April 2010 - mathfun@mailman.xmission.com

[math-fun] Stupid math joke
by Henry Baker 26 Apr '10

26 Apr '10

Q: Why did the chicken cross the MÃ¶bius strip? A: To get to the same side. (I warned you it was stupid!)

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[math-fun] Interesting combustion engine design
by Henry Baker 25 Apr '10

25 Apr '10

Animation here: http://www.ericksonmotors.com/mccsim.htm Miniature working engines here: http://www.craftsmanshipmuseum.com/KnappCol.htm

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Re: [math-fun] tilted sinewave from family of polynomials
by Bill Gosper 24 Apr '10

24 Apr '10

AARGH! All but two of the graphics have disappeared from the *notebook* from which that HTML was made! Likewise on my memory stick! Let's hope this is a Mac issue, and not WRI punting Macsyma outputs! A few hours will tell. --rwg On Fri, Apr 23, 2010 at 3:04 AM, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > Bill. > The first two .GIF files (showing your previous diagrams) display when this > is opened; subsequent formulae are all blank. Tried with Firefox and with > Safari. Fred > > On 4/23/10, Bill Gosper <billgosper(a)gmail.com> wrote: > > See end of http://www.tweedledum.com/rwg/pizza.html . Anybody recognize > > those? > > --rwg > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > >

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[math-fun] Trying to recall funny shape
by Dan Asimov 23 Apr '10

23 Apr '10

I recently read of an odd solid shape, with I think an odd name, that has recently been defined. It may be this: Take a solid cylinder whose diameter equals its height, cut it in half along a square, rotate one half 90 degrees, and glue the halves back together again. Does this shape have a name, or is there something similar that does? Apparently it has very interesting rolling behavior. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] derandomized Galton board
by James Propp 23 Apr '10

23 Apr '10

Joerg Arndt writes: >Hack attack by corrupt file? > >Any attempt to view (or convert) this file lead to >death by OOM killer. Is "OOM" short for "Out Of Memory"? Is anyone else having this problem? I created this dynamic gif file using whirlgif Rev 3.02 (c) 1997,1998 by Hans Dinsen-Hansen (c) 1995,1996 by Kevin Kadow (c) 1991,1992 by Mark Podlipec It works for me on my Mac when viewed using either Firefox or Safari. Jim

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Re: [math-fun] derandomized Galton board
by James Propp 23 Apr '10

23 Apr '10

I just created a Jing file http://jamespropp.org/Galton.swf Click on the play-recording icon in the middle to make it start. The audio quality isn't great, and I made a couple of mistakes in my narration, but it should give you the idea. Jim

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[math-fun] derandomized Galton board
by James Propp 23 Apr '10

23 Apr '10

Inspired by some joint investigations I conducted with Steve Witham a year or so ago, I recently did a simulation of an even more natural "derandomized random walk" in the quadrant: see http://jamespropp.org/quincunx.gif This shows what happens when you use a non-random Galton board in which the pins are replaced by flip-flops, so that the 1st marble to arrive at a flip-flop gets routed to the right, the 2nd gets routed to the left, the 3rd gets routed to the right, and so on, in alternation. My quasi-Galton board has 1024 rows, but the picture is rotated by 45 degrees, so the "bottom row" actually appears as a diagonal. Hence: the 1st marble to arrive at a flip-flop gets routed rightward, the 2nd gets routed downward, the 3rd gets routed rightward, and so on. White pixels corresponds to flip-flops that lie outside the board, or have not yet been visited by a marble. Yellow pixels correspond to flip-flops that most recently sent a marble rightward. Blue pixels correspond to flip-flops that most recently sent a marble downward. The nth frame (where the initial frame is considered the 0th) shows the states of the flip-flops (or, as I tend to call them, rotors) after the first n particles have passed beyond the first 1024 rows. When viewed with a suitable browser, quincunx.gif shows the first 1000 steps of what happens to the rotors. (It's kind of small at first, but wait for it: the set of non-white pixels will eventually become big enough to look at without a magnifying glass.) As you can see, patterns spontaneously form and propagate in a kind of Jacob's ladder fashion, with some triangles moving outward and others moving inward, as in Steve's "rounder-router" simulations. I believe this is new, but I won't be surprised if it isn't, since it's such a natural thing to look at. Has anyone seen this before? I've also done some variants of this simulation (changing the initial conditions and in some cases incorporating randomness); I can post them if people are interested. Note that there's another way one might do the simulation that looks superficially different, in which you drop a new marble at (0,0) at each time step, so that after n marbles have been dropped in, there's a single marble in {(i,j): i+j=k} for all k between 0 and n-1. This resulting movie, viewed as a static three-dimensional object, is essentially the same as the movie I created; you're just slicing the space-time object differently using different notions of "time". Perhaps this way of looking at things is more informative; I haven't coded it yet. The object being studied can be written as a fairly simple three- dimensional celluluar automaton with specific simple boundary conditions, so perhaps Wolfram and his crowd know about it already. Any leads or observations will be appreciated! Jim Propp

