July 2004 - mathfun@mailman.xmission.com

Re: [math-fun] Fractal coordinates
by Mike Stay 22 Jul '04

22 Jul '04

Gene Salamin wrote: > The number of coordinates needed should depend on the > topological dimension, not the Hausdorff dimension. And that's always a nonnegative integer, so there's no trouble with fractional numbers of coordinates. Thanks! -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] biroller, again
by Bill Gosper 22 Jul '04

22 Jul '04

Last week I was privileged to attend a small puzzlement including Rob Pike (organizer), Wei-Hwa Huang, David Hagelbarger, Scott Schwartz, and John Lamping. David had machined an aluminum biroller. (He also made an aluminum version of my Twubblesome Twelve.) Wei-Hwa pointed out an obvious derivation of the volume: if the biroller with its inscribed sphere rests on a pole, all the horizontal cross-sections are squares with inscribed circles. So volumes(biroller/sphere) = areas(square/circle) = 4/pi. Archimedes may have seen this and not (completely) invented integral calculus after all. The same trick applies to moments-of-inertia(biroller/sphere) = 16/3pi. John was able to spin the biroller on an "equatorial corner", so that it swept out the union of a bicone and prolate spheroid. My insight- free double integral says this axis has exactly half the moment of inertia of the polar one, 64r^5/45. The opaque region of the corner-spinning biroller resembles a prolate spheroid, but is actually the intersecion of the spheroid and the bicone. Is there a name for these "inscribed solids of rotation"? I ran a long demo on my battery-powered laptop, using a conference room projector instead of the laptop screen. At the end I was amazed to see a 100% battery reading, and concluded that the laptop had sustained itself on power sucked out of the video cable. But my hosts assured me that this was a typical laptop failure and my battery was actually down. Indeed I've seen power meter nonsense on a very similar laptop, but I then continued to run on battery at a friends' house, where it gradually dropped to 90%, as if the "100%" were legit. Thought for today: (c69) (fancy_display:false,sqrt(2^(1/3)+5^(1/6)+5^(2/3)),%%=subst([2=t,t=2],substpart(expand(piece),denest_radicals(%%),1))) 2/3 1/6 1/3 (d69) sqrt(5 + 5 + 2 ) = 5/6 2/3 1/3 1/6 5 5 1/6 5 5 7/6 - ------- + ---- + 2 sqrt(5) + ------- + ---- + 2 sqrt(2) 1/6 sqrt(2) 1/6 2 2 ------------------------------------------------------- 3 (I tried to demo my favorite, from www.tweedledum.com/rwg/idents.htm, but in speeding up other cases, I made that one take forever.) --rwg squawky castrato: disharmonic anorchidism

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Re: [math-fun] zetta, yotta, what next?
by Phil Carmody 22 Jul '04

