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[math-fun] Pixel perfect Ptolemizer playpen [was: Julian has constructed a spacefill dense with sextuple points]
by Bill Gosper 21 Mar '17

21 Mar '17

On 2016-08-31 16:55, Bill Gosper wrote: Hi Vitaliy, I have a sandbox with a general Fourier animator. It takes a formula for the coefficients (amplitudes and phases as functions of frequency), speed, and coloration. It isn't very fast because it colors one pixel at a time, emulating the Symbolics mathematically correct draw-triangle ALU-add microcode necessary for smoothly drawing moving edges. Mathematically correct means that, if two triangles share an edge, there will never be missing or overwritten pixels, and if the triangle is "inside out", the "bump" will be negated. (The trick is to consistently define exactly which pixels to bump.) With this primitive, you can then animate the movement of an edge segment simply by drawing two triangles filling the quadrilateral defined by the old and new positions of its endpoints. To draw a seamless T joint, just [include] the "zero area" triangle formed by the constituent edges, which will usually add or subtract the "bump" constant to or from a few pixels. --Bill Howard Cannon has implemented a fast, full-res re-creation of the Symbolics triangle primitive, along with several lovely Ptolemaic sweeps, plus luxurious controls: http://hichacks.is-a-geek.net/triangles/artcircle E.g., for a nice picture of 2D Gibbs ringing, change the "Circle" box to Gibbs Tri, and click >. Besser illustrates frequency (over)modulation. The vector sum runs back and forth <Amplitude> radians along the circumference of the circle Howard (hic(a)iname.com) tailored the page for Chrome, and seeks your comments and suggestions. (I'll try to supply him with the coefficients for Julian's sextuple point spacefill.) --rwg

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[math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Henry Baker 20 Mar '17

20 Mar '17

FYI -- https://www.youtube.com/watch?v=aP0Y1uAA-2Y Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic John L. Gustafson, Natl Univ of Singapore https://web.stanford.edu/class/ee380/Abstracts/170201.html A new data type called a "posit" is designed for direct drop-in replacement for IEEE Standard 754 floats. Unlike unum arithmetic, posits do not require interval-type mathematics or variable size operands, and they round if an answer is inexact, much the way floats do. However, they provide compelling advantages over floats, including simpler hardware implementation that scales from as few as two-bit operands to thousands of bits. For any bit width, they have a larger dynamic range, higher accuracy, better closure under arithmetic operations, and simpler exception-handling. For example, posits never overflow to infinity or underflow to zero, and there is no "Not-a-Number" (NaN) value. Posits should take up less space to implement in silicon than an IEEE float of the same size. With fewer gate delays per operation as well as lower silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPs using similar hardware reso urces. GPU accelerators, in particular, could do more arithmetic per watt and per dollar yet deliver superior answer quality. A series of comprehensive benchmarks compares how many decimals of accuracy can be produced for a set number of bits-per-value, using various number formats. Low-precision posits provide a better solution than "approximate computing" methods that try to tolerate decreases in answer quality. High-precision posits provide better answers (more correct decimals) than floats of the same size, suggesting that in some cases, a 32-bit posit may do a better job than a 64-bit float. In other words, posits beat floats at their own game. http://ee380.stanford.edu/Abstracts/170201-slides.pdf

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[math-fun] Insolation problem
by Keith F. Lynch 19 Mar '17

19 Mar '17

Inspired by the equinox that's coming in about seven hours: Higher latitudes get less insolation (watts of sunlight per square meter of level surface area averaged over a year) than lower latitudes. How high would the Earth's obliquity (axial tilt) have to be for the poles to get more insolation than the equator? Or could this never happen? Assuming Earth's orbit was perfectly circular and that the insolation is exactly 1 kilowatt per square meter when the sun is directly overhead, is there a simple formula for insolation as a function of latitude and obliquity?

