September 2014 - mathfun@mailman.xmission.com

[math-fun] Machinoid arcsin?
by Warren D Smith 24 Sep '14

24 Sep '14

Date: Tue, 23 Sep 2014 11:59:54 -0600 From: rcs(a)xmission.com The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239). ("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?) Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification. Rich --well... arcsin(x) = -i * ln( i*x + sqrt(1-x*x) ) arsinh(x) = ln( x + sqrt(x*x+1) ) arcsin(x) = -i * arsinh(i*x) = arctan( x / sqrt(1-x*x) ) so, from this plus Rich's arctan(N/D) = imag ln(D+iN) we can conclude arcsin(N/D) = imag( ln( sqrt(D*D-N*N) + i*N ) ) where |N|<=|D| so then the natural factorizations happen within the complex numbers with squared-integer norms [where norm(A+iB)=A*A+B*B] and real and imaginary parts which both are constructible-algebraic numbers. This is a ring, but a pretty nasty ring.

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[math-fun] Sum i^p_n/p_n ~ -1/2 - 0.334981325i
by Bill Gosper 24 Sep '14

24 Sep '14

(Finch, p95), which ISC teases is -1/2 - i ( 5*Zeta[3]/2 - 6*EulerGamma/5 - 9*Log[3]/5 -1.52145483*10^-8) We need a few more digits to rule this out. We can probably get them from Victor Miller>Also, the following paper "Zeta Expansions of Classical Constants" (for some reason never published): http://algo.inria.fr/flajolet/Publications/landau.ps < This is a bit hard to find, even under its current title "Zeta Function Expansions of Classical Constants". (Firefox absquatulated before I could grab URL.) But caution: Eqn (10) is almost certainly a copy&paste blunder from eqn (6). I.e., -1/1^m - ... - 1/(n0-1)^m should be -1/p_1^m - ... - 1/p_{n0-1}^m . On Sun, Jun 2, 2013 at 10:58 AM, Mike Stay <metaweta(a)gmail.com> wrote: http://mathworld.wolfram.com/PrimeProducts.html Lots of fun stuff here, like prod(i=1, inf, p_i^s) = (2pi)^2s On Sat, Jun 1, 2013 at 9:04 PM, Bill Gosper <billgosper(a)gmail.com> wrote: [Alan Adler, of Aeropress fame] 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 Consciously or not, I'll bet Steampunk was inspired by Hanna-Barbera's Flintstones. --rwg

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Re: [math-fun] Jörg's t-shirt rep13
by Bill Gosper 24 Sep '14

