April 2007 - mathfun@mailman.xmission.com

[math-fun] Re: unistable object
by James Propp 11 Apr '07

11 Apr '07

>Science News' Math Trek has an interesting article this week, by Julie J. Rehmeyer. >It's about a unistable object discovered by Gabor Domokos and Peter Varkonyi. > >There is a prize offered for a polyhedral solution, but it's $10K/face-count. >The authors don't expect to pay much. There's no discussion of the obvious >first move, simply taking their rounded solution and using sandpaper to >make a lot of tiny flat faces. I must be missing something. Why doesn't the 19-face polyhedron shown in http://mathworld.wolfram.com/UnistablePolyhedron.html qualify? Jim Propp

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RE: [math-fun] Re: unistable object
by Daniel Asimov 11 Apr '07

11 Apr '07

Rich asks: << What's the back side of this object look like? I count 17 visible faces. Rich >> The colored depiction of the 19-face polyhedron is rotatable with one's mouse. ==Dan -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com on behalf of James Propp Sent: Tue 4/10/2007 7:44 PM To: math-fun(a)mailman.xmission.com Subject: [math-fun] Re: unistable object >Science News' Math Trek has an interesting article this week, by Julie J. Rehmeyer. >It's about a unistable object discovered by Gabor Domokos and Peter Varkonyi. > >There is a prize offered for a polyhedral solution, but it's $10K/face-count. >The authors don't expect to pay much. There's no discussion of the obvious >first move, simply taking their rounded solution and using sandpaper to >make a lot of tiny flat faces. I must be missing something. Why doesn't the 19-face polyhedron shown in http://mathworld.wolfram.com/UnistablePolyhedron.html qualify? Jim Propp

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[math-fun] unistable object
by Richard Schroeppel 11 Apr '07

11 Apr '07

Science News' Math Trek has an interesting article this week, by Julie J. Rehmeyer. It's about a unistable object discovered by Gabor Domokos and Peter Varkonyi. http://sciencenews.org/articles/20070407/mathtrek.asp The SciNews page has a picture, and pointers to three of their tech reports. A unistable object is a 3D shape with only one stable equilibrium point. It's easy to do this if you allow the object to contain an osmium mascon under just under the surface of a curved face, or equivalently if you allow holes (or voids). The problem is interesting if you require a convex object of uniform density. We've discussed it on this list a couple of times. My particular inspiration may go back to the early space program: the astronauts would come back in a small space capsule, parachuting into the ocean. The capsule designers had to plan for two possible orientations of the capsule while it was bobbing in the ocean, waiting for recovery. The new object has curvy faces with various edges, reminiscent of a "surface of constant width". It has one stable and one unstable balance point, although the picture looks like some of the edges might also contribute more unstable balance points. It vaguely resembles a turtle, which has lead the authors to start playing with turtles. There is a prize offered for a polyhedral solution, but it's $10K/face-count. The authors don't expect to pay much. There's no discussion of the obvious first move, simply taking their rounded solution and using sandpaper to make a lot of tiny flat faces. (unrelated) news item from last week: Fermilab has been building magnets for CERN's Large Hadron Collider, and one failed a pressure test badly, "blowing up" in the tunnel. No injuries, damage assessment in progress, Fermilab appropriately embarrassed, lots of copies of the magnet will need rework. First light was expected this fall, noone's admitting to a schedule slip yet. Rich rcs(a)cs.arizona.edu

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[math-fun] Internet phone calls
by Daniel Asimov 06 Apr '07

06 Apr '07

I just read that Vonage has been enjoined from enrolling any new customers while a certain federal judge decides whether it is infringing on one or more patents held by Verizon for making phone calls over the internet. Can someone please describe just what these Verizon patents cover? (I ask because I've been able to have real-time audio conversations on the net for many years -- e.g., by being logged into the same Yahoo group as someone else at the same time.) --Dan

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[math-fun] determinant of determinants
by Richard Schroeppel 04 Apr '07

04 Apr '07

[ [A B] [A B] [A B] ] [ [G H] [I J] [K L] ] [ ] [ [C D] [C D] [C D] ] [ [G H] [I J] [K L] ] = 0. [ ] [ [E F] [E F] [E F] ] [ [G H] [I J] [K L] ] Each inner 2x2 matrix is replaced with its determinant, and then the determinant of the outer 3x3 is evaluated. It looks more prosaic when you realize that the inner determinants can instead be permanents, or Re((A+Bi)(G+Hi)) or Im, or (A,B)dot(G,H). On the other hand, it looks a little more interesting when you don't apply Det to the inner matrices, but just do Det of the outer 3x3. (The non-commutative multiplication of the inner matrices is fixed by requiring that all the products in the 3x3 det go from left to right, i.e. M1x * M2y * M3z.) Then the det is a 2x2 zero matrix. Is there a Big Book of these? Rich rschroe(a)cs.arizona.edu

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Re: [math-fun] pile of sand constant
by Daniel Asimov 03 Apr '07

