June 2015 - mathfun@mailman.xmission.com

[math-fun] Pentagonia (& new tilings?): fibonacci number system attack suggested
by Warren D Smith 16 Jun '15

16 Jun '15

I agree with JA about the "greedy method" working for the Fibonacci & golden number systems. However, for the more general number systems I was talking about, it is not obvious what the analogue of greed would be nor whether/when it would work.

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[math-fun] Pentagonia (& new tilings?): fibonacci number system attack suggested
by Warren D Smith 16 Jun '15

16 Jun '15

Possible idea: re-express/re-examine that Minsky iteration and/or the Penrose tiling using numbers/coordinates written in "fibonacci number system" or "golden number system." Likely to be helpful. Fibonacci system: write integer n>=0 uniquely as n=SUM_k F[k] * d[k] using digits d[k]={0,1} and no two consecutive digits both 1. Here F[0]=1, F[1]=2, F[2]=3, F[4]=5, F[6]=8, etc, you may prefer to shift my subscripts. Example 27 = 1001001_F = 21+5+1 22 = 1000001_F = 21+1 ------------------------- 49 = 2001002_F unreduced 49 = 2001010_F still unreduced 49 = 10011010_F still unreduced 49 = 10100010_F = 34+13+2 Example 12 = 10101_F 10 = 10010_F ------------------------- 22 = 20111_F unreduced 22 = 21001_F still unreduced 22 = 110001_F still unreduced 22 =1000001_F Note, the fibonacci numbers always have form 10000...00_F and their doubles always have form 100100...00_F except that 2=10_F. Golden system: write real x>=0 uniquely as n=SUM_k g^k * d[k] using digits d[k]={0,1} and no two consecutive digits both 1, where g=1.61803... is the golden ratio g+1=g^2. ========================================== Meanwhile this suggests other possible applications & new questions. As I think Gosper already very familiar with, you can make weird fractalish tiles using funny number systems. For example complex numbers can be represented using radix=i-1 and digits={0,1}, and then the full set of complex numbers with 0s to the left of the decimal point, is a very weird-shaped but simply-connected object that tiles the complex plane. [Or digits={0,1,2,...X^2} and radix=i-X more generally, where X>0 is any fixed integer. Also, binary with radix=(+-1+-sqrt(-7))/2 or radix=+-sqrt(-2) with any fixed sign choices.] NOW, consider a wider range of number systems where, like fibonacci system, we forbid certain substrings of digits. If we use "binary" i.e. digit set {0,1}: If the radix R is some algebraic number with minimal polynomial equation P(R)=0 and P has degree<K with all coefficients either 0 or +-1, then we forbid the bit pattern arising from the positive coefficients of P, as a substring. (And demand P have more negative than positive coefficients, and have both signs.) [In the case of the golden ratio R=g, we have g^2=g+1.] We also demand that R satisfy a polynomial equation of the form 2*R^J=Q(R) where Q has all coefficients either 0 or 1. [In the case of the golden ratio R=g, we have 2*g^2=g^3+1.] These two demands try to assure uniqueness of