January 2014 - mathfun@mailman.xmission.com

[math-fun] Microwave ovens
by Bill Gosper 29 Jan '14

29 Jan '14

"There is no reverse on a motorcycle ..." was supposedly one of Shannon's favorite test sentences. http://languagelog.ldc.upenn.edu/myl/Shannon1950.pdf --rwg From: meekerdb <meekerdb(a)verizon.net> On 1/27/2014 1:22 AM, Joerg Arndt wrote: > Most certainly a kind of motor is used where the initial > internal position (and possibly the initial current) > determines the direction. But I've noticed that mine seems to be completely consistent. It reverses every time. > > A similar thing used to be (long ago) a problem with > tractors when the motor was idling: it could start moving > backward when put into move, potentially injuring/killing > the people accompanying it (usually walking behind). A two-stroke engine will also run backwards. Even though the ignition timing is poor for running backward it can get started that way by kicking back when starting. Once at the start of a motorcycle race I saw kid have his bike lurch backwards as everyone else leaped forward. A look of puzzlement came over his face as he tried easing out the clutch again and the bike started to back up. Then he threw down the bike as if it had betrayed him. Brent Meeker > > Best, jj > > > * James Propp <jamespropp(a)gmail.com> [Jan 27. 2014 10:12]: >> Why do microwave ovens alternate between rotating the >> food-carousel clockwise and rotating it counterclockwise (or, as the Brits >> would say, anticlockwise)? >> >> Jim Propp OK, so why are μwave ovens so damn noisy? Note to James Randi: Unworthy nonbeliever, if you want to learn the true power of homeopathy, try making Earl Grey tea in a sauerkraut jar.

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[math-fun] cleaning a jar
by Warren D Smith 29 Jan '14

29 Jan '14

Q. What fraction full of water should jar be, to get max cleaning effect when shake it? Assume you shake the jar on a fixed trajectory and schedule. The jar contains two fluids, "water" and "air," of different densities. We'll model them as incompressible. These fluids each experience forces proportional to acceleration times density-difference times their volume. (To make it simplest you may assume the water density is 1 and air density is 0. And as we said the acceleration is fixed.) The amount of energy that the fluids accumulate as a result of those forces, is proportional to force times distance traveled. The maximum amount of distance the water can travel is proportional to 1-F where F is the water-fraction. This also seems true if "maximum" is replaced by "average, under randomness assumptions." Conclusion: achieved kinetic energy of water = F*(1-F)*constant. If we then guess that the cleaning rate is increasing function of kinetic energy, then max cleaning effect is got by choosing F=1/2.

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Re: [math-fun] Jumping square points
by Adam P. Goucher 29 Jan '14

29 Jan '14

Very nice! That generalises to n-dimensional hypercubes, whereas my solution is specific to the square case. Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Michael Kleber > Sent: 01/29/14 02:56 PM > To: math-fun > Subject: Re: [math-fun] Jumping square points > > I think you can do effectively the same thing as Adam without needing ideas > like Gaussian integer GCD. Eric's problem asked about forming a new square > with size "t" > "s", and that's because it's obvious you can never make a > square that's *smaller* than the unit square you started with, as all > points you ever get are part of the square lattice with the original square > as its unit. But since the point-replacing process is invertible, if it > can't give you a smaller square, it can't give you a larger one either. > > --Michael > > > > On Wed, Jan 29, 2014 at 9:46 AM, Adam P. Goucher <apgoucher(a)gmx.com> wrote: > > > That's a very beautiful problem. I've appended a short solution to the end > > of > > this e-mail. > > > > > > > > > ----- Original Message ----- > > > From: Eric Angelini > > > Sent: 01/29/14 10:30 AM > > > To: math-fun > > > Subject: [math-fun] Jumping square points > > > > > > Hello Math-Fun, > > > Four points shape a square of size "s". > > > Every minute a point randomly jumps over another point > > > and lands symmetrically behind it. > > > Is it possible that, at some stage, the 4 points shape > > > a new square having size "t" > "s"? > > > Best, > > > É. > > > > > > > > > _______________________________________________ > > > math-fun mailing list > > > math-fun(a)mailman.xmission.com > > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > ***** spoiler alert ***** > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > > > No. Operate in the ring Z[i] of Gaussian integers, and let {a,b,c,d} be > > the Gaussian integers corresponding to the vertices of the square. When > > you apply the process to obtain {a,2a-b,c,d}, the greatest common divisor > > (again a Gaussian integer, up to multiplication by the ring units) remains > > invariant (similar to Euclid's algorithm). > > > > Hence, if they form a square at any time, it must be the same size and > > orientation as the original square. > > > > > > > > Sincerely, > > > > > > Adam P. Goucher > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > -- > Forewarned is worth an octopus in the bush. > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Cleaning out a jar
by Henry Baker 29 Jan '14

