June 2014 - mathfun@mailman.xmission.com

[math-fun] Fwd: Spirography
by Bill Gosper 18 Jun '14

18 Jun '14

---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> Date: Wed, Jun 18, 2014 at 2:17 AM Subject: Re: [math-fun] Spirography To: Bill Gosper <billgosper(a)gmail.com> I don't think it can be made to lock. The arc described by any point on a tooth of the inside gear will have curvature of constant sign—i.e. it makes a little convex loop when it comes in to touch the annulus (see picture). It looks like the shape of the loop is such that you can't force locking, but can come fairly close with rounded, convex inner teeth and spiky, concave outer teeth. Of course, the precise shapes would be peculiar to the radii of the disks involved, which would make for fewer combinations. I think the best thing that could be done would be to use three dimensions to effectively add more teeth, by having several layers of teeth, making the teeth slope, or replacing the teeth with an undulating ridge-into-slot mechanism. Caveats: Nothing I said above was rigorous; it was based entirely on intuition from a model of a very special case. The model is also inaccurate, as it assumes that the inner gear rolls along a circle (so that the pen draws a hypotrochoid) rather than pivoting about successive contact points (approximating the hypotrochoid by a series of small arcs). The latter may render my intuition invalid, especially for small numbers of teeth (for lots of teeth you would probably need to make the teeth very short to see the effect), as it was based on looking at hypotrochoids. Actually, I made a model of the small-number-of-teeth version, and the pictures it makes are a neat-looking variation on spirographs: Module[{n = 8, r0 = 6, r1 = 4, r2 = 1, \[Theta]0 = 0}, Manipulate[ParametricPlot[Block[{m = Round[2*\[Pi]*t/(2*(\[Pi]/2 - \[Pi]/n - ArcCos[r0*Sin[\[Pi]/n]/r1]))], t0 = Round[2*\[Pi]*t/(2*(\[Pi]/2 - \[Pi]/n - ArcCos[r0*Sin[\[Pi]/n]/r1]))]*(1/2 - 1/n - ArcCos[r0*Sin[\[Pi]/n]/r1]/\[Pi])}, ri[r0*e[m/n] + r1*e[-(t - t0) + 1/2 + m/n] + r2*e[(t0 - t) + (\[Theta]0 - t0)]]], {t, 0, t1}, Axes -> None, PlotRange -> {{-1.1*r0, 1.1*r0}, {-1.1*r0, 1.1*r0}}, ImageSize -> 400], {t1, 1, 30}]] with e[z_] := Exp[2*I*\[Pi]*z], ri = ({Re[#], Im[#]} &), r2≤r1≤r0, and r1>r0*Sin[π/n]. Julian, who keeps typing "teetch" On Mon, Jun 16, 2014 at 4:40 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > Allan Wechsler>I want Gosper's young friend Julian's opinion on this. I'm > still betting that there are locking solutions for moderate numbers of > teeth, say, eight. In fact, I suspect there are locking solutions for all > numbers of teeth, but that for larger numbers the grip would be extremely > tenuous. On Mon, Jun 16, 2014 at 3:57 PM, meekerdb <meekerdb(a)verizon.net> > wrote: > > On 6/16/2014 10:14 AM, Allan Wechsler wrote: > > The patent office calls similar devices "rose engines". As for locking > gears, I have an intuition that the thing is possible with teeth that have > concave shanks and a widened head, which can fit between the heads of the > opposing teeth when at an angle, but not when straight on. (One would > connect them by using the third dimension, dropping a new gear onto the > surface to intermesh with one already there.) My intuition also says that > the thing becomes much more delicate when there are lots of teeth. > > I doubt that can be made to work. As the teeth unmesh they are pulled out > almost orthogonally. You could make gears that resisted unmeshing by > embedding magnets in the teeth. Brent Meeker > > gosper.org/Spirographgears.png

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[math-fun] keyword frequencies in programming languages
by Warren D Smith 17 Jun '14

