt[5] - Sqrt[10]) (2 + Sqrt[2] - 2 Sqrt[5] + Sqrt[10]) (1 + Sqrt[2] + 2 Sqrt[5] + Sqrt[10])), ((1 + 2 Sqrt[2] + 2 Sqrt[5]) (1 + 2 Sqrt[2] - Sqrt[ 10])^2 (-1 + 3 Sqrt[2] - Sqrt[10])^2 (1 + 2 Sqrt[5] - Sqrt[ 10]) (3 + Sqrt[2] + 3 Sqrt[5] - Sqrt[10]) (2 + Sqrt[2] - 2 Sqrt[5] + Sqrt[10]))/((1 + 3 Sqrt[2] - Sqrt[10]) (-1 + 2 Sqrt[5] - Sqrt[10]) (-1 + 3 Sqrt[5] - Sqrt[10])^2 (-2 - Sqrt[ 2] + 3 Sqrt[5] - Sqrt[10]) (3 + Sqrt[2] - 3 Sqrt[5] + Sqrt[ 10]) (1 - Sqrt[5] + Sqrt[10]))} In[972]:= RootReduce@% Out[972]= {1, 1, 1, 1} I wonder if this calculation was humanly possible. --rwg On 2018-07-22 07:04, Fred Lunnon wrote: > But you are not comparing like with like --- the LLL products do have > more than single digit coefficients; 43267971744 is (quite) large; > and output from LLL can hardly be considered uniformly random anyway! > > > On 7/22/18, Bill Gosper <billgosper(a)gmail.com> wrote: >> FindIntegerNullVector constructed 47 products of linear combinations of >> √2, √5, and √10, all = 1. The very first thing it found was >> >> (3 + 3 √2 - 2 √5 - √10) (3 - 3 √2 - 2 √5 + √10). >> >> Is this even obviously rational? It's 1 iff >> (3 - 3 √2 + 2 √5 - √10) (3 + 3 √2 + 2 √5 + √10) is. >> >> Things got rapidly worse: >> >> (√2 - 1) (3 + √2 - √10) (43 + 26 √2 + 10 √5 + 11 √10)/71, >> >> until bewilderment: replaced doubt: >> >> (51 + 38 √2 - 15 √5 - 16 √10) (1 + 3 √2 - √10) (-1 + 2 √5 - √10) >> (1 - √2 + 2 √5 - √10) (-1 + 3 √5 - √10)^3 (4 + 3 √2 + √10) >> (3 + √2 - 3 √5 + √10) (3 + 7√2/3 + 4√5/3 + √10) >> (17 + 6 √2 + 8 √5 + √10)^3 (-29 - 20 √2 + 18 √5 + 11 √10)/43267971744 >> >> In:= RootReduce@% >> Out= 1 >> >> Such things ought to be trivial to construct: >> Just multiply a random pile of surds by their reciprocal, with denominator >> rationalized. But I'd expect these reciprocals to have coefficients a lot >> bigger than 51: >> >> In[707]:= RandomInteger[{-9, 9}, {4, 4, 4}].Sqrt[{1, 2, 5, 10}] >> >> Out[707]= {{6 + 5 Sqrt[2] + 8 Sqrt[10], >> 6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10], -8 + 5 Sqrt[2] - >> 9 Sqrt[5] + 2 Sqrt[10], >> 5 + 3 Sqrt[2] - 8 Sqrt[5] + 5 Sqrt[10]}, {2 + 4 Sqrt[2] + >> 6 Sqrt[5] + 9 Sqrt[10], 1 - 5 Sqrt[2] + 8 Sqrt[5] - Sqrt[10], >> 1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10], -5 + 8 Sqrt[2] + 5 Sqrt[5] + >> 3 Sqrt[10]}, {-2 + 3 Sqrt[2] - 2 Sqrt[5] + 7 Sqrt[10], >> 5 - 6 Sqrt[2] - 6 Sqrt[5] - 9 Sqrt[10], -3 + 4 Sqrt[2] - >> 4 Sqrt[5] + 5 Sqrt[10], -1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]}, {1, >> 5 + 4 Sqrt[2] + 9 Sqrt[5] + 9 Sqrt[10], >> 4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10], >> 3 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]}} >> >> In[708]:= Times @@ # & /@ % >> >> Out[708]= {(6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10]) (-8 + 5 Sqrt[2] - >> 9 Sqrt[5] + 2 Sqrt[10]) (5 + 3 Sqrt[2] - 8 Sqrt[5] + >> 5 Sqrt[10]) (6 + 5 Sqrt[2] + 8 Sqrt[10]), (1 - 5 Sqrt[2] + >> 8 Sqrt[5] - Sqrt[10]) (-5 + 8 Sqrt[2] + 5 Sqrt[5] + >> 3 Sqrt[10]) (1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10]) (2 + >> 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]), (5 - 6 Sqrt[2] - 6 Sqrt[5] - >> 9 Sqrt[10]) (-1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]) (-3 + >> 4 Sqrt[2] - 4 Sqrt[5] + 5 Sqrt[10]) (-2 + 3 Sqrt[2] - 2 Sqrt[5] + >> 7 Sqrt[10]), (4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10]) (3 + >> 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]) (5 + 4 Sqrt[2] + 9 Sqrt[5] + >> 9 Sqrt[10])} >> >> made four quadruples with single digit coefficients. Reciprocating, >> In[709]:= RationalizeDenominator /@ (1/%) >> >> Out[709]= {(-8740961902019984 - 5440587070445184 Sqrt[2] + >> 2455653039709912 Sqrt[5] + 1573532047536568 Sqrt[10])/ >> 510888132844109317136, (4549175633470544 + >> 1131438893999820 Sqrt[2] - 1605347641058392 Sqrt[5] + >> 706649915148436 Sqrt[10])/ >> 968178139395832169104, (508007526470048 - 194043325345184 Sqrt[2] - >> 158757589376544 Sqrt[5] + 99676189957856 Sqrt[10])/ >> 20517561741702420224, (-102601486842 + 402200749082 Sqrt[2] + >> 176375285826 Sqrt[5] - 343223324388 Sqrt[10])/7676200432663996}