29 Jan '14

At 03:06 AM 1/29/2014, Bill Gosper wrote: > And the human nose is pathetic compared to a bloodhound's. Perhaps not: http://news.bbc.co.uk/2/hi/health/6183379.stm [Human] Sense of smell 'underestimated' The sensitivity of the human sense of smell has been significantly underestimated, a study suggests. US research had confounded the established belief that people have a poorer sense of smell than animals. The work, published in the journal Nature Neuroscience, asked people to follow scents on the ground, as a dog would do, and found they were as good. A UK expert said the findings were "intriguing" and would aid better understanding of the sense. The researchers from University of California Berkley laid scent trails, including one of chocolate essential oil, in a grassy field, and asked 32 people to find the 10 metre trail and track it to the end. Those who took part were blindfolded and wore thick gloves and earplugs to force them to rely exclusively on smell. Two thirds were able to follow the scent. And while they remained slower than the animals at tracking scents, their performance improved over time. In other tests, it was found that humans required both nostrils to be working to be able to track scents. 'Highly developed' Writing in Nature Neuroscience, the researchers led by Dr Noam Sobel showed the human sense of smell was more powerful than previously believed and that, with training, humans might be capable of tasks which had been thought to be the exclusive province of non-human animals. Dr Peter Brennan, a physiologist at the University of Bristol, said: "It's certainly an intriguing piece of research. "It shows that although the sense of smell is less important for humans than it is for many other animals, it is nonetheless a highly developed and sophisticated sensory system. "It is not the second class system that has been the traditional assumption." He added: "There has been previous evidence that scent can elicit orientation and movement towards maternal odours by new-born babies, but this is the first time that adult humans have been shown to follow a scent trail." Dr Brennan said the findings could enable specific areas of research into the human sense of smell. "For example, it would be interesting to study the extent to which blind people make use of their sense of smell for finding their way around their environment." Story from BBC NEWS: http://news.bbc.co.uk/go/pr/fr/-/2/hi/health/6183379.stm

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[math-fun] Jumping square points
by Eric Angelini 29 Jan '14

29 Jan '14

Hello Math-Fun, Four points shape a square of size "s". Every minute a point randomly jumps over another point and lands symmetrically behind it. Is it possible that, at some stage, the 4 points shape a new square having size "t" > "s"? Best, É.

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[math-fun] Cheap (40 euro?) FPGA hacking board
by Henry Baker 29 Jan '14

29 Jan '14

If you want to hack FPGA's for cool stuff (bit patterns, arithmetic, etc.) cheaply, then the PSHDL language & system might work for you. http://pshdl.org/pshdl.html Here's a YouTube video about PSHDL: http://www.youtube.com/watch?v=G2f3Vf74qAI Here's a crowd-funded project to produce the cheap FPGA hacking board: http://boards.pshdl.org/ (I have no connection with this person or organization, but I've always been intrigued by the possibilities of FPGA's.

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[math-fun] Greedily packing disks in a quadrant
by James Propp 28 Jan '14

28 Jan '14

Given a quadrant, and a unit disk tightly wedged in its corner, and some other unit disks touching that one, let's play a game where we add more unit disks, always adding a new disk that's as close as possible to the original disk (the one that's tightly wedged in the corner). For instance, suppose (A) we add a second unit disk that's tangent to the first disk and to one of the rays bounding the quadrant, and a third disk that's tangent to the first two disks (so that the centers of the three disks form an equilateral triangle one of whose sides is parallel to the ray). See the attachment for pictures of this case and the other three cases described below. If we now continue adding disks greedily, I claim (but have not proved) that you get a subset of the standard close-packing of the plane by unit disks. (Proof, anyone?) Things are more interesting in some other cases, though, such as (B) The second disk is tangent to the first disk and to one of the rays bounding the quadrant, and the third disk is tangent to the first disk and to the other ray bounding the quadrant (so that the centers of the three disks form an isosceles right triangle). (C) The centers of the three disks form an equilateral triangle (as in (A)), but one that is symmetrical with respect to the line that bisects the quadrant. (D) We start with just two disks, and the second disk (tangent to the first) has its center lying on the line that bisects the quadrant. In the event of a tie between two locations where one might add the next disk, I'd like to follow Yogi Berra's Imperative ("When you see a fork in the road, take it") and follow both branches of the tree. Actually, "tree" is the wrong word, since hopefully there'll be some convergence going on as well. Note that if we start with just one initial disk, then there are infinitely many ways to add the second disk at distance 0 from the initial disk (or distance 2, if you measure from center to center). So I won't ask the question in this case, at least not yet; following the Yogi Berra Imperative would give us too many possibilities to consider. My belief is that in cases (B)-(D), something approximating the standard dense-packing of the plane by unit disks will emerge, but with interesting defects, and that in each case the picture as the number of disks goes to infinity will be unique (up to a flip). It should be possible to explore this computationally, but my programming chops aren't up to the task. One would need to find a good way to represent the arrangement of the disks, and then find a way to solve the problem of where to place the next disk, using exact arithmetic so that ties can be detected. But I'm guessing that some of you (Bill Gosper and Veit Elser in particular) have already dealt with problems like this one, and may even have some off-the-shelf code that would be useful. Note: This is a reboot of my thread "Greedily packing disks in a sector" from a year ago. I asked that question in the context of a 72 degree sector, but I now realize that the 90 degree sector is worth looking at too, and might yield some pretty pictures and pretty mathematics. Jim Propp