17 Jun '14

I got some large programs in a few languages and counted how often various words appeared. Sometimes I cheated a bit. QUESTION: I would appreciate it is anybody could supply corresponding data for some large programs in some other languages such as LISP, ML, or whatever. Here are my results: PASCAL chess0.5 by P.Frey+L.Atkin 1978: ":="=666, END=316, BEGIN=287, IF=205, THEN=205, DO=140, PROCEDURE=115, OF=107, VAR=80, ARRAY=74, ELSE=67, TO=63, FOR=66, FALSE=55, TRUE=48, WHILE=43, GOTO=42, WRITE=39, WRITELN=37, AND=32, NOT=32, WITH=31, CASE=27, FUNCTION=21, ABS=14, LABEL=11, DIV=9, DOWNTO=7. PACKED=6, RECORD=6, UNTIL=5, TYPE=2, NAME=2, ONLY=1, CONST=1. C++ Gull-II chess program by Vadim Demichev: " = "=2189, IF=1409, "{"=1006, INT=498, DEFINE=362, ELSE=327, RETURN=282, FOR=260, UINT=205, DO=168, LSB=203, VOID=81, CONST=80, BOOL=78, TEMPLATE=72, GOTO=69, FPRINTF=65, ENDIF=63, STDOUT=59, MIN=58, CONTINUE=57, IFDEF=56, SIZEOF=51, DO=49, CHAR=41, BREAK=32, UNDEF=30, POPCOUNT=28, VOLATILE=26, INLINE=22, WHILE=20, MEMSET=19, STRCMP=18, FILE=18, DOUBLE=15, TYPEDEF=15, STRUCT=14, CASE=14, UNSIGNED=13, RAND=12, DEFAULT=8, MAX=7, ENUM=4. C++ Senpai1.0 chess program by Fabian Letouzey: " = "=1299, INT=1293, "{"=1188, IF=441, RETURN=426, CONST=308, ASSERT=286, FOR=206, VOID=179, STD=159, BOOL=138, FILE=131, FALSE=128, ELSE=119, TRUE=93, POPCOUNT=59, UINT=45, LSB=43, STATIC=27, CASE=26, MAX=26, BREAK=22, MIN=21, RAND=21, CHAR=15, DOUBLE=19, WHILE=18, ENUM=14, STRUCT=13, TYPEDEF=12, VOLATILE=11, SIGNED=11, CONTINUE=8, DEFAULT=7, SIZEOF=6, DEFINE=6. C Bzip2 de/compress program by Julian Seward: " = "=1144, IF=701, "{"=573, INT=451, RETURN=276, DEFINE=218, FOR=208, UCHAR=151, VOID=134, UINT=134, BREAK=110, FILE=98, ELSE=96, WHILE=94, STDERR=89, FPRINTF=86, STATIC=83, TRUE=82, CHAR=80, CASE=79, GOTO=46, BOOL=45, MAX=41, CONTINUE=40, UNDEF=35, ENDIF=35, UNSIGNED=32, RAND=32, SIZEOF=27, INCLUDE=26, QSORT=24, ASSERT=24, STDOUT=21, DO=21, FCLOSE=19, EOF=19, FOPEN=15, TYPEDEF=15, STDIN=15, CONST=13, STRLEN=10. C: Gcc C-compiler (now dead version circa 2004): " = "=83693, "{"=59732, IF=54646, RETURN=31658, RAND=29586, CASE=28527, FOR=26467, STR=22012 (count includes many functions), ELSE=21005, CONST=16284, GOTO=15923, INT=13078, VOID=12765, STATIC=10312, DEFINE=10004, BREAK=8035, CHAR=7578, ENDIF=6755, UNSIGNED=6512, MIN=4374, ENUM=4181, TRUE=3328, DO=3271, PRINTF=2884, DEFAULT=2519, FALSE=2366, BOOL=2335, MAX=2278, SWITCH=2215, WHILE=2177, SIZEOF=1924, INLINE=1552, FILE=1302, CONTINUE=1301, UNDEF=1159, FLOAT=986, INCLUDE=845, CLEAR=687, UNION=669, SIGNED=656, DOUBLE=621, TYPEDEF=513, VOLATILE=430, STDERR=400, SWAP=346, MEMSET=335, UCHAR=278, SORT=275, UINT=191, ASSERT=129, POPCOUNT=46, STDOUT=21. Fortran: 140 programs collected by Don Knuth 1971, I just repeat his counts from 1st column of table 1 in http://www.cs.tufts.edu/~nr/cs257/archive/don-knuth/empirical-fortran.pdf : (Assignment)=41%, IF=14.5%, GOTO=13, CALL=8, CONTINUE=5, WRITE=4, FORMAT=4, DO=4, DATA=2, RETURN=2, DIMENSION=2, COMMON=1.5, END=1, BUFFER=1, SUBROUTINE=1, REWIND=1. Others: Christian S. Collberg, Ginger Myles, Michael Stepp: An empirical study of Java bytecode programs, Software Practice and Experience 37,6 (2007) 581-641. http://goto.ucsd.edu/~mstepp/publications/empirical.pdf They got 1132 java programs in bytecode, not source, form, and counted the bytecode frequencies plus much more. One interesting finding is, if you look at floating point constants in programs, 44% of them are in {0, 1, 0.5, 2.0} and you can boost 44% up to 56% if you also put in {255.0 100.0 -1.0 4.0 5.0 -inf 10.0 0.9 0.75 1000.0 64.0 3.0 pi NaN 20.0 4.0 90.0 0.25 8.0 180.0 2*pi 0.1 6.0 HUGE 360.0 1.0e-4 -2.0 pi/2 sqrt(0.5) +inf } thus proving the real numbers have a lot less entropy than you thought :) Michael D. Ernst, Greg J. Badros, and David Notkin: An empirical analysis of C preprocessor use, IEEE Transactions on Software Engineering 28,12 (2002) 1146-1170. http://homes.cs.washington.edu/~mernst/pubs/c-preprocessor-tse2002.pdf Robert P. Cook & Insup Lee: A contextual analysis of Pascal programs, Software: Practice and Experience 12,2 (February 1982) 195-203 [can anybody supply PDF?] examined 264 pascal programs, partial info here: http://warriors.eecs.umich.edu/old_schedules/Readings/2001-03-28-Pascal.pdf