the representation, and to assure you can add. We could also try to assure you can subtract by demanding a polynomial equation of form -R^J=S(R) exist where S has all coefficients either 0 or 1. I doubt these conditions really suffice. Probably more are needed. For example any polynomial multiple of P whose coeffs are 0 and +-1 only, yields possibly-additional forbidden substrings but I think you always get only a finite set of forbiddances in this way. Also, just because 2 or -1 are re-expressible does not necessarily suffice to make addition or subtraction be a finite process; and if it is an infinite process that might sometimes be ok. Also you might be able to add even without being able to express 2 in any finite-length way. For example, if 1 is expressible as 0.xxxxxx where xxxxx is some unique nonforbidden infinitely long string of 1s and 0s then maybe that means you can add. So the question of determining these number systems is a currently-confusing, but interesting topic. Anyhow, suppose one found such a "binary with forbidden substrings" number system with complex non-real radix R, and P(R)=0 with P monic and all non-leading coefficients -1 or 0. Let the set of numbers with only 0s to left of decimal point be called T. It seems to me that T usually will be fractally shaped and that with a finite set of tiles -- more precisely, a finite set of scalings of T by integer powers of |R| (the powers come from the locations of nonleading coefficients of P) -- you can tile the plane. Specifically, T will be tiled by a finite set of such scaled versions of itself, then recurse. If that is right, then we've got something kind of like the Penrose tiling in the sense that a finite set of tiles suffice, but it is even cooler than Penrose in the sense that they are all the "same" tile (just rescaled) and they have a very funny fractal shape. Also like Penrose, I think such a tiling will inherently be aperiodic in most cases (e.g. if irrational tile-count ratios). Also, in some cases these tilings should also yield, as an immediate side-benefit, interesting new planefilling curves. (I just said binary for simplicity, but the same ought to be true for other integers besides 2.) OK, so can examples be found? My first try: Let R^4=R+1 with R=-0.248126+1.03398*i and |R|=1.0633. Binary digits but forbid "11" and "101001" as substrings. (Are there any other forbiddances?) 1=0.0011(forbidden)=0.0001001001001...(allowed). Therefore it appears we can add and subtract. Tile T is tiled by T/R^2 (adjoin 10) and T/R^3 (adjoin 010). R*T is tiled by T/R and T/R^2. R^2*T is tiled by T and T/R. etc. Hence we apparently can tile the entire plane with just 3 tiles: T, T/R, and T/R^2. And this happens only aperiodically provided all the numbers that look irrational, are. And these tiles should have a fractal appearance. Caveat emptor: I have not actually drawn pictures confirming this, and I could have made errors. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Mathematical Enchantments video
by Henry Baker 16 Jun '15

