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Re: [math-fun] Space-filling curves again
by Bill Gosper 19 Oct '16

19 Oct '16

The animation I intended was perhaps too computationally ambitious. Meanwhile, gosper.org/FHilbert.gif is just conventional plots of the first 288 approximations. I dislike such plots due to the finite line thickness, which someone might imagine contributed to spacefilling. If you read all the frames into Preview, say, you can see that the texture of frame n repeats at frame 4n. This gives us a way to interpolate the nonexistent frame n+1/4 from frame 4n+1. Also uploaded: gosper.org/outcirc.gif made by flipping a sign in incirc.gif. --rwg Stay tuned for a Ptolemaic Hilbert sweep from a multi-hour computation. On 2016-09-26 18:42, James Propp wrote: > I hope Bill will write a lively document (or create a lively video) that > explains what's wrong with so many accounts of spacefilling curves. > > (I myself wish that more accounts started in a "Not Knot"-ish vein, > explaining why the "kindergartner's space-filling curve" --- scribbling > one's crayon back and forth until the square is filled --- isn't a solution > to the mathematician's problem.) > >> >> > Is there a way to relax an approximation to a space-filling curve in >> > continuous time so that it works out its kinks and regresses to simpler >> > approximations? >> > >> > (No interim self-intersections please!) >> > >> > Jim Propp >> >> http://gosper.org/FDrags128.mp4 >> >> If people really want to see it, Julian's recent Fourier matrix product can >> produce >> the analogous animation for Hilbert's "curve". > > > I would like to see it. > > But it is a seductive thought crime to view a spacefilling function as some >> kind of >> limit of "spacefilling curves"! Those curves Henry sketches are mere >> schematics, >> of no mathematical consequence. This leads inevitably to embarrassed >> hemming >> and hawing about how the area jumps from 0 to 1 at the very last moment, >> when >> both interior and exterior suddenly disappear and become boundary. >> Successive partial sums of the Fourier series are even more seductive. But >> no matter how many terms you take, you're still infinitely far from the >> end. >> --rwg > > > That's one way to look at it. But it depends on the notion of path-space > you use and what metric you put on it, doesn't it? In particular, it's > crucial to look at paths equipped with a parametrization. Then you really > can get convergence to a limit. And the limiting object is a continuous > function from a line segment onto a square. If you just look at the range > of a path and not the parametrization, you can't describe the square as the > limit in any meaningful way. Which I gather is part of Bill's point. > > Jim Propp > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Torque on a spinning differential gearbox
by Jason Holt 18 Oct '16

18 Oct '16

Imagine I have a lossless rear wheel drive car with a 1:1 differential gearbox in the back. I apply the brakes so the wheels are locked. I apply 1Nm of torque to the input shaft, and observe +/-0.5Nm of torque on each wheel. Now I spin the massive left rear wheel so it's moving freely at 100Hz in one direction, and the massive right rear wheel 100Hz in the other direction. The rotations cancel, and the drive shaft thus remains stationary. Now I apply a 1Nm torque to the drive shaft. Do the rear wheels experience: A: 0.5Nm torque, or B: Some amount of torque inversely proportional to the rotation rate of the wheels? If it's the former, then a small amount of work on the drive shaft (due to low RPM) causes a lot of work to happen in the two wheels (due to the same torque * high RPM) and would be a good work-magnifier (or work-transferrer, since it can't break conservation). If it's the latter, then a differential should make a good torque-magnifier (and thus a great CVT).

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Re: [math-fun] Eyjafjallagurglehole
by Henry Baker 18 Oct '16

