November 2008 - mathfun@mailman.xmission.com

Re: [math-fun] polyhedra with all faces congruent
by Dan Asimov 13 Dec '08

13 Dec '08

Here's the Math Reviews review of a paper by Branko Grunbaum that answers the "Odd?" question in the negative. << Branko Grunbaum On polyhedra in E^3 having all faces congruent. Bull. Res. Council Israel Sect. F 8F 1960 215–218 (1960) The polyhedra having all faces congruent include the reciprocals (duals) of all the “uniform” polyhedra, namely, the five Platonic solids, the dipyramids, the reciprocals of the thirteen Archimedian solids, and the reciprocals of the antiprisms. These last have, for any n, 2n kite-shaped faces. (It is unfortunate that the author uses, instead of “kite”, the name “deltoid”, which belongs more properly to a curve, the three-cusped hypocycloid.) The author proves that, if all the faces of a convex polyhedron are congruent, their number is even. Since the faces cannot all have more than five sides, and the result is obvious when the faces are triangular or pentagonal, the problem is quickly reduced to the case of quadrangular faces, and then to kites or parallelograms. The kites are dealt with by an ingenious combination of metrical and topological arguments. Finally, if the faces are parallelograms, their number is equal to the product of two consecutive integers [Coxeter, Regular polytopes, Methuen, London, 1948, p. 27; MR0027148 (10,261e)]. > _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

7 14

Re: [math-fun] Doin' the Hilbert walk
by Steve Witham 27 Nov '08

27 Nov '08

>From: "Fred lunnon" <fred.lunnon(a)gmail.com> > >[Nor unfortunately, has anyone else attempted to discuss exactly >what exactly constitutes a Hilbert-like curve in dimension > 2.] Okay, now I feel qualified to ask what you mean. Probably, "in math language rather than computer code," right? But "exactly?" One sense of an exact constitution of a >2D Hilbert curve requires making some arbitrary choices, which would sound even uglier in math than it does in code. (Since a certain level of arbitrariness and noise is expected in computer code.) Or...do you just want a *class* of curves? That's easier, although I don' speaka good maths.... You would say that the curve passes through the 2^d subcubes in Gray-code order, and the subcubes' curves are miniatures of the whole curve oriented so they connect, i.e., The first subcube's start matches the cube's start--which is a corner. The last subcubes's end matches the cube's end--an adjacent corner. The ends and starts of adjacent subcubes in the Gray code sequence are...right next to each other--you have to put in that connecting step, but so did Hilbert. To me, any arrangement that meets those constraints is a "Hilbert- like" map from one version of the walk to the next. The curve is the limit of the sequence...am I missing something? >From: rwg(a)sdf.lonestar.org > >Looking at stills of these is like trying to parse protein molecules. I drew some "exploded views" of 3D Hibert walks: http://www.tiac.net/~sw/2008/10/Hilbert/hilbert_pic.pdf (If you need a page to download that from: http://www.tiac.net/~sw/2008/10/Hilbert about 4/5 of the way down.) This just takes the binary of each coordinate and reinterprets it in base 4.125. --Steve

3 5

[math-fun] Very cool math pix
by Henry Baker 26 Nov '08

26 Nov '08

This fellow has developed very interesting ways to color the complex plane. His stained glass window is the cover of MAA Focus magazine, Nov. 2008. http://math.scu.edu/~ffarris/