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[math-fun] Strange Gasket Sums
by Bill Gosper 19 Mar '17

19 Mar '17

I've just been reminded that Julian and I wrote them up about four years ago for G4G10: gosper.org/Article.pdf If you don't like seriously freaky formulas, just look at the pictures. --rwg

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[math-fun] http://www.conwaylife.com/wiki/R-pentomino
by Bill Gosper 19 Mar '17

19 Mar '17

relates how it led our RKG to discover the glider. I just added a line explaining why it SHOULD have led the MIT AILab group to discover the glider gun. --rwg

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[math-fun] Panel puzzle
by James Propp 17 Mar '17

17 Mar '17

You can join four isosceles right triangles along hinges (I use tape) to form a square, and then join four such square panels (again with tape) to form a long rectangle, and lastly join the first and last panels together with tape to form a closed band. If you color the two sides of the band with different colors, it's a fun puzzle to turn the band inside out, exchanging the colors. I learned about this in the 80s, and amused myself by learning how to do the moves behind my back. Does anyone know the name and/or provenance of this puzzle? Jim

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[math-fun] OECP? music-fun?
by James Propp 15 Mar '17

15 Mar '17

This is off-topic but since so many mathematicians love music I thought some of you might be able to provide useful leads: Is there anything like an Online Encyclopedia of Chord Progressions and/or a forum for "recreational musicology"? Of course, trying to list occurrences of IV - V - I in Western music would be as impossible as listing occurrences of 1,3,5,7,9,... in the mathematical literature, but less common chord progressions might be trackable. For instance, if you listen to the "Blazing Saddles" theme ( https://m.youtube.com/watch?v=hzRmp-O496Y ; 00:41-00:42 and elsewhere) and the "Big Country" theme (https://m.youtube.com/watch?v=AQTH3a0mjR8 ; 00:04-00:05 and 00:44-00:45 and elsewhere) you'll hear an C-major chord going to an A-major chord with a suspended E in the melody (before the song goes back to the D major tonic). How did VII-flat - V - I become a cowboy movie cliche? Does it trace back to actual cowboy music, or is it purely a Hollywood creation? And what are the antecedents in classical music? There may be a fun story here, but how would one search the internet to find it? Jim Propp

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[math-fun] 3rd Law of Thermodynamics proved mathematically
by Henry Baker 15 Mar '17

15 Mar '17

FYI -- "[We] place ultimate bounds on the ***speed*** at which information can be erased." "This tape with self-destruct in five seconds. Good luck, Jim." -- Mission Impossible TV series. We should now be able to determine an upper bound on how much information was stored on the tape. http://www.nature.com/articles/ncomms14538 http://www.nature.com/articles/ncomms14538.pdf A general derivation and quantification of the third law of thermodynamics LluÃs Masanes & Jonathan Oppenheim Nature Communications 8, Article number: 14538 (2017) doi:10.1038/ncomms14538 Quantum information Thermodynamics Received: 31 March 2016 Accepted: 09 January 2017 Published online: 14 March 2017 Abstract The most accepted version of the third law of thermodynamics, the unattainability principle, states that any process cannot reach absolute zero temperature in a finite number of steps and within a finite time. Here, we provide a derivation of the principle that applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any temperature, and translate these resources into the minimal time or number of steps, by considering the notion of a thermal machine that obeys similar restrictions to universal computers. We generally find that the obtainable temperature can scale as an inverse power of the cooling time. Our results also clarify the connection between two versions of the third law (the unattainability principle and the heat theorem), and place ultimate bounds on the speed at which information can be erased.