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[math-fun] tilted sinewave from family of polynomials
by Bill Gosper 23 Apr '10

23 Apr '10

See end of http://www.tweedledum.com/rwg/pizza.html . Anybody recognize those? --rwg

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Re: [math-fun] Fwd: Trying to recall funny shape
by Henry Baker 22 Apr '10

22 Apr '10

It _might_ be slightly easier to deal with a "discrete" version of this problem, instead of your "continuous" version. The idea is that the object has a large number of "faces", and transitions from sitting on one face to sitting on another face, but retains enough momentum to continue to move. The problem is to keep the momentum from falling too low or from "snowballing" & getting too large. One can take a cue from the little walking robots -- they still take discrete steps, but if the inclined plane is a bit steeper, they make larger excursions. One needs to incorporate some sort of automatic throttling mechanism to remove energy when the object starts to move too fast. Thus, we need some sort of _dissipation_ mechanism -- a dashpot or perhaps a fan (dissipate energy into moving air). I can imagine some sort of clockwork mechanism inside such a device with faces, such that a moving weight provide just enough of an impulse to get the object to move to the next face in the sequence. This requires a clever design of the faces to make sure that the faces are large enough so that the object can stabilize itself, but not so large that it can't make the next transition. Furthermore, the angle between the faces can't be very large, or it will take too much energy to move the object onto the next face. At 02:54 PM 4/22/2010, Allan Wechsler wrote: >Mistakenly sent this to Henry alone. > >---------- Forwarded message ---------- >From: Allan Wechsler <acwacw(a)gmail.com> >Date: Thu, Apr 22, 2010 at 5:53 PM >Subject: Re: [math-fun] Trying to recall funny shape >To: Henry Baker <hbaker1(a)pipeline.com> > >I'm attracted to the simpler problem: no internal moving parts, constant >density (though if hollows are allowed, I think we don't lose any generality >there), and a level surface. Within these constraints, I wonder what is >possible. I suspect that in theory you can make it do almost anything (as >long as you don't mind it rolling very slowly). > >Here's a problem: Can you design a roller that goes in a straight line >forever? It would have a cylindrical ring for the steady state, and a >nearby hillock to provide the initial extra potential energy, but designing >the transition so that the ground-trace would be straight would be a big >challenge. > >On Thu, Apr 22, 2010 at 5:38 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Very cool idea! >> >> However, you'd have to control the speed & angle very carefully to allow >> the shape to fall to one side or the other at the appropriate time. >> >> You might also have to engineer non-homogeneity into the mass to give you >> some more degrees of freedom. >> >> Of course, if you also allow for internal mechanisms, where weights can >> move around within while "rolling" without, the possibilities seem endless. >> >> If you also give some of the internal parts the ability to start with a >> spin, then gyroscopic mechanisms can also be employed. >> >> There are already simple "walking" robots with no moving parts that can >> maneuver themselves down an inclined plane, so these ideas would subsume >> those robots. >> >> At 12:00 PM 4/22/2010, Allan Wechsler wrote: >> >This business of weird rolling shapes got me musing. One could design a >> >"ball" that, released from rest in the proper orientation on a >> sufficiently >> >flat surface, would roll along a predetermined path. In principle, if one >> >were parsimonious enough with potential energy, the path could be >> >arbitrarily ornate, though the object would roll very slowly along it. >> > >> >So, in practice, how complicated a path could be programmed into, say, a >> >machined steel roller? Could a skilled machinist create a "signature" >> >roller that wrote the machinist's name? >> > >> >On Thu, Apr 22, 2010 at 2:38 PM, <rcs(a)xmission.com> wrote: >> > >> >> [another SBG contribution trapped by the mystery mail filter -- rich] >> >> ----------- >> >> >> >> Date: Wed, 21 Apr 2010 15:46:57 -0700 >> >> Subject: Re: [math-fun] Trying to recall funny shape >> >> From: "Stephen B. Gray" <stevebg(a)roadrunner.com> >> >> To: math-fun(a)mailman.xmission.com >> >> >> >> On 4/21/2010 12:50 PM, Dan Asimov wrote: >> >> >> >> I recently read of an odd solid shape, with I think an odd name, >> >>> that has recently been defined. It may be this: >> >>> >> >>> Take a solid cylinder whose diameter equals its height, cut it in half >> >>> along a square, rotate one half 90 degrees, and glue the halves back >> >>> together again. >> >>> >> >>> Does this shape have a name, or is there something similar >> >>> that does? Apparently it has very interesting rolling >> >>> behavior. >> >>> >> >>> --Dan >> >>> >> >>> It's called a gefubuwichmuttle. >> >> Next question? >> >> >> >> -Anonymous