22 Jul '04

3-in-1 > From: "David Wilson" <davidwwilson(a)comcast.net> > > Back in 1977, when I first set my hands on a computer, a gigabyte of storage > would have been a phenominal luxury. Today, it is as common as dirt to hear > "gig of disk." Most computer users see a "gig" as a unit of disk space or > memory, > much as a cup is a unit of liquid. They neither know nor care that "giga" > means > 10^9, that a "gig" or gigabyte is actually 2^30 = 1,073,741,824 bytes, or This is a common misconception. It has only ever been true for RAM where the technology itself has been physically growing by powers of two. Ah, Microsoft has propagated it too, and it's also been true in the uninformed media, alas, hence the commonness of the misconception. In telecommunications, they've been working with 1000 octets a second for their kb/s, and the units multipliers all scales by powers of 10. E.g. my Nokia BB2M is rated as 2.048Mb/s, meaning 2048000b/s, to match G.702, and the entire plesiochronous digital hierarchy (US Sonet) is based on multiples of that measurement. SDH also followed the same well-entrenched tradition as PDH. The use of multipliers as powers of ten in communications to measure large quantities of data predates other computer uses of the term. Hard disk technology is where the real confusion arises. Hard disks simply aren't limited by the same power of two increases as RAM technology, and hard disks have had arbitrary sizes. Most, but not all, hard disk manufacturers have always described the size of their disks using decimal powers, rather than binary ones. The popular media has incorrectly called this deceitful, because their OS choses, equally arbitrarily, to divide by powers of 2 and report a smaller size. Considering the raw data capacity of a hard disk has _never_ been reflected by the formatted capacity of same, due to boot sectors, partition tables, FATs, superblocks, reserved space, and simple cluster/sector wastage, the popular press' arguments are mostly without merit. Why do they not complain at the OS for chosing the 2^20 scaling factor rather than the HD manufactors for using the 10^6 one? To guarantee confusion, there _is_ a real reason, alas. It's basically that hard disks have had power-of-2 based sector sizes - typically 512 bytes, and so there has been _some_ binary tradition. Recently, 1996 IIRC, some utterly horrible prefixes were invented as an attempt to reduce the confusion. Pedantically, _only_ Gibi is 2^30 nowadays. This keeps the telecomms guys and the hard disk guys happy, at least, as the sanity of ISO 31 is restored at least on paper. > Allan Wechsler wrote: > > > Kilo, Mega, Giga, Tera, Peta, Exa, Heta, Oto, Nea > > > Starting from Tera, they are derived by doing some sort of surgery on the > > Greek word for the log base-1000. For Tera, Peta, and Exa, I believe this > > is just a coincidence; the idea behind the extension to Heta, Oto, and Nea > > is to pretend that the coincidence was intentional. Indeed. The etymology I have for tera is Gk. teras, monster (giga was Gk. gigas, giant of course). The term was coined in 1947 by the UIC. When a prefix for a larger multiplier was required it has been suggested, but not verified, that in 1975 the CIPM created the pretension tera <- tetra, and extended it to peta <- penta and exa <- hexa. If the full OED can't be sure of this derivation, I fear only the people who were in that room in 1975 can. Some have claimed that it was the CGPM who coined the terms, but the CGPM sat in December, and the CIPM sat in March. The reducing multipliers were given their scandinavian numerically-based prefixes, femto- and atto-, much earlier; in 1961 by the IUPAP. (pico- being nearly a hundred years old now, and not number-based) And back to the subject line: <<< In 1993, hacker Morgan Burke proposed, to general approval on Usenet, the following additional prefixes: groucho (10^-30), harpo (10^-27), harpi (10^27), grouchi (10^30). This would leave the prefixes zeppo-, gummo-, and chico- available for future expansion. Sadly, there is little immediate prospect that Mr. Burke's eminently sensible proposal will be ratified. >>> http://burks.brighton.ac.uk/burks/foldoc/44/92.htm And the somewhat joking, but in some ways reasonably soundly justified: <<< xona, weka, vunda, uda, treda, sorta... >>> http://jimvb.home.mindspring.com/unitsystem.htm (Basically, scan back through the alphabet for the first character, and make the rest of the prefix resemble a greco- latin word corresponding to the power of 1000, such as x+nono->xono, w+deka->weka, etc.) It's in the hands of the standards bodies. If gibi and yotta are possible, then pretty much anything is! Phil ===== When inserting a CD, hold down shift to stop the AutoRun feature In the Device Manager, disable the SbcpHid device. http://www.cs.princeton.edu/~jhalderm/cd3/ __________________________________ Do you Yahoo!? Vote for the stars of Yahoo!'s next ad campaign! http://advision.webevents.yahoo.com/yahoo/votelifeengine/

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[math-fun] Humor: Weapons of Math Instruction
by Fred Ballard 21 Jul '04

21 Jul '04

I believe this is just a light-hearted spoof, but my apologies in advance to anyone who might take offense. This was forwarded to me without attribution: At New York's Kennedy Airport today, an individual, later discovered to be a public school teacher, was arrested trying to board a flight while in possession of a ruler, a protractor, a setsquare, a slide rule, and a calculator. At a morning press conference, Attorney general John Ashcroft said he believes the man is a member of the notorious Al-Gebra movement. The FBI is charging him with carrying weapons of math instruction. "Al-Gebra is a fearsome cult," Ashcroft said. "They desire average solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like 'x' and 'y' and refer to themselves as 'unknowns', but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say, 'there are 3 sides to every triangle'." When asked to comment on the arrest, President Bush said, "If God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes."

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[math-fun] Question about differential equations
by Eugene Salamin 21 Jul '04

21 Jul '04

Can someone help me out with a problem involving differential equations? The equation u"(x) = (V(x) - E) u(x), with V(x+P) = V(x) is the Schroedinger wave equation for an electron in a 1-dimensional crystal. V(x) is the periodic potential, and E is the energy of the electron. If u1(x) and u2(x) are linearly independent solutions, so are u1(x+P) and u2(x+P). Since the solution space is 2-dimensional, it follows that [u1(x+P)] = [A B] [u1(x)] [u2(x+P)] [C D] [u2(x)] for some constants A, B, C, D. Call the matrix M. The absence of a u'(x) term implies that the Wronskian is constant, from which it follows that det(M) = 1. A change of basis in the solution space induces a similarity transformation on M. The differential equation is real, so either u1 and u2 are real, or u2 can be chosen to be the complex conjugate of u1. Either way, tr(M) = A+D is real. So the eigenvalues of M are either (exp(+i t), exp(-i t)) or (r, 1/r), with r and t real. In the first case, an electron of energy E can exist in the crystal, while in the second case the energy is forbidden. Thus the band structure is characterized by the function tr(M(E)). The allowed energy bands are given by -2 <= tr(M(E)) <= 2, and energies E for which this inequality is false lie in the gap between bands. Now for my question. Suppose I'm only interested in the energy bands, and I don't care about the actual wave function u(x). Then, given some value of E, is there a way to calculate tr(M(E)) without having to solve the differential equation? Gene __________________________________ Do you Yahoo!? Yahoo! Mail Address AutoComplete - You start. We finish. http://promotions.yahoo.com/new_mail