18 Oct '16

I can't comment on the volcano, per se, but many waves in non-linear media have wavelength-dependent velocities that can "un-disperse" (my terminology) as they propagate -- i.e., pulses actually sharpen up instead of dispersing. In some media -- e.g., long-distance water canals in Victorian Scotland -- there are also *solitons* which bunch up into a characteristic shape which neither disperses nor sharpens. See also "pulse compression", including "chirped pulse amplification"; radar, laser. At 11:18 AM 10/18/2016, Bill Gosper wrote: >https://www.youtube.com/watch?v=3ydnLkVvfZM 2:44 fuzzy bright flash. Commenters >offer unconvincing conjectures. Same volcano, >https://www.youtube.com/watch?v=BicT13ecUbc 0:19 et al, visible "shock" waves. >About as far as possible from Iceland, >https://www.youtube.com/watch?v=BUREX8aFbMs 0:12, much larger "shock" wave. >Also note ocean impacts ~ 0:34. >I'm puzzled by how such a large source can make such a sharp noise. A significant >volume of gas has to conspire to come out of solution all at once. The grandaddy was >Krakatoa, the loudest noise in recorded history, blowing out sailors' eardrums many >miles away. Why "bang" and not "boom"? Was there ear damage to witnesses of >thermonuclear atmospheric tests? In the videos, when buildings and stuff suddenly >move away from, or get flattened by the explosion, those are actual shock waves-- >pistons of air moving faster than sound. > >Maybe that explains the "bang". The supersonic trailing gas piles up against the >leading edge, which is trying to slow down to Mach 1. Sort of like how tsunamis >trip over themselves. --rwg

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[math-fun] Eyjafjallagurglehole
by Bill Gosper 18 Oct '16

18 Oct '16

https://www.youtube.com/watch?v=3ydnLkVvfZM 2:44 fuzzy bright flash. Commenters offer unconvincing conjectures. Same volcano, https://www.youtube.com/watch?v=BicT13ecUbc 0:19 et al, visible "shock" waves. About as far as possible from Iceland, https://www.youtube.com/watch?v=BUREX8aFbMs 0:12, much larger "shock" wave. Also note ocean impacts ~ 0:34. I'm puzzled by how such a large source can make such a sharp noise. A significant volume of gas has to conspire to come out of solution all at once. The grandaddy was Krakatoa, the loudest noise in recorded history, blowing out sailors' eardrums many miles away. Why "bang" and not "boom"? Was there ear damage to witnesses of thermonuclear atmospheric tests? In the videos, when buildings and stuff suddenly move away from, or get flattened by the explosion, those are actual shock waves-- pistons of air moving faster than sound. Maybe that explains the "bang". The supersonic trailing gas piles up against the leading edge, which is trying to slow down to Mach 1. Sort of like how tsunamis trip over themselves. --rwg

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[math-fun] A strange kind of base two representation
by James Propp 17 Oct '16

17 Oct '16

I'm thinking of submitting an original problem for use in a precollege math contest, and I'm wondering if it's really new. Please don't share the problem with others at this time, since it may end up as one of the contest problems. Show that every positive integer n can be written as a sum of exactly n powers of two each of which appears at most twice. Here a power of 2 means 2 to the power of any integer. For example 5 can be written as 2 + 1 + 1 + 1/2 + 1/4 + 1/4. (it's possible that I discussed a variant of this problem on math-fun years ago; if so, please forgive the repetition.) Jim Propp

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[math-fun] "Punctured" uniform discrete distributions
by Henry Baker 16 Oct '16

16 Oct '16

The discussion about g^X mod p made me start thinking about a uniform discrete distribution that had a small number of "holes" -- elements where the probability was zero. E.g., prob(y = g^x mod p, x in Z) = if y=0 then 0 else if 0<y<p then 1/(p-1) If one is sampling from a distribution X' with exactly one hole, where |X'|=n, how many samples would be required to start to distinguish between the uniform distribution X (|X|=n) with zero holes and the uniform distribution X' (|X'|=n) with one hole? By "start to distinguish", I mean "become suspicious" or "statistically significant".

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Re: [math-fun] mult. group of GF(p) uniformly distributed?
by Henry Baker 14 Oct '16

14 Oct '16

Short answer: NO! p pidgeon holes, p pidgeons. p-2 pidgeon holes get 1 pidgeon each; pidgeon hole "0" gets no pidgeons; pidgeon hole "1" gets 2 pidgeons. So the distribution over the non-"0" pidgeon holes isn't uniform, but has a spike at "1". At 01:08 PM 10/14/2016, Henry Baker wrote: >Is this known or conjectured with a conjecture name? > >If X is a uniform random distribution over [0,p-1], p prime, >is g^X a uniform random distribution, for a generator g ?