24 Sep '14

This is a resend. Friggingidiot xmission software embargoed the original because I neglected to remove Julian's pretty inline pic, even though it was going to delete it anyway! On Wed, Sep 24, 2014 at 1:21 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > * Bill Gosper <billgosper(a)gmail.com> [Sep 23. 2014 11:25]: > > * Bill Gosper <billgosper(a)gmail.com> [Sep 22. 2014 15:55]: [...] > > That's a beautiful clump of three rep-13s. But it's not obviously a clump > of 13 stars. > > jj>It should be (I could create it as a complex numeration system), but > time is up for now; have to postpone it. Indeed _all_ my "tiles" should > correspond to a numeration system, being rep-tiles, being doable as an IFS. > > rwg>Julian sent a possible recipe that might not be tested before you > leave. If it works, Betsy Ross can thank her lucky stars no Revolutionary > thought of it in her time. Julian also sent an amazing codeberg that > threatens to collect your t-shirt, if the offer survives dimacs. > > jj>Great, offer will stand. I have a student exploring ways to present > these curves in a setting like math-museum; the "amoeba flow" would be a > very welcome addition (we thought of it, but I found no time so far and she > doesn't want to program). Best, jj > > --rwg > > Lest there remain any doubt: > ---------- Forwarded message ---------- > From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> > Date: Tue, Sep 23, 2014 at 2:36 PM > Subject: Re: [math-fun] Jörg's t-shirt rep13 > To: Bill Gosper <billgosper(a)gmail.com> > > > L-system for the boundary: axiom F+F+F+F+F+F+, rewrite rule > F->F-F-F-F+F+F+F, angle=π/3 (this took me way too long to figure out). I > wanted this because it lets me make nicer-looking 1-into-13s than the > base-system method, because I can use Polygon rather than Point. Here's > how I generated the attached (not, obviously, all in one go; it took a > little debugging): > > Lsys2fns["F", "F-F-F-F+F+F+F", \[Pi]/3`30] > > Outer[#1[#2[#3[#4[0]]]] &, #, #, #, #] &[%]; > > Join @@ ((Function[L, L*#] /@ Exp[I*\[Pi]*Range[0, 5/3, 1/3]]) &@Flatten[% > - (7/2 - (I Sqrt[3])/2)]); > > Outer[Plus, (9/2 - 5*I*Sqrt[3]/2)*N[Join[{0}, Exp[I*\[Pi]*Range[0, 5]/3], > 2*Exp[I*\[Pi]*Range[0, 5]/3]], 30], %]; > > Polygon[ri /@ #] & /@ %; > > Graphics[Transpose[{Prepend[Hue /@ Accumulate[RandomReal[{1/6, 1/3}, 12]], > Black], %}], ImageSize -> 1000] > > gosper.org/13-flake order 5.png > <http://gosper.org/13-flake%20order%205.png> > You'll notice a couple of magic constants; they are ComplexExpand[z0 /. > (Solve[(y - z0)/(x - z0) == w, z0] /. {x -> 0, y -> 1 - 2*I*Sqrt[3], w -> > Exp[I*\[Pi]/3]})[[1]]] and (1-2*I*Sqrt[3])*jj13[1]/jj13[7/12], where > 1-2*I*Sqrt[3] is the last point in the fractal generated by the L-system > (with axiom F), jj13 is the original (non-flake) fractal, and jj13[7/12] is > the point where three of them touch when making a flake. > > Julian > > On Tue, Sep 23, 2014 at 12:32 PM, Bill Gosper <billgosper(a)gmail.com> > wrote: > >> On Tue, Sep 23, 2014 at 2:01 AM, Bill Gosper <billgosper(a)gmail.com> >> wrote: >> >>> * Bill Gosper <billgosper(a)gmail.com> [Sep 22. 2014 15:55]: >>> >>> jj>Extra points if you do that gif-ery for the curve >>> http://jjj.de/tmp-ryde/t-shirt-R13-15-hires-cropped.png >>> (L-system at bottom, turns are by 120 degrees). >>> Total curve should be light gray, and the moving part >>> should be 1/13 of the total (so the same shape will >>> appear 13 times, as indicated in the image). >>> ... plus you get a T-shirt! (If you use the image to, >>> well, print one. I got one and it looks quite OK). >>> >>> Best, jj >>> >>> That highly symmetric (almost dihedral 6) supersnowflake from >>> three of these in a loop--is that also rep13? Yours? >>> >>> >>> I assume you mean page 10 of >>> http://jjj.de/tmp-rwg/all-r13-tiles.pdf >>> Here is generation 4 of the same: >>> http://jjj.de/tmp-rwg/r13-15-plus-tile-4.pdf >>> >>> >>> It has rotational 6-symmetry, which (apparently) can only happen >>> for orders of the form 6*k+1. >>> No flip-symmetry (so indeed not D_6 as you mention), and I am >>> fairly sure none of my curves has D_6. >>> >>> Mine? I'd think so. >>> >>> Note I'll be leaving tomorrow morning (that's in 24 hours from now) for >>> http://dimacs.rutgers.edu/Workshops/OEIS/announcement.html >>> and will be away from my mail until 12.10. >>> >>> Best, jj >>> >>> That's a beautiful clump of three rep-13s. But it's not obviously a clump >>> of 13 stars. Julian sent a possible recipe that might not be tested before >>> you leave. If it works, Betsy Ross can thank her lucky stars no >>> Revolutionary thought of it in her time. Julian also sent an amazing >>> codeberg that threatens to collect your t-shirt, if the offer survives dimacs. >>> >>> --rwg >>> >>> Here's Julian's evidence that it is indeed rep13: >> gosper.org/jjstar.png (site just froze--can't test) >> To get Julian's original solid fill, {}->{0}. >> Why wasn't this found earlier? Mandelbrot would have amygdalated with >> envy. >> It easily rivals the Kochflake and Franceflake, and will soon(?) have a >> large Googleprint. >> --rwg >> >> -- >> >