16 Jun '15

At 11:58 AM 6/16/2015, Mike Stay wrote: >I can't listen to audiobooks for the same reason: it's like going back >to a 56k modem after having fiber. The software I use on my mp3 player can also speed up playback w/o changing pitch. I highly recommend it. It's called "RockBox", and it is free/open source SW that you can put on a number of mp3 players. I use it on a Sansa player which cost me < $50. http://www.rockbox.org/download/ I've also downloaded & compiled the Rockbox sources myself, but I would recommend doing this in a virtual machine environment (& also allow yourself 12 hours for the first time you try to do this!). I think you can also compile it successfully under Cygwin. The software also supports codecs other than mp3, including .m4a (the "AAC" codec) and Apple's lossless codec. The software on Sansa HW supports uSD cards up to 32GB (it may work with bigger uSD cards, but I haven't tried it).

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Re: [math-fun] Mathematical Enchantments video
by Henry Baker 16 Jun '15

16 Jun '15

Hi Jim: Some additional thoughts: * Your camera person kept moving the camera to follow your movements. Not only does this motion take an enormous number of bits to encode, it is also quite distracting (motion sickness, anyone?). Have him/her zoom out a bit & leave the camera still. (I mean _really_ still; use a large heavy tripod to make sure that there's no vibration whatsoever -- any vibration increases the bit rate unnecessarily. This vibration can be eliminated in post-production using various kinds of filters, but these filter will also kill the crispness of the highdef video.) * You do use some high motion video clips which won't work at low frame rates. But the originals of these videos was considerably lower resolution than your 1920x1080/30fps HD. Rather than taking a picture of the video playing, edit in the original video -- this looks considerably better & also takes many fewer bits. * Consider using _variable bit rate_ encoding with a "constant quality" setting (see the ffmpeg documentation). Under these conditions, the compressor will notice that certain material is more challenging for the encoder than other material, and allocate more bits. Thus, when there's no motion -- e.g., a picture of a slide -- the compressor will use very few bits. However, if you include some video clips, the compressor will ramp up the bit rate so as to continue to provide high quality even on challenging material. The problem with "variable bit rate" ("VBR") is that it may cause glitching during real-time playback when the bit rate exceeds the bit rate of the network. Using extremely large playback buffers can help (i.e., use the excellent "VLC" player & adjust its parameters) smooth over these humps in VBR videos. * I rarely watch videos downloaded in real time, but prefer to download them in advance. This enables me to use the VLC player to _speed up_ the video during boring parts -- sometimes to 2X the normal rate. The VLC player is capable of speeding up the audio without affecting the pitch; the speaker doesn't sound like he/she is breathing helium. At 07:40 AM 6/16/2015, James Propp wrote: >Thanks for the suggestions, Henry! > >There's still some editing work being done on the file, so it'd be premature to create a low-rate version, except as a proof of principle. I'll guessing that when the final edits are done, I'll avail myself of your kind offer and make your low-rate version available alongside the high-rate version. > >I'd be the last person to be able to judge how much value my facial expressions and body language add to the presentation. I expect that Kahn is right about the comparative unimportance of these things when the material is highly technical and there's not really an authorial point-of-view at work. But in more personal presentations (like some of the more impassioned TED Talks I've seen), seeing the speaker adds a lot. My talk falls somewhere in the middle. > >I like the suggestion of getting rid of stereo audio, but I just told the video editor "These are my edits, there won't be others", so I'm reluctant to go back on my word. But I'll certainly keep this issue in mind for future videos I make. > >Jim > >On Tue, Jun 16, 2015 at 10:11 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >Re your 36-minute video: > >Excellent/outstanding math content. However: > >This HD (1080p/30fps) video data rate is ~5Mbits/sec, which far exceeds the capabilities of many home networks -- e.g., DSL. > >Here are some ways to get the data rate down to a reasonable rate. > >1. A math talk never requires 30fps unless you're doing animations. The physics lectures at UCSB KITP use 0.1fps (yes, that's 1/10 of a frame per second = 1 frame every 10 seconds). 0.1fps is perhaps a bit extreme, but even 1 fps is just fine for many/most lectures. > >2. Alternatively, put the picture of the lecturer in a little "picture-in-picture" in one of the corners, and use the rest of the screen for the slides. Due to the way video compression works, the slides themselves take almost zero bits to encode, with the bulk of the bits being used to represent the moving picture -- the lecturer. > >3. Follow the advice of Sal Khan (of the Khan Academy) and dispense with the picture of the lecturer entirely. He has actually done tests that show that for complex mathematical concepts, looking at the lecturer instead of the blackboard or slides is actually counterproductive. Khan's "disembodied" lectures have been proven extremely effective. > >4. Cut down on the resolution of the video. UCSB KITP uses 960x540, which is 1/4 of the number of pixels of 1920x1080. This resolution seems to handle all but the most challenging slides just fine. > >5. Try out the various options of ffmpeg to get better compression. I use WinFF (I think this is also available on Linux) as a GUI for ffmpeg, and you can edit the XML files of WinFF to create your own video & audio compression recipes. > >6. You don't need stereo audio for most lectures, which can also save a few bits. UCSB KITP uses 16 KHz sampling for a 32.5Kbits/sec "AAC" audio rate. > >Bottom line: UCSB KITP gets their video lectures into 65.5 Kbits/sec (half video, half audio), which is a factor of 76x smaller bandwidth than your 3.8 GByte video file. > >If you'd like, I can re-encode your video using these parameters to show you what it would look like. However, since much of the video focuses on the speaker, rather than the slides, it won't look as good as the KITP material.

