April 2010 - mathfun@mailman.xmission.com

[math-fun] Fwd: Trying to recall funny shape
by Allan Wechsler 22 Apr '10

22 Apr '10

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] Trying to recall funny shape
by Henry Baker 22 Apr '10

22 Apr '10

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] Trying to recall funny shape
by rcs＠xmission.com 22 Apr '10

22 Apr '10

[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 > > ________________________________________________________________________________________ > "Outside of a dog, a book is man's best friend. Inside of a dog, > it's too dark to read." --Groucho Marx > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > It's called a gefubuwichmuttle. Next question? -Anonymous

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Re: [math-fun] Trying to recall funny shape
by Dan Asimov 22 Apr '10

22 Apr '10

Thanks, Seb, very nice. ________________________________________________________________________________________ "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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[math-fun] IMPORTUNATE PERMUTATION
by Bill Gosper 22 Apr '10

22 Apr '10

While flagitating some kids with my usual sutra of always replacing Bernoulli numbers with Bernoulli polynomials in some new variable, I derived (c47) intosum(sum(2*(-1)^i*x^(2*i-1)*bernpoly(y,2*i)/(2*i)!,i,0,inf) = sin(x*y)+cot(x/2)*cos(x*y)) inf ==== i 2 i - 1 \ 2 (- 1) x bernpoly(y, 2 i) (d47) > ---------------------------------- = / (2 i)! ==== i = 0 x sin(x y) + cot(-) cos(x y) 2 and (c48) intosum(sum(2*(-1)^i*x^(2*i)*bernpoly(y,2*i+1)/(2*i+1)!,i,0,inf) = cot(x/2)*sin(x*y)-cos(x*y)) inf ==== i 2 i \ 2 (- 1) x bernpoly(y, 2 i + 1) (d48) > ---------------------------------- = / (2 i + 1)! ==== i = 0 x cot(-) sin(x y) - cos(x y) 2 generalizing the expansions of cot(x/2) and -+1(!) respectively. Putting y=0 and y=1 in the first one must be equal, giving the cot halfangle formula. --rwg COURT PAINTERS PERSCRUTATION

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[math-fun] IMPORTUNATE PERMUTATION
by Bill Gosper 21 Apr '10

21 Apr '10

On 4/21/10, Bill Gosper <billgosper(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote: > While flagitating some kids with my usual sutra of always replacing > Bernoulli > numbers with Bernoulli polynomials in some new variable, ... Don't people get locked up for that sort of thing these days? (Or is that only in Switzerland ...) > Putting y=0 and y=1 in the first one must be equal, giving the cot halfangle > formula. --rwg Maybe should have read "y=0 and y=pi" ? Nope. All the Bernpolys save B_1 are equal at y=0 and 1. Ah, you've been had by Macsyma's egregious argument order! Sorry about that. --rwg

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Re: [math-fun] Trying to recall funny shape
by Dan Asimov 21 Apr '10

21 Apr '10

Thanks, Clark and Thane! Nice links. Didn't know about all those sphericons. It would be fun to play with solid models of them. Now that I see them, I'm sure the shape I was trying to think of was in fact the (4,1) sphericon: Rotate a square about a diagonal, cut it in half through that diagonal, rotate one half 90 degrees with respect to the other and glue them back together again. That nice and simple sphericon would be especially fun to play with. There are probably plenty of games waiting to be invented that could be based on this shape. --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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[math-fun] pop-up books
by James Propp 21 Apr '10

21 Apr '10

What's the biggest pop-up any of us has seen in a pop-up book? What are the mathematical or physical limitations on what a book can do when you turn a page? (I should probably just ask Eric Demaine, but it seems like a fun enough question that I thought I'd post it in this forum and see what people come up with.) Jim Propp

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[math-fun] Rhombic triacontahedron dissection
by Fred lunnon 20 Apr '10

20 Apr '10

The dissection of a rhombic triacontahedron interior into 10 obtuse and 10 acute hexahedral "rhombs" --- sheared cubes, with face diagonals in the golden ratio --- is well-known, apparently going back to Kepler. (A) Modulo symmetries of the icosahedron, is the dissection unique? (B) How many isomorphic solutions are there? George Hart introduces an elegant 5-colouring of the rhombic faces: see http://www.georgehart.com/virtual-polyhedra/dissection-rt.html. (C) Does 5-colouring rhombs make the dissection easier? harder? It is elementary that the triacontahedron exterior has 30 faces, 60 edges, 32 corners; Coxeter (Regular Polytopes, sect 2.71 p26) asserts that 21 faces belong to 7 obtuse rhombs, and 9 to 9 acute rhombs. (D) How many more (non-coincident) corners, edges, faces, solids are hidden in the dissected interior? Guy Inchbald has a page showing some exploded views of the dissection: see http://www.steelpillow.com/polyhedra/quasicr/quasicr.htm These are difficult to interpret, largely as a result of the tendency of (technically oriented) human vision to re-interpret perspective rhombs as (weirdly oriented) rectangles. I have prepared a Maple worksheet which allows the exploded dissection to be colour-coded, rotated interactively, and (with some difficulty) selectively dis-assembled; the results are notably easier to interpret. The script is available on email request. Fred Lunnon

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[math-fun] Packing of regular tetrahedra
by W. Edwin Clark 17 Apr '10

17 Apr '10

On Sat, Aug 15, 2009 at 6:45 PM, Stephen B. Gray <stevebg(a)roadrunner.com> wrote: > Packing of regular tetrahedra reaches 78.2 %. See > www.sciencedaily.com/releases/2009/08/090812143943.htm . It has now reached 100/117 ~ 0.8547 Moreover this packing by Yoav Kallus, Veit Elser and Simon Gravel is periodic with just 4 tetrahedra in the unit cell. See http://arxiv.org/abs/0910.5226 --Edwin Clark

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