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[math-fun] keyword frequencies in programming languages
by Warren D Smith 17 Jun '14

17 Jun '14

> From: Andy Latto <andy.latto(a)pobox.com> > What would you consider the "keywords" for lisp; the special forms? > Seems somewhat arbitrary; for example, common lisp specifies that IF > is a special form and COND is a macro, but could easily have made the > opposite choice, and only a small number of programs that do analysis > of other programs would need to be any different. > > Andy --I do not know. As Andy correctly points out, there is a lot of arbitrariness is defining exactly what we want these statistics to be ... library calls, common idioms, and macros seem fair game even though perhaps technically not "keywords" ... so the vague answer is "just do something reasonable." You could try macsyma as a big LISP program, I guess (I'm assuming various math-funners have the LISP source for it and tons of expertise, which I don't) And to do a really good job, you really should instrument a compiler, or something, to gain a high degree of understanding of the programs you are collecting stats on. However, I'm way too lazy to do that. I used fairly crude word counting scripts and made some arbitrary decisions about what to count. So my counts are not as accurate as they could be, but it does not matter all that much. I redid my counts for gcc, by the way, by making my counter scripts remove comments before counting (which for some programs makes little difference, but in the case of gcc it made a considerable difference) and also added counts for some rather-arbitrarily selected characters and character combinations. The new gcc results are: " = "=82943, "-"=73769, IF=72024, "{"=59278, ">"=39119, "=="=33504, " & "=32425, RETURN=26285, RAND=29586, "&&"=26653, CASE=26098, STR=15288 (count includes many functions), ELSE=20575, CONST=16284, GOTO=15666, " + "=14668, "<"=13228, VOID=11991, INT=11824, FOR=10148, "!="=10095, STATIC=9980, DEFINE=8750, BREAK=7925, ENDIF=6755, CHAR=6524, UNSIGNED=6512, "++"=4869, ENUM=3844, PRINTF=2798, DEFAULT=2519, TRUE=2352, BOOL=2263, FALSE=2105, MIN=1961, MAX=1927, WHILE=1929, SIZEOF=1924, SWITCH=1915, "+="=1617, "/"=1327, CONTINUE=1185, UNDEF=1159, INLINE=1138, FILE=1129, INCLUDE=845, SIGNED=656, FLOAT=541, TYPEDEF=513, DOUBLE=457, UNION=438, DO=438, CLEAR=395, STDERR=360, MEMSET=316, "^"=303, "&="=296, SORT=275, UCHAR=274, SWAP=211, VOLATILE=194, UINT=181, ASSERT=129, "*="=57, CLZ=46, CTZ=36, "^="=29, POPCOUNT=28, STDIN=24, FCLOSE=18, STDOUT=14, LSB=11. and you will notice that indeed, some of the stuff I counted like MIN, POPCOUNT, RAND, SWAP, MEMSET are not C keywords, they are macros or library calls.