03 Apr '07

Joshua writes: << I read that the angle of repose for dry sand is about 30 degrees, but there are some other sites out there that tell me it depends on the grain size. . . . >> If the grain size is roughly comparable to the particles of material that constitute the base surface (say, earth), it makes sense that the angle would depend on the grain size. But where the grain size is much larger than the particles making up the base surface (e.g., table salt on linoleum), I'd expect the situation would be virtually independent of the scale of the stuff being piled. --Dan

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[math-fun] FW: New Repunit R(109297) (Harvey Dubner)
by Schroeppel, Richard 03 Apr '07

03 Apr '07

-----Original Message----- From: Number Theory List on behalf of Harvey Dubner Sent: Tue 4/3/2007 11:15 AM To: NMBRTHRY(a)LISTSERV.NODAK.EDU Subject: New Repunit R(109297) New Probable Prime Repunit, R(109297) On March 28 I found a new probable prime repunit, R(109297), which consists of 109297 1's. It is effectively impossible to prove true primality at this time. The only known repunit primes are R(2), R(19), R(23), R(317), and R(1031). In 1999 I found the PRP repunit R(49081), and in 2000 Lew Baxter found the PRP repunit R(86453). I plan to try to raise the test limit to 200000. My continued interest in repunits is due primarily to the fact that my first project on a home computer (1980) was searching for repunit primes. References: H.C, Williams and H. Dubner, The Primality of R1031, Math. Comp. 47(176), Oct 1986, 703-711. Harvey Dubner, Repunit R49081 is a Probable Prime, Math. Comp. 71(238), (2002), 833-835. A004023, On-line Encyclopedea of Integer Sequences. Harvey Dubner

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Re: [math-fun] pile of sand constant
by Henry Baker 03 Apr '07

03 Apr '07

The angle of inclination of several of the pyramids is in the range of 51.5 - 53.2 degrees. http://www.greatbuildings.com/buildings/Great_Pyramid.html Apparently, the Egyptians tried higher angles of inclination, but some of these pyramids collapsed -- e.g., Meidum. http://en.wikipedia.org/wiki/Egyptian_pyramids I would guess that the term "angle of repose" comes from the fact that these pyramids became graves? :-) :-) A pile of sand would not have the same shape _regardless of size_, as the increasing weight of the central part eventually changes the shape of the grains to cause a collapse into a smaller angle of inclination. And we in California are well aware that even relatively small amounts of water can cause spectacular collapses -- e.g., Route 1. At 09:01 AM 4/3/2007, Joshua Zucker wrote: >On 4/3/07, Simon Plouffe <simon.plouffe(a)gmail.com> wrote: >>Hello, >> >> I have this information about a certain constant >>associated with a pile of sand. Apparently the ratio >>of the half the base to the height is approx. 4/3. > >I think the geological term you are looking for is the "angle of repose". > >>I don't remember where I saw that for the first time, >>empirically, well at least for ordinary and dry sand, >>it looks like it is. In other words, a pile of sand >>will always have the same shape regardless of the size, >>or the slope is constant. > >I read that the angle of repose for dry sand is about 30 degrees, but >there are some other sites out there that tell me it depends on the >grain size. With 30 degrees, that would be sqrt(3), not 4/3. But I >think that's a pretty rough number, and I see numbers ranging up to >just under 40 degrees, which would give a ratio of about 1.2, so your >value of 4/3 seems to be in the range. The angle should be steeper >for smaller sand particles. > >--Joshua Zucker

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Re: [math-fun] pile of sand constant
by Daniel Asimov 03 Apr '07

03 Apr '07

Simon writes: << I have this information about a certain constant associated with a pile of sand. Apparently the ratio of the half the base to the height is approx. 4/3. I don't remember where I saw that for the first time, empirically, well at least for ordinary and dry sand, it looks like it is. In other words, a pile of sand will always have the same shape regardless of the size, or the slope is constant. Is there anybody that has any information about this phenomena? Is this known ? >> Thanks to the title of a noel by Wallace Stegner, I learned that the angle of the steepest pile of [whatever] is called its "angle of repose"; here it would be atan(3/4). (Cf. < http://en.wikipedia.org/wiki/Angle_of_repose >.) The angle depends heavily on the kind of stuff that is piled up. The angle of repose would be low for glass marbles, and higher for gravel of about the same size. So I doubt that the number for a pile of sand is intentionally trying to be 3/4 (or any fraction with a small denominator). --Dan

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[math-fun] pile of sand constant
by Simon Plouffe 03 Apr '07

03 Apr '07

Hello, I have this information about a certain constant associated with a pile of sand. Apparently the ratio of the half the base to the height is approx. 4/3. I don't remember where I saw that for the first time, empirically, well at least for ordinary and dry sand, it looks like it is. In other words, a pile of sand will always have the same shape regardless of the size, or the slope is constant. Is there anybody that has any information about this phenomena? Is this known ? Simon Plouffe

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