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[math-fun] Pentagonia (& new tilings?): fibonacci number system attack suggested
by Warren D Smith 16 Jun '15

16 Jun '15

J Arndt: > * Warren D Smith <warren.wds(a)gmail.com> [Jun 15. 2015 07:58]: >> Possible idea: >> re-express/re-examine that Minsky iteration and/or the Penrose tiling >> using numbers/coordinates written in "fibonacci number system" or >> "golden number system." Likely to be helpful. >> >> Fibonacci system: >> [...] > > For those "sparse" expansions, just use the greedy method. --WDS: uh, what is "the greedy method"? I mean, the problem with number systems like mine is, unless you forbid substrings, the representation will be very nonunique. So are you claiming, just walk right to left, each time you can add a "1" without using a forbidden and without exceeding the target number, then do it? And that this works? And in what circumstances does this work? Always? JA: 1 http://jjj.de/tmp-rd/rd-p1m1-twindragon-sign-d012.png 2 http://jjj.de/tmp-rd/rd-naf-m1p1-d1.png <--= I really like this one 3 http://jjj.de/tmp-rd/rd-naf-w17.4p0ps2-terdragon-d012.png 4 http://jjj.de/tmp-rd/rd-naf-m1p1-ver2-d012.png --WDS: so are you saying, you already thought of my whole tiling idea? The above pictures, 1 seems to be a "conventional" binary with some complex radix, probably 1-i since it seems tile T is tiled by two sqrt(1/2)-scaled, 90-degree-rotated copies. Pictures #2-4, I'm not sure what they are. IF they constitute a tiling of T by means of several rescaled copies of T, and the angles of rotation of the pieces are irrational in degrees, then that means you tile the plane aperiodically. But in pictures 3, 4, apparently only rotation angles divisible by 90 are involved, in picture 2 no rotations at all. So based on these pictures alone, it appears you did not think of my whole idea, or if you did then you did not find examples of the coolest phenomena. But you're ahead of me in the sense you have some software tools for manufacturing pictures. I could be wrong, since I have not done any graphical computing of my own, but I suspect some very cool tilings can be got via my forbidden substrings + funny complex radix number systems approach, including inherently aperiodic ones with only "one" tile type, albeit present in several scaled versions. (And what is the minimum number of scaled versions? I do not know.) Also, which seems not very related(?), the question of how the hell Gosper, Ziegler-Hunt etc found Penrose-like tiling using Minsky-like iteration, probably has a proof that explains the mystery, and probably that proof will be based on examining what happens to their numbers when you look at them in Fibonacci and Golden number systems. But again, I have not actually done that examination and produced such a proof -- this is just a suggestion that's likely to go somewhere if you try. The over-arching question would be: to try to classify all possible such tilings and number systems. Which might be doable. Or at least to find all the "small" examples, which ought to be doable via computer searches aided by some human feedback.

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Re: [math-fun] Mathematical Enchantments video
by Henry Baker 16 Jun '15

16 Jun '15

One last thought on video compression: Under normal circumstances, I never use the "two pass" encoding feature of ffmpeg, because I rarely need the absolute lowest bit rate, and multiple passes takes a lot more encoding time. However, when you're done with all editing, and you're ready to publish your best & final video, you should use ffmpeg's "two pass" capability. By using 2 passes, the encoder can "look ahead" to see what the video during the next few seconds is going to look like, and to tailor its decisions based on that additional information. Just as Knuth's TeX can optimize its font caching by looking ahead, video compressors can do roughly the same thing.

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Re: [math-fun] Mathematical Enchantments video
by Henry Baker 16 Jun '15