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Re: [math-fun] Math-fun is discussing spirographs
by Bill Gosper 17 Jun '14

17 Jun '14

[Goddam idiot xmission.com quarantined my first try as overlong, due to the attachments I left in for eavesdroppers, even though it deletes all attachments!] On Mon, Jun 16, 2014 at 11:08 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > --------- Forwarded message ---------- > From: John Edmark <edmark(a)stanford.edu> > Date: Mon, Jun 16, 2014 at 10:40 PM > Subject: Re: Math-fun is discussing spirographs > To: Bill Gosper <billgosper(a)gmail.com> > Cc: Julian Ziegler Hunts <julianj.zh(a)gmail.com>, Neil Bickford < > techie314(a)gmail.com> > > > Hi Bill, > > I have not yet put my hyper-spirograph info on my website. I've been > meaning to, so perhaps I'll use this as an impetus to do so. In the > meantime, I've attached some images of the drawings it can make, > > [now http://gosper.org/edmark1.jpg, http://gosper.org/edmark2.jpg] > > and included a link to a movie of it in action: > > https://www.dropbox.com/s/ckrzsctrjofb85w/Spirograph%20Drawing%20%28Short%2… > > I have .dxf files of my hyper-spirograph that I'm happy to make available. > They can be used to laser-cut 1/8" acrylic. Some very simple assembly with > acrylic glue produces a working mechanism with several gears and annuli. > > This version of a Spirograph, solves the disengagement problem by making > the inner gears just one tooth smaller than the annulus. It's a clever, > albeit option-limiting, idea: > > http://www.amazon.com/Klutz-Book-Spiral-Doug-Stillinger/dp/0545459923/ref=s… > > (BTW, the above was actually the inspiration for an idea that can be found > at www.JohnEdmark.com. choose Videos, category "half baked" and then > "Nested Gears") > > There are a number of virtual Spirograph apps out there on the web. > > Julian created a hyper-spirograph app, but has chosen not to publish it, > since it doesn't absolutely precisely model what happens with the actual > hyper-spirograph. (In my view it is still a very cool app, and I wish he > would make it public.) > > Cheers, > > -John > > > > On 6/16/14, 11:06 AM, Bill Gosper wrote: > > Hi John, is there anything on the Web about your recursive one? > --Bill Gosper > ----------- > Henry (and others), > > I know you wanted a physical object, but well-constructed virtual objects > can be just as much fun, especially in today's world of touch screens. > > Actually, I'd also be happy with a cool iPhone app, and I was going to > post on this subject eventually, if nobody else did, once the subject of > physical spirographs had been adequately discussed. More on virtual objects > below. > > It's also only a matter of a few years before binocular VR glasses & > haptic gloves are available, at which point you will be able to manipulate > your virtual objects as if they were real. > > I look forward to that. (I recently gave an address at a math department's > awards ceremonies, in which I told the graduating math majors that it was > their generation that would make pure mathematics tangible via virtual > reality interfaces, and that I was counting on them.) > > Also, we should be able to utilize 3D printers today that have the > resolution to build our own spirograph, complete with gear teeth that work. > > Well, the gear teeth in Hasbro's spirographs work fine, as long as forces > are applied in the right direction, and NO vanilla gear mechanism will work > if you pull the gear apart from each other! So the question is, how can we > modify the gear mechanism to resist forces that would otherwise pull the > gears apart? Getting back to Henry's suggestion, I ask you all: What sort > of spirography software exists (or has Hasbro's copyright exerted a > dampening effect on people's creativity)? One thing I can imagine is > software that lets the user draw two freehand closed curves. The first > curve will remain stationary throughout the drawing process; the second > curve will roll around the first, maintaining tangency and avoiding > slippage. The user also specifies one or more points relative to the second > curve that will serve as drawing tips, and associated colors and > tip-thicknesses. The computer will perform rescalings of the hand-drawn > curves if the user wishes the lengths of the two curves to have a certain > specified ratio. And hey, if you prefer a parametric curve, or a curve > specified as (a component of) the locus of some two-variable equation over > the reals, you can specify that too. Does this exist, or can someone create > it? I don't think I'm the only person who'd love a toy like this. Jim > > At 08:54 AM 6/16/2014, James Propp wrote: > > I bought my kids a spirograph (a favorite toy of mine in my youth), and it > seems that the manufacturers have had a few fun ideas in the intervening > decades. In particular, there's a wider variety of gear-toothed laminae one > can now roll inside or outside of a gear-toothed annulus, resulting in a > wider variety of pictures. But one of the basic problems of spirography > circa 1970 remains a problem in 2014: slippage. The gear-teeth only serve > their stabilizing function > > if > > one continually makes sure that one is applying force to the pen (inserted > into a hole in the moving lamina) in such a way that the normal component > between the lamina and the annulus has the correct sign, while with the > other hand steadily pushes down on the annulus to keep it stationary. > > This > > is hard work for my five-year-old daughter, who would probably use the > spirograph more if slippage weren't such a problem. (One could argue that > the main purposes of a spirograph never was mathematical enlightenment but > the encouragement of hand-eye coordination, and that the slippage thing is > a feature rather than a bug, but I'm going to pretend I didn't say that.) > Has anyone designed a mechanism that permits the spirographer to focus on > circumferential force? I'm imagining something like a latchable/unlatchable > zipper. Once you've zipped the lamina and the annulus together (making use > of the fact that they're in three > > dimensions), > > they stick together like the two halves of a zipper, and permit only a > single, circumferential degree of freedom. If someone could solve that > problem, then I think the other problem (holding the annulus stationary) > wouldn't be as big an issue (though it might still be good to have a wax > tray and some pushpins to hold the annulus and the paper to the wax tray, > so that one-handed people could do spirographs). Does anyone know of > progress that's been made along these lines, or does anyone have any good > ideas for mechanisms that would prevent slippage? Jim Propp P.S. Wikipedia > tells me that "spirograph" is a registered trademark of Hasbro, so please > pretend I capitalized it, etc. > > > -- > John Edmark > Lecturer in Design > Dept. of Art & Art History > Stanford University > 650.564.4884 > edmark(a)stanford.edu > www.JohnEdmark.com > >