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Re: [math-fun] mult. group of GF(p) uniformly distributed?
by Henry Baker 14 Oct '16

14 Oct '16

[corrected: replace p by p-1] "X is a uniform random distribution over [0,p-1], p prime" should have been "X is a uniform random distribution over [0,p-2], p prime", although [0,p-1]=[0,p-2] (mod p-1), so they are the same set. It isn't an off-by-one error in the range set but a clumsy (but still accurate) definition of the domain set. At 04:55 PM 10/14/2016, Mike Speciner wrote: >Sure there is, unless the question is rephrased to say [0,p-1), or [1,p-1], or [0,p-2], ... > >On 14-Oct-16 19:49, Henry Baker wrote: >>"0" isn't a member of the "multiplicative *group* of GF(p)", so there is no off-by-one problem. >> >>I guess since x->g^x is 1-1, it maps uniform distributions to uniform distributions. >> >>But I guess this reasoning only works for finite (countable??) sets. >> >>At 03:12 PM 10/14/2016, Allan Wechsler wrote: >>>I did misunderstand. >>> >>>However, I think we have an off-by-one problem: g^k will never, in fact, take on the value 0. >>> >>>It will be equally distributed over the nonzero values, of which there are only (p-1). >>> >>>On Fri, Oct 14, 2016 at 6:02 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >>>>But 2 is not a generator of [0,6] = Z/7. For a generator g of Z/p, the powers >>>>g^k will be all distinct for 0 <= k <= p-1; hence {g^k | 0 <= k < p} = Z/p. >>>> >>>>So g^X is in fact uniform on Z/p when X is. >>>> >>>>--Dan >>>> >>>>On Oct 14, 2016, at 2:25 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: >>>> >>>>Let p=7, g=2. For X in [0,6], g^X takes on the values 1, 2, 4, 1, 2, 4, 1. >>>>This isn't uniform: it takes the value 1 thrice, the values 2 and 4 twice, >>>>and 3, 5, and 6 never. >>>> >>>>Am I misunderstanding your question? >>>> >>>>On Fri, Oct 14, 2016 at 4:08 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>>>>Is this known or conjectured with a conjecture name? >>>>> >>>>>If X is a uniform random distribution over [0,p-1], p prime, >>>>>is g^X a uniform random distribution, for a generator g ?

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Re: [math-fun] mult. group of GF(p) uniformly distributed?
by Henry Baker 14 Oct '16

14 Oct '16

"0" isn't a member of the "multiplicative *group* of GF(p)", so there is no off-by-one problem. I guess since x->g^x is 1-1, it maps uniform distributions to uniform distributions. But I guess this reasoning only works for finite (countable??) sets. At 03:12 PM 10/14/2016, Allan Wechsler wrote: >I did misunderstand. > >However, I think we have an off-by-one problem: g^k will never, in fact, take on the value 0. > >It will be equally distributed over the nonzero values, of which there are only (p-1). > >On Fri, Oct 14, 2016 at 6:02 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> But 2 is not a generator of [0,6] = Z/7. For a generator g of Z/p, the powers >> g^k will be all distinct for 0 <= k <= p-1; hence {g^k | 0 <= k < p} = Z/p. >> >> So g^X is in fact uniform on Z/p when X is. >> >> --Dan >> >> On Oct 14, 2016, at 2:25 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: >> >> Let p=7, g=2. For X in [0,6], g^X takes on the values 1, 2, 4, 1, 2, 4, 1. >> This isn't uniform: it takes the value 1 thrice, the values 2 and 4 twice, >> and 3, 5, and 6 never. >> >> Am I misunderstanding your question? >> >> On Fri, Oct 14, 2016 at 4:08 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>> Is this known or conjectured with a conjecture name? >>> >>> If X is a uniform random distribution over [0,p-1], p prime, >>> is g^X a uniform random distribution, for a generator g ?