1 0

[math-fun] Looks fishy; Smells OK.
by Bill Gosper 22 Jul '18

22 Jul '18

FindIntegerNullVector constructed 47 products of linear combinations of √2, √5, and √10, all = 1. The very first thing it found was (3 + 3 √2 - 2 √5 - √10) (3 - 3 √2 - 2 √5 + √10). Is this even obviously rational? It's 1 iff (3 - 3 √2 + 2 √5 - √10) (3 + 3 √2 + 2 √5 + √10) is. Things got rapidly worse: (√2 - 1) (3 + √2 - √10) (43 + 26 √2 + 10 √5 + 11 √10)/71, until bewilderment: replaced doubt: (51 + 38 √2 - 15 √5 - 16 √10) (1 + 3 √2 - √10) (-1 + 2 √5 - √10) (1 - √2 + 2 √5 - √10) (-1 + 3 √5 - √10)^3 (4 + 3 √2 + √10) (3 + √2 - 3 √5 + √10) (3 + 7√2/3 + 4√5/3 + √10) (17 + 6 √2 + 8 √5 + √10)^3 (-29 - 20 √2 + 18 √5 + 11 √10)/43267971744 In:= RootReduce@% Out= 1 Such things ought to be trivial to construct: Just multiply a random pile of surds by their reciprocal, with denominator rationalized. But I'd expect these reciprocals to have coefficients a lot bigger than 51: In[707]:= RandomInteger[{-9, 9}, {4, 4, 4}].Sqrt[{1, 2, 5, 10}] Out[707]= {{6 + 5 Sqrt[2] + 8 Sqrt[10], 6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10], -8 + 5 Sqrt[2] - 9 Sqrt[5] + 2 Sqrt[10], 5 + 3 Sqrt[2] - 8 Sqrt[5] + 5 Sqrt[10]}, {2 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10], 1 - 5 Sqrt[2] + 8 Sqrt[5] - Sqrt[10], 1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10], -5 + 8 Sqrt[2] + 5 Sqrt[5] + 3 Sqrt[10]}, {-2 + 3 Sqrt[2] - 2 Sqrt[5] + 7 Sqrt[10], 5 - 6 Sqrt[2] - 6 Sqrt[5] - 9 Sqrt[10], -3 + 4 Sqrt[2] - 4 Sqrt[5] + 5 Sqrt[10], -1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]}, {1, 5 + 4 Sqrt[2] + 9 Sqrt[5] + 9 Sqrt[10], 4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10], 3 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]}} In[708]:= Times @@ # & /@ % Out[708]= {(6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10]) (-8 + 5 Sqrt[2] - 9 Sqrt[5] + 2 Sqrt[10]) (5 + 3 Sqrt[2] - 8 Sqrt[5] + 5 Sqrt[10]) (6 + 5 Sqrt[2] + 8 Sqrt[10]), (1 - 5 Sqrt[2] + 8 Sqrt[5] - Sqrt[10]) (-5 + 8 Sqrt[2] + 5 Sqrt[5] + 3 Sqrt[10]) (1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10]) (2 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]), (5 - 6 Sqrt[2] - 6 Sqrt[5] - 9 Sqrt[10]) (-1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]) (-3 + 4 Sqrt[2] - 4 Sqrt[5] + 5 Sqrt[10]) (-2 + 3 Sqrt[2] - 2 Sqrt[5] + 7 Sqrt[10]), (4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10]) (3 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]) (5 + 4 Sqrt[2] + 9 Sqrt[5] + 9 Sqrt[10])} made four quadruples with single digit coefficients. Reciprocating, In[709]:= RationalizeDenominator /@ (1/%) Out[709]= {(-8740961902019984 - 5440587070445184 Sqrt[2] + 2455653039709912 Sqrt[5] + 1573532047536568 Sqrt[10])/ 510888132844109317136, (4549175633470544 + 1131438893999820 Sqrt[2] - 1605347641058392 Sqrt[5] + 706649915148436 Sqrt[10])/ 968178139395832169104, (508007526470048 - 194043325345184 Sqrt[2] - 158757589376544 Sqrt[5] + 99676189957856 Sqrt[10])/ 20517561741702420224, (-102601486842 + 402200749082 Sqrt[2] + 176375285826 Sqrt[5] - 343223324388 Sqrt[10])/7676200432663996} --rwg