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Re: [math-fun] Spirography
by Henry Baker 16 Jun '14

16 Jun '14

I know you wanted a physical object, but well-constructed virtual objects can be just as much fun, especially in today's world of touch screens. It's also only a matter of a few years before binocular VR glasses & haptic gloves are available, at which point you will be able to manipulate your virtual objects as if they were real. Also, we should be able to utilize 3D printers today that have the resolution to build our own spirograph, complete with gear teeth that work. At 08:54 AM 6/16/2014, James Propp wrote: >I bought my kids a spirograph (a favorite toy of mine in my youth), and it >seems that the manufacturers have had a few fun ideas in the intervening >decades. In particular, there's a wider variety of gear-toothed laminae >one can now roll inside or outside of a gear-toothed annulus, resulting in >a wider variety of pictures. > >But one of the basic problems of spirography circa 1970 remains a problem >in 2014: slippage. The gear-teeth only serve their stabilizing function if >one continually makes sure that one is applying force to the pen (inserted >into a hole in the moving lamina) in such a way that the normal component >between the lamina and the annulus has the correct sign, while with the >other hand steadily pushes down on the annulus to keep it stationary. This >is hard work for my five-year-old daughter, who would probably use the >spirograph more if slippage weren't such a problem. (One could argue that >the main purposes of a spirograph never was mathematical enlightenment but >the encouragement of hand-eye coordination, and that the slippage thing is >a feature rather than a bug, but I'm going to pretend I didn't say that.) > >Has anyone designed a mechanism that permits the spirographer to focus on >circumferential force? I'm imagining something like a >latchable/unlatchable zipper. Once you've zipped the lamina and the >annulus together (making use of the fact that they're in three dimensions), >they stick together like the two halves of a zipper, and permit only a >single, circumferential degree of freedom. > >If someone could solve that problem, then I think the other problem >(holding the annulus stationary) wouldn't be as big an issue (though it >might still be good to have a wax tray and some pushpins to hold the >annulus and the paper to the wax tray, so that one-handed people could do >spirographs). > >Does anyone know of progress that's been made along these lines, or does >anyone have any good ideas for mechanisms that would prevent slippage? > >Jim Propp > >P.S. Wikipedia tells me that "spirograph" is a registered trademark of >Hasbro, so please pretend I capitalized it, etc.