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[math-fun] Fwd: PolyhedronData["TriangularHebesphenorotunda", "Faces"]
by Bill Gosper 14 Oct '16

14 Oct '16

---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Fri, Oct 14, 2016 at 12:20 PM Subject: PolyhedronData["TriangularHebesphenorotunda", "Faces"] To: Wolfram Technical Support <support(a)wolfram.com> <Johnson solid png censored courtesy xmission.com> (PolyhedronData["TriangularHebesphenorotunda"] default view hides C3 symmetry.) GraphicsComplex[{{-0.9601329154398749, 0.715228827607169, -0.03894034047821841}, {-0.8187282992850485, 0.1613051767267678, -0.8594117437126204}, ...}] Bleahh. GraphicsComplex[{{-1, 0, 1/32 (Sqrt[3] + 5 Sqrt[15])}, {-(1/2), -(1/(2 Sqrt[3])), -(( 45 + Sqrt[5])/(32 Sqrt[3]))}, {-(1/2), -(Sqrt[3]/2), 1/32 (Sqrt[3] + 5 Sqrt[15])}, {-(1/2), Sqrt[3]/2, 1/32 (Sqrt[3] + 5 Sqrt[15])}, {-(1/2), (2 + Sqrt[5])/( 2 Sqrt[3]), -((13 + Sqrt[5])/(32 Sqrt[3]))}, {0, 1/Sqrt[ 3], -((45 + Sqrt[5])/(32 Sqrt[3]))}, {0, -((3 + Sqrt[5])/( 2 Sqrt[3])), (-29 + 15 Sqrt[5])/(32 Sqrt[3])}, {1/ 2, -(1/(2 Sqrt[3])), -((45 + Sqrt[5])/(32 Sqrt[3]))}, {1/ 2, -(Sqrt[3]/2), 1/32 (Sqrt[3] + 5 Sqrt[15])}, {1/2, Sqrt[3]/2, 1/32 (Sqrt[3] + 5 Sqrt[15])}, {1/2, (2 + Sqrt[5])/( 2 Sqrt[3]), -((13 + Sqrt[5])/(32 Sqrt[3]))}, {1, 0, 1/32 (Sqrt[3] + 5 Sqrt[15])}, {1/4 (-3 - Sqrt[5]), 1/12 (Sqrt[3] - Sqrt[15]), -((13 + Sqrt[5])/(32 Sqrt[3]))}, {1/ 4 (-3 - Sqrt[5]), (3 + Sqrt[5])/(4 Sqrt[3]), (-29 + 15 Sqrt[5])/( 32 Sqrt[3])}, {1/4 (-1 - Sqrt[5]), -((5 + Sqrt[5])/( 4 Sqrt[3])), -((13 + Sqrt[5])/(32 Sqrt[3]))}, {1/ 4 (1 + Sqrt[5]), -((5 + Sqrt[5])/(4 Sqrt[3])), -((13 + Sqrt[5])/( 32 Sqrt[3]))}, {1/4 (3 + Sqrt[5]), 1/12 (Sqrt[3] - Sqrt[15]), -((13 + Sqrt[5])/(32 Sqrt[3]))}, {1/ 4 (3 + Sqrt[5]), (3 + Sqrt[5])/(4 Sqrt[3]), (-29 + 15 Sqrt[5])/( 32 Sqrt[3])}}, Polygon[{{14, 5, 6, 2, 13}, {5, 14, 4}, {14, 1, 4}, {1, 14, 13}, {1, 13, 15, 3}, {15, 13, 2}, {7, 15, 2, 8, 16}, {3, 15, 7}, {9, 3, 7}, {9, 7, 16}, {17, 16, 8}, {9, 16, 17, 12}, {6, 8, 2}, {6, 11, 18, 17, 8}, {18, 12, 17}, {18, 10, 12}, {10, 18, 11}, {5, 4, 10, 11}, {5, 11, 6}, {10, 4, 1, 3, 9, 12}}]] In[776]:= PolyhedronData["TriangularHebesphenorotunda", "Volume"] Out[776]= 5.10875 Bleahh, 5/2 + (7 Sqrt[5])/6 In[458]:= PolyhedronData[%453[[-1]], "Centroid"] Out[458]= {0., 0., 0.} Bleahh. {0, 0, 0}. --rwg >So [ElongatedPentagonalRotunda] just barely won't stand on a square face. --rwg >You could use one as a carpenter's level. It must tip over, and >again when rotated 180º. But then why not just keep in your pocket a scrap of copper or PVC pipe?

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