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Re: [math-fun] Unsymmetrical Wankel engine
by Henry Baker 28 Jan '14

28 Jan '14

Actually, the miniature .049 glow-plug model airplane engines _are_ diesels, as they don't have any ignition. I seem to recall that in the good ol' days before cars had electric starters, you had to crank the engine to start it. If you weren't aggressive enough, the car would kick back & possibly run backwards for a bit, breaking your arm in the process. Ditto for farm tractors. The problem with jet engines is that they aren't symmetrical: the blades have an orientation. Furthermore, trying to stuff air backwards through the exhaust into the intake won't work, either, because the fuel injectors aren't in the right place. So-called scramjets don't appear to have any moving parts at all, much less rotating parts, so there is no chirality. At 11:45 AM 1/28/2014, Bill Gosper wrote: >What keeps diesels from starting backwards--valving? What about external >combustion (jet) engines? An engine with perhaps similar problems can be >made from http://toobnix.org/?p=741 >http://vimeo.com/38945542 >generating compression from increasing the twist (decreasing the pitch) >along the length. Further along, the rotors can continuously deform to >circularity >to accommodate porting.

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[math-fun] Unsymmetrical Wankel engine
by Bill Gosper 28 Jan '14

28 Jan '14

What keeps diesels from starting backwards--valving? What about external combustion (jet) engines? An engine with perhaps similar problems can be made from http://toobnix.org/?p=741 http://vimeo.com/38945542 generating compression from increasing the twist (decreasing the pitch) along the length. Further along, the rotors can continuously deform to circularity to accommodate porting. --rwg "Minnesota: Come for the culture. Stay, because your car won't start." It just took me a minute to get the ps on my original mention of the toobnix: No manual entry for man overboard wds> two-stroke engine symmetrical and can get deranged on start and then run backwards The Wankel engine also is symmetrical, as least as depicted here: http://en.wikipedia.org/wiki/File:Wankel_Cycle_anim_en.gif Can one devise an unsymmetrical Wankelian engine with a preferred direction of time? To do so, it seems we want to replace the Reuleaux triangle with a chiral, but still constant-width, shape. http://en.wikipedia.org/wiki/Reuleaux_triangle The Reuleaux triangle is the curve of constant width=1 with least area (the circle has the greatest area). But consulting http://en.wikipedia.org/wiki/Curve_of_constant_width we see that any triangle ABC, including a general and hence chiral triangle, can be converted into a curve of constant width as explained in this picture: http://en.wikipedia.org/wiki/File:How_to_make_mathematical_roller_curve_bas… (The width is 2x+2y-a-b-c.) I believe this allows us to construct a Wankelian engine from any such arising from a triangle is sufficiently close to equilateral. Specifically, imagine the const-width thing (call it "CW") rolling around inside the "stadium" (outer boundary of the Wankel combustion compartment). As it does so, inside CW some small fixed circular ink-pad inks out a black shape inside CW. Cut that shape out of CW's inside, leaving a hole. As CW rolls round the stadium the hole will roll round the fixed circle. It thus can be used to transmit torque to a rotating rod with fixed axis. -- Warren D. Smith

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[math-fun] Unsymmetrical Wankel engine
by Warren D Smith 28 Jan '14

28 Jan '14

>two-stroke engine symmetrical and can get deranged on start >and then run backwards The Wankel engine also is symmetrical, as least as depicted here: http://en.wikipedia.org/wiki/File:Wankel_Cycle_anim_en.gif Can one devise an unsymmetrical Wankelian engine with a preferred direction of time? To do so, it seems we want to replace the Reuleaux triangle with a chiral, but still constant-width, shape. http://en.wikipedia.org/wiki/Reuleaux_triangle The Reuleaux triangle is the curve of constant width=1 with least area (the circle has the greatest area). But consulting http://en.wikipedia.org/wiki/Curve_of_constant_width we see that any triangle ABC, including a general and hence chiral triangle, can be converted into a curve of constant width as explained in this picture: http://en.wikipedia.org/wiki/File:How_to_make_mathematical_roller_curve_bas… (The width is 2x+2y-a-b-c.) I believe this allows us to construct a Wankelian engine from any such arising from a triangle is sufficiently close to equilateral. Specifically, imagine the const-width thing (call it "CW") rolling around inside the "stadium" (outer boundary of the Wankel combustion compartment). As it does so, inside CW some small fixed circular ink-pad inks out a black shape inside CW. Cut that shape out of CW's inside, leaving a hole. As CW rolls round the stadium the hole will roll round the fixed circle. It thus can be used to transmit torque to a rotating rod with fixed axis. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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