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Re: [math-fun] Jörg's t-shirt rep13
by Bill Gosper 23 Sep '14

23 Sep '14

* Bill Gosper <billgosper(a)gmail.com> [Sep 22. 2014 15:55]: jj>Extra points if you do that gif-ery for the curve http://jjj.de/tmp-ryde/t-shirt-R13-15-hires-cropped.png (L-system at bottom, turns are by 120 degrees). Total curve should be light gray, and the moving part should be 1/13 of the total (so the same shape will appear 13 times, as indicated in the image). ... plus you get a T-shirt! (If you use the image to, well, print one. I got one and it looks quite OK). Best, jj That highly symmetric (almost dihedral 6) supersnowflake from three of these in a loop--is that also rep13? Yours? I assume you mean page 10 of http://jjj.de/tmp-rwg/all-r13-tiles.pdf Here is generation 4 of the same: http://jjj.de/tmp-rwg/r13-15-plus-tile-4.pdf It has rotational 6-symmetry, which (apparently) can only happen for orders of the form 6*k+1. No flip-symmetry (so indeed not D_6 as you mention), and I am fairly sure none of my curves has D_6. Mine? I'd think so. Note I'll be leaving tomorrow morning (that's in 24 hours from now) for http://dimacs.rutgers.edu/Workshops/OEIS/announcement.html and will be away from my mail until 12.10. Best, jj That's a beautiful clump of three rep-13s. But it's not obviously a clump of 13 stars. Julian sent a possible recipe that might not be tested before you leave. If it works, Betsy Ross can thank her lucky stars no Revolutionary thought of it in her time. Julian also sent an amazing codeberg that threatens to collect your t-shirt, if the offer survives dimacs. --rwg

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[math-fun] online GKP
by Bill Gosper 23 Sep '14

23 Sep '14

MJCollins> This can also be found in chapter 7.5 of Graham/Knuth/Patashnik "Concrete Mathematics". WD> Which I find available online at https://notendur.hi.is/pgg/(ebook-pdf)%20-%20Mathematics%20-%20Concrete%20M… <https://notendur.hi.is/pgg/%28ebook-pdf%29%20-%20Mathematics%20-%20Concrete…> Whit This pdf is a case of getting what you pay for. Yes, it will probably teach you Raney's lemma, but it is an OCR of a book that someone has scribbled in, and is full of dangerous errors (e.g., missing minus signs) and hopeless garbles. In citing their reasons for creating the controversial Euler font for this book, GK&P cagily forgot to mention confounding piracy! However much you may dislike the Euler font, it is bliss compared to the insanely random caca from the OCRware trying to imitate it. --rwg The only thing nearly as ugly was the preview copy of his latest edition Addison Wesley sent DEK after punting their fine old printer and switching to phototypesetting. It was laughable. I wonder if anyone (in the Free World) would have paid money for it. It was why he had to drop everything and build TeX and MetaFont.

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[math-fun] Jörg's t-shirt rep13
by Bill Gosper 23 Sep '14

23 Sep '14

jj>Extra points if you do that gif-ery for the curve http://jjj.de/tmp-ryde/t-shirt-R13-15-hires-cropped.png (L-system at bottom, turns are by 120 degrees). Total curve should be light gray, and the moving part should be 1/13 of the total (so the same shape will appear 13 times, as indicated in the image). ... plus you get a T-shirt! (If you use the image to, well, print one. I got one and it looks quite OK). Best, jj That highly symmetric (almost dihedral 6) supersnowflake from three of these in a loop--is that also rep13? Yours? --rwg

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[math-fun] "wolfram language" video
by Warren D Smith 22 Sep '14

22 Sep '14

http://www.wolfram.com/language/intro-video.html?KeepThis=true&TB_iframe=tr… fascinating. I guess this is as close to creating an artificial intelligence as I've heard of anyone coming (or so the propaganda makes it appear).