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[math-fun] biggest flop of all time
by Richard Schroeppel 19 Jul '04

19 Jul '04

passed along from the Mersenne Prime mailing list ... Rich ---------------------- List-Post: <mailto:prime@hogranch.com> Did anyone notice we are coming up on a new prefix? GIMPS has performed over 800 quintillion floating point operations in the seven years of its existence, 800 exaflop. I calculate that in about four months, we will have collectively performed a sextillion floating point operations, allowing me to legitimately use a prefix I have never actually seen in use, even to describe the absurdly large quantities often associated with computing. GIMPS will soon complete the first zettaflop. spike

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[math-fun] Possible virus or Trojan horse
by Gosper 19 Jul '04

19 Jul '04

Indeed, that's a forgery. Beware a possibly even more dangerous one: 41KB "From: rwg(a)tc.spnet.com", and now, sadly, "From: rwg(a)spnet.com". Someone is desperately trying to invade spnet.com, and, as an indirect result, most isps are rejecting my email from there. To top things off, this MIT alternative has been down for a couple of weeks. --rwg

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[math-fun] Niemiec's virus alert
by Richard Schroeppel 19 Jul '04

19 Jul '04

(A note of possible reassurance, at the risk of starting a flood of "my spam experiences" -- remember, if you must speak, to say something new/different.) I've been getting spam & worse with forged "From" lines for months. Occasionally it's purportedly from a friend with a ten-year-old address. The most startling one claimed to be from one of my own less-used email accounts. I suspected serious foul play, but it turned out to be simple luck on the forger's part: none of the rest of the header was consistent with true origination from the other account. I suspect Niemiec's forgery from "RWG" is the same: he hasn't sent from the Macsyma address in years. Rich rcs(a)cs.arizona.edu

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[math-fun] Possible virus or Trojan horse.
by Mark D. Niemiec 19 Jul '04

19 Jul '04

I just received the following message, allegedly from Bill Gosper. I find it highly suspect, since 1) it's in HTML (and I can't imagine Bill sending mail in HTML), 2) the headers are all wrong 3) it contains a one-word explanation (typical of trojan horses, atypical of Bill) 4) contains a .cpl attachment (Windows control panel applet, most of which are trojan horses instead). What is curious, however, is that a fully up-to-date Norton Antivirus does not detect this as a known trojan horse or virus. Nevertheless, I suspect that someone on the list has got a bug, so I suggest that anyone running Windows do a scan. The Received: header seems forged, since DNS resolution is inconsistent: HOBBITON.net -> unresolvable 80.55.203.130 -> e-wroc.net -> 198.65.102.86 -> unresolvable) Whois reports that 80.55.203.130 belongs to RIPE in Amsterdam, while 198.65.102.86 beloings to Verio, in Englewood CO. Here's the message: Return-Path: <rwg(a)NEWTON.macsyma.com> Received: from HOBBITON.net ([80.55.203.130]) by montgomery.mail.atl.earthlink.net (EarthLink SMTP Server) with SMTP id 1bMBkK40p3Nl3qB0 for <mniemiec(a)interserv.com>; Mon, 19 Jul 2004 12:49:41 -0400 (EDT) Date: Mon, 19 Jul 2004 18:57:44 +0100 To: "Mniemiec" <mniemiec(a)interserv.com> From: "Rwg" <rwg(a)NEWTON.macsyma.com> Subject: Re: Message-ID: <khaigvpygaksxspcbjm(a)interserv.com> MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="--------mkdnyyiyrrpbudfadypg" X-ELNK-AV: 0 ----------mkdnyyiyrrpbudfadypg Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: 7bit <html><body> >Animals<br> <br> </body></html> ----------mkdnyyiyrrpbudfadypg Content-Type: application/octet-stream; name="Doll.cpl" Content-Transfer-Encoding: base64 Content-Disposition: attachment; filename="Doll.cpl" <snip> -- Mark D. Niemiec <mniemiec(a)interserv.com>

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[math-fun] Hilbert 3 in 8D
by N. J. A. Sloane 17 Jul '04

17 Jul '04

Hilbert's 3rd problem (solved by Dehn) asks for a proof that a regular tetrahedron cannot be dissected into a finite number of pieces that can be put together to form a cube of the same volume. I believe the problem is easier in even dimensions. I want to dissect a tetrahedron (not necessarily regular, in fact a right simplex or orthoscheme is my favorite) into a brick (not necessarily regular) in 2d dimensions, for d >= 2. Has anyone seen any work on such higher-dimensional dissections? Neil Sloane (njas(a)research.att.com)

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