1 0

[math-fun] canonic paleontology
by rwg＠sdf.lonestar.org 25 Nov '08

25 Nov '08

Ed Pegg suggested jotting down some glider gun recollections. I'm a little fuzzy on the chronology of the period 46 guns and p32n and p20n puffers vs Corderman's switch engine and Schick's p12n puffers. Mike B, Mike S, RTW? Someone with a good set of LIFELINEs? The glider gun shuttle appears during generations 774-791 of the r pentomino. I had glimpsed something suspicious thereabouts, and had made a mental note to investigate, but the software was so primitive that I procrastinated the long(!) search and manual copying. Before I got around to it, Robert W. April, an analytical chemist friend of Schroeppel's, visited the (MIT) AI Lab, noodled on the teletype, and found that two rows of 15, separated by 1 blank line, promptly produce two shuttles which successfully bounce off each other before choking on their own eggs. I immediately built a (then record) p30 oscillator by arranging four shuttle pairs in a square. At dinner that evening (Joyce Chen's 500 Memorial Drive?), I expressed optimism for an imminent glider gun. (And got some scoffing.) But there were so many x, y, and time offets for both the corner and side collisions that one of them *had* to work. Exactly one did. I expected to make a large square with 8 shuttles, or even more if a glider came the wrong way out of a corner collision. But before the search even began, Richard P. Howell of the RLE PDP-1 project discovered that a block could eat shuttle eggs, reducing the hypothetical gun to two shuttles. We hastily telegraphed "<28(?) coordinate pairs> is a glider gun" to Martin Gardner c/o Scientific American, fully expecting numerous other discoverers. Western Union (or we) garbled one coordinate, which Martin was able to correct by symmetry. It took a while for the telegram to reach Martin's then-unknown Hastings-on-Hudson address, and we were disappointed that he had not been able to test and confirm it, questioning whether we really understood that a gun must produce infinitely many gliders, not just a large number. (But we were also relieved at his failure to mention other discoverers.) Then out of nowhere we get a really excited congratulatory letter from an unknown guy named Robert T. Wainwright (soon to be editor and publisher of LIFELINE), to whom Martin apparently sent the pattern for confirmation. Scientific American wrote the $50 check. Conway now claims that he phrased his conjecture in expectation of guns and puffertrains (his names) but I recall the opposite, suspecting him of purposely choosing the less plausible outcome to provoke refutation. --rwg Postlude: Noodling on that same teletype a few months later, Jan Kok, then in high school, found the period 8 galaxy Golly logo, starting with a fylfot of diagonal hooks. Before "other discoverers" finally got into the act, I found puffers of period 32n and then 20n, and many guns of period 46n and 30n. Charles Corderman then found his switch engine puffers and Paul Schick found his p12n puffers, well before I found the period 900+200n (middleweight) spaceship guns. Since then, people have built guns of true periods 22, 24, 36, 44, 48, 50, 54, 55, 56, and *all* periods >=60. By clever interlacing, these can produce glider streams of any period >=14, the theoretical density limit. About two years ago, Nick Gotts found a *quadratically* growing pattern starting with only 26 dots, probably unbeatable.

2 2

[math-fun] incredible computing machine from NVIDIA
by Simon Plouffe 24 Nov '08

24 Nov '08

Hello, The graphic card company NVIDIA just announced recently an incredible machine : The NVIDIA Tesla Personal Supercomputer. price ; 9995$ US or 7000 Euros. computing power ; 4 teraflops. I don't know if you realize this, it means that a relatively fortunate but still ordinary citizen can buy a machine which is big as an ordinary pc and which represent the computing power of the TOP500 computer list of 2001. In other words, if we were in 2001, that 'home' computer would be 2nd in the world! This is purely incredible. Well, here is the second announcement, it can also run mathematica, I want one like that. http://www.top500.org/list/2001/06/100 http://www.nvidia.com/object/personal_supercomputing.html http://tech.slashdot.org/article.pl?sid=08/11/23/068234&from=rss in plain english, that machine can run programs up to at the least 100 times faster. Simon Plouffe