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Re: [math-fun] mile-high (or more) towers?
by Henry Baker 22 Sep '14

22 Sep '14

Bill: Do you have any more details and/or references on McCarthy's & Moravec's schemes? Thx. At 08:32 PM 9/21/2014, Bill Gosper wrote: >John McCarthy once computed that an affordable column of tungsten (I forget his dimensions, >something like a hundred meters by a few inches) would be a crust penetrator. > >Hans Moravec designed towers many miles high rigidified by a circulating >stream of superconducting "niobium cannonballs". You could also build >very large "vacuum balloons" thus stiffened. >--rwg

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Re: [math-fun] mile-high (or more) towers?
by Henry Baker 22 Sep '14

22 Sep '14

As you can imagine, this "highest building" problem has been addressed by the best & brightest for the last 5000+ years. I can't come up with the correct reference right now, but the ideal shape for a tall structure made out of a uniform material is a long oval shape (curved at both the bottom & top) & braced with guy wires. It sits on one end point, while the taut wires keep it upright. It isn't particularly pretty! The "waist" has the largest circumference, which is necessary to counteract Euler buckling. You can build a similar tower without the guy wires, but it will be only 1/2 the height, because it sits on its "waist". If you move away from a uniform material, then an ideal structure is fractal. Most TV towers are simple triangular lattice structures. To make one into a fractal, simply replace each beam with another (smaller) triangular lattice. The reference below shows what the result looks like: http://www.london-institute.org/people/farr/fractals.shtml I don't believe that this reference (or any others that I can find) found the optimum number of fractal levels, nor the optimum scaling of each fractal level. There's also no mathematical proof that the resulting structure is optimal. In particular, it may be that a 2-material solution might be much more efficient, where one of the materials is good at compressive loads and the other material is good at tensile loads. Note that these fractal structures only address Euler buckling; you still have the problem of the basic compressive strength of the material, which can be crushed in a non-Eulerian manner. I was trying to come up with a "dynamic programming" (recursive descent) model to compute such things a few years ago, but I got diverted into another project. I have the ".stl" file for this "Farr" structure which I can email you if you would like. ".stl" files are used for 3D printing. So-called "tensegrity" structures have been investigated in great detail by NASA, because it wants to put large, light structures into space, but send them up in a "compressed" manner, and then pull them taut into their final shape. NASA has also looked at origami-like structures for the same reasons. At 01:19 PM 9/21/2014, Warren D Smith wrote: >Suppose we wanted to create an extremely high tower (think of it as a >very long vertical rod) by combining the facts that > (a) compressive strength of materials is large > (b) "buckling" instabilities can be prevented by active control. >Another idea is an extremely large-span arch (if a constant-diameter rod is bent into >the shape of a "cycloid" curve, then the stresses in this arch are purely compressive). >(Of course, it'd be a bummer if you had a power or software failure... or an airplane hit it...) >In fact back when I was an undergrad at MIT, some engineering prof was assuring us >gullible students this was a revolutionary idea that would completely >change structural engineering, and he was procuring big grants to do it. > >These are fine fantasies; now let's think a little about reality. (The results >of my analysis will suggest said professor was completely full of shit, albeit apparently a >successful con-man.) > >For a vertical tower, you could keep the stabilization turned on as you built it. However >for an arch, I'm not seeing how to do that. So it seems as though an actively-stabilized arch would >be pretty much impossible to build. (You could build with reinforcement which later was removed, but >not if we are doing this for an enormous arch beyond the size attainable without active >stabilization...) > >Percy Bridgeman, in his investigations of high pressures, was able to reach >up to 5 GPa pressures using high strength steels and 10 GPa pressures >(100,000 Kgforce per square cm) using tungsten-carbide/cobalt (6 wt% Co) composite material. >Later, 300 GPa pressures (comparable to those at Earth's core) >were achieved using "anvils" made of single crystal diamond >and 600 GPa using "2 stage" diamond anvils with the second (very small) stage made of >"nanocrystalline poly-crystal diamond." >Unfortunately, all of these apparatus eventually fail, one reason being "creep." >Even at only 0.5 GPa, today's commercial steel high pressure equipment >says you have to replace the "Bridgeman seal" every 2 years. > >In view of this, I doubt it is possible to maintain non-hydrostatic compressive stresses >of more than about 300 MPa (equals 30000 kgforce/cm^2) in any large structure. >That stress level would be sufficient to support a vertical steel rod, >of uniform diameter, 38 km high. The shrinkage of steel at this stress level >is