2 1

[math-fun] Everyone's a winner! (was Re: What is the simplex best at being best at?)
by Keith F. Lynch 21 Jul '18

21 Jul '18

James Propp <jamespropp(a)gmail.com> wrote: > In no sense did I intend a real math question. But I suspect that > the meta-game of asking ?Are there ten things I can do that nobody > else can do all of? How about nine? How about eight?? (or a > variant) has a name in pop (nerd) culture, and I was hoping to > elicit it from one of you. That would be a very difficult question to answer. Not the question of whether your first question has a name (though if it does, I don't know it), but your first question itself. (Speaking of questions, the digest option on this list turns so-called "smart" quotation marks and all other non-ASCII characters into question marks. So I wish people would stick to ASCII. Thanks.) A few weeks ago, I was in a discussion with friends, and they started speculating on which historical person had the highest-ever IQ. For instance did Archimedes have a higher IQ than Newton? I think that just might depend on what language the test was in. More generally, I think that's a profoundly misguided way of looking at the world, not much more meaningful than asking who would do best at playing your favorite video game. I made the perhaps over-bold claim that since there are more possible tasks than there are people, everyone can be the best in the world at *something* if they work at it diligently. I'll moderate it to say that almost anyone without any major disability can be in the top ten in something if they're willing to put in the effort. I also mentioned United Flight 232, which lost all its hydraulics, making it unflyable. The pilot managed to land it anyway. It wasn't a particularly good landing, but the majority of passengers did survive. In all other cases of a passenger jet losing all its hydraulics, everyone died. And there have been countless computer simulations of that situation, all of which ended in a crash. The Flight 232 passengers should be thankful that their pilot was someone very good at piloting rather than someone very good at taking IQ tests. How many things are there that it would be possible to be the best at the world in? You could start with the Olympic records and with the Guinness Book. Or with one of those lists of astonishing human achievements, such as Lynne Cox, who swam from Alaska to Russia through miles of freezing-cold water, wearing just an ordinary bathing suit. Or just look at some relatively narrow field such as ham radio, in which there are records for lowest-power trans-Atlantic contact (less than one microwatt), most countries communicated with in one day/week/month/year, shortest time to communicate with all countries, highest data rate, lowest data rate (I think the current record is one bit per day), lowest frequency, highest frequency, most contacts in one day, lowest power for bouncing a signal off the moon, etc. Then multiply the total by the number of hobbies, professions, sports, games, etc., that exist. Also, of course, for everything that's ever been done, someone holds the record for having done it first. Like most people here, there are a few math problems I was the first or only person to solve, and a few others for which I think I have the best algorithm for searching for solutions. This is of course doesn't mean I'm the person best able to solve such problems, but that they didn't happen to catch the attention of anyone better suited to solve them who had the free time available. I may hold the record for the geekiest ever Wikipedia update. I was casually browsing some of its math-related articles when I happened to notice that the twin primes constant was wrong in the 7th decimal place, so I fixed it. There are records that most of us would like to set, such as being the world's oldest person ever, beating Jeanne Calment, who lived to be 122. There are records that none of us would like to set, such as being the American wrongfully imprisoned for the longest. Richard Dan Phillips of Michigan holds the record for being exonerated and freed after 45 years (1972-2017), but the wrongfully-imprisoned record is almost certainly held by someone who was never exonerated.

2 1