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[math-fun] Spirography
by James Propp 16 Jun '14

16 Jun '14

I bought my kids a spirograph (a favorite toy of mine in my youth), and it seems that the manufacturers have had a few fun ideas in the intervening decades. In particular, there's a wider variety of gear-toothed laminae one can now roll inside or outside of a gear-toothed annulus, resulting in a wider variety of pictures. But one of the basic problems of spirography circa 1970 remains a problem in 2014: slippage. The gear-teeth only serve their stabilizing function if one continually makes sure that one is applying force to the pen (inserted into a hole in the moving lamina) in such a way that the normal component between the lamina and the annulus has the correct sign, while with the other hand steadily pushes down on the annulus to keep it stationary. This is hard work for my five-year-old daughter, who would probably use the spirograph more if slippage weren't such a problem. (One could argue that the main purposes of a spirograph never was mathematical enlightenment but the encouragement of hand-eye coordination, and that the slippage thing is a feature rather than a bug, but I'm going to pretend I didn't say that.) Has anyone designed a mechanism that permits the spirographer to focus on circumferential force? I'm imagining something like a latchable/unlatchable zipper. Once you've zipped the lamina and the annulus together (making use of the fact that they're in three dimensions), they stick together like the two halves of a zipper, and permit only a single, circumferential degree of freedom. If someone could solve that problem, then I think the other problem (holding the annulus stationary) wouldn't be as big an issue (though it might still be good to have a wax tray and some pushpins to hold the annulus and the paper to the wax tray, so that one-handed people could do spirographs). Does anyone know of progress that's been made along these lines, or does anyone have any good ideas for mechanisms that would prevent slippage? Jim Propp P.S. Wikipedia tells me that "spirograph" is a registered trademark of Hasbro, so please pretend I capitalized it, etc.

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[math-fun] test, please ignore
by rcs＠xmission.com 15 Jun '14

15 Jun '14

Please ignore this test message. -- Rich

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[math-fun] Further toward automating 𝜗 function special values
by Bill Gosper 15 Jun '14