16 Jun '15

Re your 36-minute video: Excellent/outstanding math content. However: This HD (1080p/30fps) video data rate is ~5Mbits/sec, which far exceeds the capabilities of many home networks -- e.g., DSL. Here are some ways to get the data rate down to a reasonable rate. 1. A math talk never requires 30fps unless you're doing animations. The physics lectures at UCSB KITP use 0.1fps (yes, that's 1/10 of a frame per second = 1 frame every 10 seconds). 0.1fps is perhaps a bit extreme, but even 1 fps is just fine for many/most lectures. 2. Alternatively, put the picture of the lecturer in a little "picture-in-picture" in one of the corners, and use the rest of the screen for the slides. Due to the way video compression works, the slides themselves take almost zero bits to encode, with the bulk of the bits being used to represent the moving picture -- the lecturer. 3. Follow the advice of Sal Khan (of the Khan Academy) and dispense with the picture of the lecturer entirely. He has actually done tests that show that for complex mathematical concepts, looking at the lecturer instead of the blackboard or slides is actually counterproductive. Khan's "disembodied" lectures have been proven extremely effective. 4. Cut down on the resolution of the video. UCSB KITP uses 960x540, which is 1/4 of the number of pixels of 1920x1080. This resolution seems to handle all but the most challenging slides just fine. 5. Try out the various options of ffmpeg to get better compression. I use WinFF (I think this is also available on Linux) as a GUI for ffmpeg, and you can edit the XML files of WinFF to create your own video & audio compression recipes. 6. You don't need stereo audio for most lectures, which can also save a few bits. UCSB KITP uses 16 KHz sampling for a 32.5Kbits/sec "AAC" audio rate. Bottom line: UCSB KITP gets their video lectures into 65.5 Kbits/sec (half video, half audio), which is a factor of 76x smaller bandwidth than your 3.8 GByte video file. If you'd like, I can re-encode your video using these parameters to show you what it would look like. However, since much of the video focuses on the speaker, rather than the slides, it won't look as good as the KITP material.

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[math-fun] magnetic field strength confusion
by Warren D Smith 15 Jun '15

15 Jun '15

The typical saturation magnetic field of stuff like iron, is about 1 tesla. (0.3 to 2.2 teslas more precisely for a selection of commercial materials.) However, the obvious tesla-unitted quantities you construct from the right fundamental constants are far larger: (bohr magneton)*(bohr radius)^(-2)*(elementary charge)^(-2)*(electron mass) =117.526 kiloTesla. Also, e^(-3) * m_e^2 * c^3 * epsilon_0 = 48.1 gigaTesla. If you tried using nuclear masses instead of electron masses, or electron Compton wavelengths instead of bohr radii, these discrepancies would only get huger. (They'd get smaller if you used nuclear magnetons instead of bohr magnetons, but that would definitely be physically wrong since we know ferromagnetism has nothing to do with nuclear magnetism since there is no isotope effect.) Why these enormous discrepancies? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Don't cry for me, pentagonia.
by Warren D Smith 13 Jun '15

13 Jun '15

so in view of my impossibility theorem for integer coordinates for any affined regular pentagon, I presume that means the picture by Gosper et al of a Penrose-like raster pattern, must only be approximately 5-symmetric and approximately Penrose, and presumably the approximations have something to do with the golden ratio approximations by Fibonacci ratios. I attempted to count pixels by hand, and the sizes of the mystic pentagrams appear to be 1,5,34... which are fibos, but I might easily be mis-counting. You know, it was quite difficult for Penrose to invent his tiling, but the Gosper et al computer construction is invented by accident by a moron. Except we are currently too dumb to figure out why it succeeded. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Don't cry for me, Pentagonia
by Warren D Smith 13 Jun '15

13 Jun '15

AG/RWG: Adam Goucher has made a startling observation (quoted without permission): ---- I was reading the third edition of 'Minskys and Trinskys' by Gosper, Holloway and (Ziegler Hunts)^2: http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition The pattern of disjoint D10-symmetric snowflakes (or 'Rastermen', as they're referred to in the book) shown in Figure 31 appears to be affinely isomorphic to the pattern obtained by marking Penrose tilings: http://condellpark.com/kd/penrmark24bg.gif This suggests that it should be possible to create a very integer-esque method of drawing an affined Penrose tiling on a square grid... -- 1. I point out there are a number of 5-symmetric-looking pictures in this book, but page 31 is not one of them. 2. Is it even possible for anything drawn on an integer grid to be affine to a Penrose tiling? Related question: for which N, can the vertices of a regular N-gon be affined to distinct integer coordinates?

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