about 1.5 parts per thousand. The infrastructure (sensors and actuators) >required to perform the active control would add additional weight -- >so in view of that plus the desire for some safety margin, perhaps >the upper limit really is only about 20 km. > >In contrast, the highest tower yet built is 830 meters in Dubai. This "Burj Khalifa" >is a non-guyed skyscraper containing residences and hotels. Its cross-section is >a 3-pointed star at base about 170 meters wide. >(Some other recent tall buildings have parts with separate towers linked >by girder/walkways and with wind-turbines attached to those girders which can provide >perhaps up to 20% of building's electrical needs.) It is mainly made of steel-reinforced concrete >and appears to be a technical success but economic failure. Stresses caused by the blowing of the wind >cause the top to sway about 2 meters horizontally, and there is a swinging "tuned mass damper" >inside the tower near the top which is something of a semi-active control to prevent >resonant swaying. As far as I understand this device is not controlled by any >intelligent mechanism, but it is an intentionally designed dynamic non-rigid mechanism >which I suppose could be considered as an "analog computer." Also the exterior of the building >is intentionally shaped in funny ways that computer simulations of wind flow think >make it unlikely to get into a destructive resonance with large wind vortices. > Previously the highest were radio masts made of metal periodic "lattices" (triangles) >with guy wires and dampers. Specifically the previous record holder was >the "Warsaw radio mast": 646 meters high, had equilateral triangle base 5 meters >on each side, and weighed about 420 metric tons, of which the guy wires were >about 80 tons. It was completed in the early 1970s but collapsed in 1991. >The immediate cause of the collapse was a maintenance error, >but it suffered from metal fatigue caused by wind stresses, and from difficulty trying to >keep it painted. This tower had 3 vertical steel tubes linked by latticework. Each of >the 3 main tubes was 245mm in diameter with wall thickness varying from 8 to 34 mm. >Thus the cross-sectional area of the tower near the base was > 3 * 245 * pi * 34 mm^2 = 78.5 cm^2 >for a compressive stress of 420000 kgforce / 78.5 cm^2 = 5350 kgforce/cm^2 = 53 MPa. > >So... first of all, passive prevention of buckling, via guy wires, only cost >about 20% of the Warsaw tower weight. If you also count all the latticework, >then maybe we are up to 50%. If active control cost more weight than that, what >would be the point? Well, I suppose there are circumstances >where it is infeasible to guy a structure... and the mass of the guy >wires presumably grows quadratically or faster with tower height while the mass of an >active control system might be hoped to grow only linearly... > >In a speed-V wind, assuming drag coefficient of 1, the total torque near bottom on a height-H >vertical tube of radius R, is about H^2*R*V^2 kg*sec^2/m^5. >Note the H^2, quadratic, growth term, indicating that linear-growth hope is eventually dead meat. > >If V=200 km/hr = 55.556 meter/sec and H=650 meter and R=0.245 meter, we find >torque = 320 million newton*meters. If this is 10^9 in all (3 tubes) >then the force on one tube that needs to be applied to counter this torque is >about 2*10^8 newtons which is 2*10^7 kgforce which is 20000 metric tons >of force. This far exceeds the weight of the tower supported by that tube >at only 420/3=140 metric tons. So actually, maximum wind-caused bending stresses >far exceed the gravitational stresses, hence the guy wires are very important. > >Active control could be done by measuring torque >(bending stress) on the rod, and applying compensating torques. >In this way, the bending stresses everywhere could permanently be kept below some constant. >Glued-on silicon and metal-foil strain gauges could be used for this, but you need >temperature compensation and compensation for stresses caused by the solidification >of the glue. The correct tower shape is not necessarily exactly vertical >if, e.g. one side gets hotter. For example if one of the three Warsaw tubes got >5C hotter than the other two tubes, it would expand 65 parts per >million which would be 4.2 cm of extra tower height. This would bend >the tower with a radius of curvature of about 77 km, >causing the top of the tower to move 2.7 meters horizontally away from vertical. >I believe that the highest winds also were expected to cause about that same amount of bend. >Thus the correct shape is not "vertical"; it is "the one which keeps all bending >stresses bounded." > >With the 3-tube Warsaw-type design, >torques could be applied by pulling cables between the 3 tubes. >If there were no wind then I would say this is feasible. However, >counteracting the torques from wind seems to be to require far too large >torques to be feasibly applied in this way. In short removing >the guy wires from the Warsaw mast and using active control instead, >would be asinine. > >SUMMARY: >I'm not seeing advantages for active stabilization of structures the size >of the largest yet built -- up to 800 meters -- and in fact the whole idea seems to be >ridiculous in view of wind loads. > >But if we were building on the moon (vacuum, no wind) >then it might be the best way to build huge towers.