15 Jun '14

It is presumably well known that, e.g., 𝜗n(a π,e^-(b π)) = algebraic Γ(1/4)/π^(3/4), a,b ∋ Q, so the only problem is to find the algebraic. Instead of PSLQing a big polynomial of unknown degree to unknown precision and then struggling to identify and simplify its relevant root, we can exploit the (Ramanujan's?) observation that these algebraics are often of a peculiar form, i.e. products of rational powers of "oligonomials". For example, targeting 𝜗₁(π/6,e^-(π/4)): In[146]:= Log[{(\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/Gamma[1/4], 2, 3, Sqrt[2] - 1, Sqrt[3] - 1, Sqrt[3] - Sqrt[2], Sqrt[6] - 1, 2^(1/4) - 1, 3^(1/4) - 1, 3^(1/4) - 2^(1/4), 6^(1/4) - 1, Sqrt[3] - Sqrt[2] + 2, Sqrt[2] - 3^(1/4)}]; In[147]:= FindIntegerNullVector[%].% Out[147]= 17 Log[2] - 6 Log[-1 + Sqrt[2]] - 3 Log[Sqrt[2] - 3^(1/4)] - 12 Log[-1 + 3^(1/4)] + 11 Log[-1 + Sqrt[3]] + 6 Log[-Sqrt[2] + Sqrt[3]] + 24 Log[(\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/ Gamma[1/4]] In[152]:= Solve[0 == ReplacePart[%147, {-1, 2} -> x]] Out[152]= {{x -> 1/24 (-17 Log[2] + 6 Log[-1 + Sqrt[2]] + 3 Log[Sqrt[2] - 3^(1/4)] + 12 Log[-1 + 3^(1/4)] - 11 Log[-1 + Sqrt[3]] - 6 Log[-Sqrt[2] + Sqrt[3]])}} So the answer is In[153]:= %147[[-1, 2, 1]] == Simplify[E^x /. %[[1]]] Out[153]= (\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/ Gamma[1/4] == ((-1 + Sqrt[2])^(1/4) (Sqrt[2] - 3^(1/4))^(1/8) Sqrt[-1 + 3^(1/4)])/( 2^(17/24) (-1 + Sqrt[3])^(11/24) (-Sqrt[2] + Sqrt[3])^(1/4)) In[154]:= N[List @@ %, 22] Out[154]= {0.3173112547318581725643, 0.3173112547318581725643} So we just collect a bunch of "logarithmically independent oligonomials". Unfortunately, when q=e^-(surd π), things get a bit hairier: Out[155]= EllipticTheta[1, \[Pi]/3, E^(-((Sqrt[19] \[Pi])/3))] == (1/( 4 2^(17/76) Sqrt[ 3] \[Pi]))(-2 + (-5 + Sqrt[57])/(1 + 3 Sqrt[57])^(1/3) + ( 4 (1 + 3 Sqrt[57])^(1/3))/(-5 + Sqrt[57])) (( Gamma[1/38] Gamma[5/38] Gamma[9/38] Gamma[11/38] Gamma[7/19] Gamma[ 17/38] Gamma[23/38] Gamma[25/38])/( 19 Gamma[13/38] Gamma[8/19]))^(1/4) In[156]:= N[List @@ %, 22] Out[156]= {0.5533022123819674888146, 0.5533022123819674888146} --rwg

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Re: [math-fun] Before I make a fool of myself to Wolfram Support
by Axel Vogt 15 Jun '14

15 Jun '14

> From: Bill Gosper <billgosper(a)gmail.com> > Subject: Re: [math-fun] Before I make a fool of myself to Wolfram ... > Axel Vogt> > I do not understand that Mathematica notations: do you mean > cot(k*(Pi)/p)*sin(k*(Pi))/p for p = 1 ... 4, k integer? <AV > > Exactly. E.g., for p=4, > Out[1745]= 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]] > > In[1746]:= Table[Limit[%, k -> n], {n, 0, 8}] > > Out[1746]= {1, 0, 0, 0, 1, 0, 0, 0, 1} > > (which certainly isn't identically 0). ... ... > In any event, the following is definitely wrong: > > Out[1500]= Cot[(\[Pi] x)/3] Sin[\[Pi] x] > > In[1501]:= Assuming[n \[Element] Integers, Limit[%, x -> n]] > > Out[1501]= 0 > > In[1502]:= Limit[%%, x -> 3] > > Out[1502]= -3 > > --rwg Maple 18 has problems with the general task as well and needs help to get it.

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Re: [math-fun] math-fun Digest, Vol 136, Issue 28
by Axel Vogt 15 Jun '14

15 Jun '14

> From: Bill Gosper <billgosper(a)gmail.com> > Subject: Re: [math-fun] Before I make a fool of myself to Wolfram ... > Axel Vogt> > I do not understand that Mathematica notations: do you mean > cot(k*(Pi)/p)*sin(k*(Pi))/p for p = 1 ... 4, k integer? <AV > > Exactly. E.g., for p=4, > Out[1745]= 1/4 Cot[(k \[Pi])/4] Sin[k \[Pi]] > > In[1746]:= Table[Limit[%, k -> n], {n, 0, 8}] > > Out[1746]= {1, 0, 0, 0, 1, 0, 0, 0, 1} > > (which certainly isn't identically 0). ... ... > In any event, the following is definitely wrong: > > Out[1500]= Cot[(\[Pi] x)/3] Sin[\[Pi] x] > > In[1501]:= Assuming[n \[Element] Integers, Limit[%, x -> n]] > > Out[1501]= 0 > > In[1502]:= Limit[%%, x -> 3] > > Out[1502]= -3 > > --rwg Maple 18 has problems with the general task as well and needs help to get it.

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