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Re: [math-fun] Fwd: Trying to recall funny shape
by Henry Baker 22 Apr '10

22 Apr '10

I forgot to mention that _loaded dice_ (die?) are also subsumed by this class of automata, so there is a several thousand year history of technology to check out. Of course, much of this technology isn't written down anywhere! ;-) At 02:54 PM 4/22/2010, Allan Wechsler wrote: >Mistakenly sent this to Henry alone. > >---------- Forwarded message ---------- >From: Allan Wechsler <acwacw(a)gmail.com> >Date: Thu, Apr 22, 2010 at 5:53 PM >Subject: Re: [math-fun] Trying to recall funny shape >To: Henry Baker <hbaker1(a)pipeline.com> > >I'm attracted to the simpler problem: no internal moving parts, constant >density (though if hollows are allowed, I think we don't lose any generality >there), and a level surface. Within these constraints, I wonder what is >possible. I suspect that in theory you can make it do almost anything (as >long as you don't mind it rolling very slowly). > >Here's a problem: Can you design a roller that goes in a straight line >forever? It would have a cylindrical ring for the steady state, and a >nearby hillock to provide the initial extra potential energy, but designing >the transition so that the ground-trace would be straight would be a big >challenge. > >On Thu, Apr 22, 2010 at 5:38 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Very cool idea! >> >> However, you'd have to control the speed & angle very carefully to allow >> the shape to fall to one side or the other at the appropriate time. >> >> You might also have to engineer non-homogeneity into the mass to give you >> some more degrees of freedom. >> >> Of course, if you also allow for internal mechanisms, where weights can >> move around within while "rolling" without, the possibilities seem endless. >> >> If you also give some of the internal parts the ability to start with a >> spin, then gyroscopic mechanisms can also be employed. >> >> There are already simple "walking" robots with no moving parts that can >> maneuver themselves down an inclined plane, so these ideas would subsume >> those robots. >> >> At 12:00 PM 4/22/2010, Allan Wechsler wrote: >> >This business of weird rolling shapes got me musing. One could design a >> >"ball" that, released from rest in the proper orientation on a >> sufficiently >> >flat surface, would roll along a predetermined path. In principle, if one >> >were parsimonious enough with potential energy, the path could be >> >arbitrarily ornate, though the object would roll very slowly along it. >> > >> >So, in practice, how complicated a path could be programmed into, say, a >> >machined steel roller? Could a skilled machinist create a "signature" >> >roller that wrote the machinist's name? >> > >> >On Thu, Apr 22, 2010 at 2:38 PM, <rcs(a)xmission.com> wrote: >> > >> >> [another SBG contribution trapped by the mystery mail filter -- rich] >> >> ----------- >> >> >> >> Date: Wed, 21 Apr 2010 15:46:57 -0700 >> >> Subject: Re: [math-fun] Trying to recall funny shape >> >> From: "Stephen B. Gray" <stevebg(a)roadrunner.com> >> >> To: math-fun(a)mailman.xmission.com >> >> >> >> On 4/21/2010 12:50 PM, Dan Asimov wrote: >> >> >> >> I recently read of an odd solid shape, with I think an odd name, >> >>> that has recently been defined. It may be this: >> >>> >> >>> Take a solid cylinder whose diameter equals its height, cut it in half >> >>> along a square, rotate one half 90 degrees, and glue the halves back >> >>> together again. >> >>> >> >>> Does this shape have a name, or is there something similar >> >>> that does? Apparently it has very interesting rolling >> >>> behavior. >> >>> >> >>> --Dan >> >>> >> >>> It's called a gefubuwichmuttle. >> >> Next question? >> >> >> >> -Anonymous

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