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Re: [math-fun] Hallowe'en spacefill
by Bill Gosper 22 Sep '14

22 Sep '14

Jörg>Pretty please, could you make an effort to be better understandable to (semi-)complete idiots like me? I just discovered by accident that you found Ventrella's asymmetric order 7 curve (triangular grid) like 40 years ahead of him... (1970's vs. 2012). http://gosper.org/IMG_0245.JPG < Yeah, that old 3-ring binder probably holds a few more unpleasant surprises for today's curvists. If you think of the "Franceflake" as the unit cell in the complex plane covered by the base&digit system (2 + (-1)^(1/3)) & (-1)^Range[0,5/3,1/3] , that trifid thing is (2 + (-1)^(1/3)) & (-1)^Range[0,4/3,2/3] *{1,1+(-1)^(1/3)} http://www.tweedledum.com/rwg/tlir7.htm I think there's yet another one which is "all boundary", as suggested by this booboo: gosper.org/oops.png jj> And: not everybody speaks Mathematica (some even because they do not like to).< To which SW doubtless replies (cue Zarvox): YOU WILL BE ASSIMILATED. > Best, jj (and THANKS, btw.!) SW's weakness for tiny programs that do interesting things has metastasized to http://blog.wolfram.com/2014/09/18/introducing-tweet-a-program/ Ugly as it may appear, the power is hard to deny. --rwg * Bill Gosper <billgosper(a)gmail.com> [Sep 21. 2014 19:54]: Taking half frequency instead (every other point) (at higher order), gosper.org/kochlea7^6.png gosper.org/kochlea7^6.svg You may need Inkscape to zoom and pan the svg, and you may need a new Inkscape for Mavericks. Note: All angles are multiples of π/6, including π/2 in bursts of five. (No π/4.) --rwg Eight hours?! This is that disastrous caching performance NeilB and I reported earlier. Clearing and redefining Julian's piecewiserecursivefractal (underlying kochlea) probably would have helped a lot. On Sat, Sep 20, 2014 at 5:14 AM, Bill Gosper <billgosper(a)gmail.com> wrote: Simply sampling Mandelbrot's Kochfill at triple frequency makes a nice spiderweb. gosper.org/halloween.png --rwg Re Cambridge U. Press, Finch's Mathematical Constants is certainly a marvel.

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