3 3

Re: [math-fun] shuffling
by Steve Witham 22 Nov '08

22 Nov '08

>From: James Propp <jpropp(a)cs.uml.edu> > >Does the shuffle operation on in iPod choose a genuinely pseudorandom >song from the playlist (so that, if there are n items, there's a 1/n >chance that the next song will be the one you just heard)? 1) "Shuffle" suggests something better: a random permutation of what's in there. That means many songs should have zero odds of coming up next. What you do when you get through the first permutation I have wondered about. 2) You can go back as many songs as you like in the random sequence, and it remembers. Suggests it has enough memory to store a permutation (although there are ways to generate fair permutations without storing them). 3) I remember a release note once saying "fixed a bug where it would play two songs in a row from the same album too often." Makes me wonder about their notion of random. 4) I have a complicated set of interdependent "smart playlists" set up so my 1GB iPod gets loaded with songs that haven't been played... or skipped... in 18 months or so. What you play or skip with the iPod updates iTunes when you sync. What I mean is, I wasn't happy with the randomness, but they finally provided a tool to do something about it. 5) No erosion. Oops, wrong question. --Steve

1 0

Re: [math-fun] hiking v. computer generated images
by Henry Baker 22 Nov '08

22 Nov '08

I thought about that, but erosion should generally "round off" things, rather than sharpen them. If you look at many mountain ranges, they look like heavily crinkled paper, with flat sections and sharp corners. It's like I wadded up a piece of paper, and then tried to press it flat. What's that all about? At 03:37 PM 11/22/2008, Hilarie Orman wrote: >> But there aren't any ridges on Mandelbrot's hills. Where did they >> go? What's wrong with the model? > >No erosion. > >Hilarie

1 0

[math-fun] hiking v. computer generated images
by Henry Baker 22 Nov '08

22 Nov '08

Mandelbrot's book had some pix of fractal "hills that never were". If you do a Google search for this phrase, Google will show you the actual pictures from Mandelbrot's book. My problem is that these hills are not as "hikeable" as real mountains, so something is wrong with this picture (or actually, the model generating the mountains). If you hike a lot, or if you stare intently out of airplane windows while taking off or landing in mountainous areas, or if you spend a lot of time with Google "Earth", you notice that there are a lot of trails & roads on mountain "ridges". But there aren't any ridges on Mandelbrot's hills. Where did they go? What's wrong with the model?

1 0

[math-fun] shuffling
by James Propp 22 Nov '08

22 Nov '08

Does the shuffle operation on in iPod choose a genuinely pseudorandom song from the playlist (so that, if there are n items, there's a 1/n chance that the next song will be the one you just heard)? Or does it pick the new song pseudorandomly from among the n-1 songs other than the one just heard? Or does it do something more interesting to make sure that a song isn't heard twice in too-close proximity? I only mention the iPod as one example, so if you happen to know about analogous "shuffle" operations on other platforms (and not just in the domain of music), I'd be interested. Jim P.S. I love having you guys around so I can get thoughtful and/or well-informed and/or fun answers to such questions five minutes after they occur to me!

2 1

[math-fun] automated proof checking
by Schroeppel, Richard 22 Nov '08

22 Nov '08

The December Notices of the AMS has four articles about automated checking of mathematical proofs. Taken together, they assert that the tools are nearly ready for prime time. Download & startup details are included for the adventurous. Julie Rehmeyer (Math Trek/Science News) has an introductory article at http://www.sciencenews.org/view/generic/id/38623/title/ Math_Trek__How_to_(really)_trust_a_mathematical_proof that includes pointers to the AMS articles. The articles make a plausible case that most individual math results could be verified with merely an unreasonable dedication (i.e., less than an impossible effort). The corollary is that (eventually) unverified results should not be submitted for review. There's lip service to the notion that math is more than (definition,theorem,proof)*, but not much more. One of the articles is about the proof of the Four Color Theorem -- the effort of formalization, and new mathematics developed along the way. The article demotes diagrams to second class objects, with character strings and syntax trees being "real". :-! There's serious work underway on things like checking the classification of simple groups and the Odd Order Theorem. The Prime Number Theorem has been verified both ways (the Erdos-Selberg elementary proof, and the Riemann Zeta Function based proof). This uses a considerable base of results about integrals and complex analytic functions. There's even some attention given to making systems that are somewhat usable, and working with theorems and proofs that are sort of readable. This seems to be progress, at the level of being able to (someday soon) check our results. The downside is that these folks have a hammer, and non-nails don't exist. Rich

2 1