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Re: [math-fun] OECP? music-fun?
by Henry Baker 14 Mar '17

14 Mar '17

I've always wanted to use Google to search for pieces of melodies as well as chords. The idea would be to type in a short midi sequence, and Google would find it (or a close match) in another midi file. More advanced: Google would find the sequence in mp3 files, mp4 files, etc. Perhaps Google has already done this? Perhaps Google has been screwed by the music industry? After all, who wants to make it trivial for copyright lawyers to find musical precedents for every big hit song? --- Another thought that's been in the back of my mind for at least 20 years: *midi sound "compression"*. The idea isn't to get the greatest compression, but to get the best musical approximation from a .wav file. As you know, a digital midi player basically cues up a finite-length "sound" from a "sound font" to be added into the output at the time and volume specified by the midi note. There are additional complexities, but this gives you a minimal midi player. [Mathematically, the sound output .wav file is a *polynomial*, where the sample "A" is the coefficient of A*x^t, and the sample A appears at time t. A midi score of one octave is a polynomial in 13 variables: 12 notes and the time "t". A sound font contains 12 polynomials in "t" which provide the samples for that particular note. The score is "played" by evaluating the score at the 12 different note polynomials. Note variables are C,B,B_,A,A_,G,G_,F,E,E_,D,D_; time variable is t. (%i1) a; 2 (%o1) a2 T + a1 T + a0 [rather short 440 A sample] (%i2) b; 2 (%o2) b2 T + b1 T + b0 [rather short B sample] (%i3) c; 3 (%o3) c2 T + c1 T + c0 [rather short C sample] (%i4) LittleLamb; 9 8 7 6 5 4 3 2 (%o4) B T + B T + B T + C T + C T + C T + B T + A T + B T + C (%i5) LittleLamb,A=a,B=b,C=c; 6 3 5 3 4 3 3 (%o5) T (c2 T + c1 T + c0) + T (c2 T + c1 T + c0) + T (c2 T + c1 T + c0) + c2 T 9 2 8 2 7 2 + T (b2 T + b1 T + b0) + T (b2 T + b1 T + b0) + T (b2 T + b1 T + b0) 3 2 2 2 2 + T (b2 T + b1 T + b0) + T (b2 T + b1 T + b0) + T (a2 T + a1 T + a0) + c1 T + c0 (%i6) ratsimp(%,T); 11 10 9 8 (%o6) b2 T + (b2 + b1) T + (c2 + b2 + b1 + b0) T + (c2 + b1 + b0) T 7 6 5 4 + (c2 + c1 + b0) T + (c1 + c0) T + (c1 + c0 + b2) T + (c0 + b1 + a2) T 3 2 + (c2 + b2 + b0 + a1) T + (b1 + a0) T + (c1 + b0) T + c0 (%i7) ] A Midi sound compression algorithm would be given an a priori "sound font" file which is an a priori "dictionary" of sounds. The sound compression algorithm would be charged with figuring out which sequence of these sounds at what volumes would produce the closest sound to the given sound file. Does this program exist? At 08:54 AM 3/14/2017, James Propp wrote: >This is off-topic but since so many mathematicians love music I thought >some of you might be able to provide useful leads: Is there anything like >an Online Encyclopedia of Chord Progressions and/or a forum for >"recreational musicology"? > >Of course, trying to list occurrences of IV - V - I in Western music would >be as impossible as listing occurrences of 1,3,5,7,9,... in the >mathematical literature, but less common chord progressions might be >trackable. For instance, if you listen to the "Blazing Saddles" theme ( >https://m.youtube.com/watch?v=hzRmp-O496Y ; 00:41-00:42 and elsewhere) and >the "Big Country" theme (https://m.youtube.com/watch?v=AQTH3a0mjR8 ; >00:04-00:05 and 00:44-00:45 and elsewhere) you'll hear an C-major chord >going to an A-major chord with a suspended E in the melody (before the song >goes back to the D major tonic). How did VII-flat - V - I become a cowboy >movie cliche? Does it trace back to actual cowboy music, or is it purely a >Hollywood creation? And what are the antecedents in classical music? There >may be a fun story here, but how would one search the internet to find it? > >Jim Propp

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[math-fun] Acta Mathematica
by Hans Havermann 12 Mar '17

12 Mar '17

It appears that since the end of January, by arrangement with the Institut Mittag-Leffler, International Press now provides fully open online access to the entire content of Acta Mathematica — from its first issue of 1882 to the most recent: http://projecteuclid.org/all/euclid.acta

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