March 2003 - mathfun@mailman.xmission.com

[math-fun] Cool Lego math sculptures...
by Henry Baker 23 Mar '03

23 Mar '03

at http://www.lipsons.pwp.blueyonder.co.uk/lego.htm >From MAA's "Focus" mag., 23,3 (Mar 2003).

1 0

[math-fun] imagining numbers
by Helger Lipmaa 18 Mar '03

18 Mar '03

Has anybody read Mazur's book "Imagining numbers"? Any comments? Helger

1 0

[math-fun] (No) Math-Fun rule for posting news articles
by Richard Schroeppel 15 Mar '03

15 Mar '03

We rely on the judgment of list members as to what is acceptable. The Math-Fun list has no particular rules about posting news articles. We do have formatting rules (stick to plain text), and the list-server will hold anything over 40KB for approval. If the subset of people who want to argue about copyright will get in touch with me, I'll see if we can set up another list. I'll kick off the discussion with a funny story about my attempts to subscribe to the NYT in rural Utah. Rich rcs(a)cs.arizona.edu

1 0

Re: [math-fun] Extension of a theorem on n-gons
by Richard Schroeppel 15 Mar '03

15 Mar '03

I've edited Steve Gray's message to kill the PDF attachment. You can get the paper directly from him (stevebg(a)adelphia.net) Rich ----- Gene Salamin asked: > Could you provide a statement of the 100 year old result, and of your > extention to it? Steve Gray replied: The 98-year-old result has been called the Douglas-Neumann theorem, but was first found by K. Petr in 1905, so I call it the Petr-Douglas-Neumann theorem, after Jesse Douglas and B.H. Neumann. It concerns layers of isosceles triangles constructed on the sides of an n-gon. After n-2 layers have been constructed, the resulting n-gon is regular, and after one more layer, a single point results. The first layer of triangles have apex angle 360/n, the second twice that, the third 3x360/n, etc. In my generalization the triangles used are not isosceles but are derived from an arbitrary-shaped "model" n-gon which is in general not similar to the one the triangles are placed on. After n-2 layers one gets an n-gon similar to the model, and after one more layer, a single point results. This is the first of several generalizations I have found and proved. I am currently working on another generalization which, if I ever find and prove it, will be the basis of another paper. However I already have enough material for at least one more paper, and probably two or three. One interesting thing about all these theorems is that they are "angle-only" theorems, that is, the constructions and results can be stated in natural fashion without reference to distances or lengths, which appear only secondarily. They are among the few theorems involving n-gons, where n is arbitrary. In some of my variations n must be prime, and in some others, certain of the construction parameters must have (simple) number-theoretic relationships. I'm enclosing a PDF of the final paper. If you have any more specific questions, please write back, although I do want to keep certain details of the other theorems private until I send in the next paper. Coincidentally, the same issue of the Monthly contains a geometry problem I submitted in 9/2001 that I had forgotten about. Steve Gray

1 0

[math-fun] small prime gaps
by Richard Schroeppel 14 Mar '03

14 Mar '03

Date: Thu, 13 Mar 2003 23:11:22 -0500 Reply-To: Hugh Montgomery <hlm(a)UMICH.EDU> Sender: Number Theory List <NMBRTHRY(a)LISTSERV.NODAK.EDU> From: Hugh Montgomery <hlm(a)UMICH.EDU> Subject: small gaps between primes To: NMBRTHRY(a)LISTSERV.NODAK.EDU Oberwolfach 13 March, 2003 This week at Oberwolfach, a meeting on Elementary and Analytic Number Theory is taking place, organized by Bruedern, Montgomery, and Vaughan. At this meeting, today, Dan Goldston spoke about his new joint work with Yildirim concerning primes in short intervals. They seem to have proved that liminf (p_{n+1} - p_n)/log p_n = 0. Indeed, it looks like they will be able to show that p_{n+1} - p_n is infinitely often as small as (log p_n)^{4/5}. It may be that the method can be applied also to long gaps between primes, with the prospect of possibly breaking the Erdos--Rankin order of magnitude. I would say that this is the biggest excietement that prime number theory has seen since the Bombieri--Vinogradov theorem was proved in 1966. Hugh Montgomery

1 0

Re: [math-fun] Ulam(1,2)
by Dan Hoey 14 Mar '03

14 Mar '03

"David Wilson" <davidwwilson(a)attbi.com> wrote: > I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and > including every subsequent number which is a unique sum of distinct > earlier terms. This is Sloane's A002858. > ... Can anyone confirm my data? Did you intend to use zero-based indexing? If u(0)=1, u(1)=2, I concur that u(1000)=12336, u(1999)=25511. Otherwise one of us has a bug. I also get u(10000)=132790, u(50000)=676043, u(74084)=1000002. But there's a pretty definite bug in the histogram. My histogram of f(n)=(u(n) mod 21.6)/21.6 agrees with yours in its general trend, but differs in particulars. For f(1000...1999) in 20 buckets I get 113 101 119 98 87 70 58 40 31 8 5 0 0 2 4 14 22 64 66 98 rather than your 110 101 120 97 90 67 58 40 31 9 4 0 0 2 7 11 22 64 66 101. As a test, I find that the five numbers in the [.5,.55) bucket arise from u(1091)=13533 (f=0.527771) u(1210)=14894 (f=0.53704834) u(1730)=21870 (f=0.5) u(1748)=22151 (f=0.50927734) u(1947)=24851 (f=0.50927734). What four numbers do you have? I agree exactly with Klaus Brockhaus's data for f(1000...1259), so I think I'm slightly likelier to be correct, but only slightly, and I'm pretty suspicious about the zero-based indexing. My data suggests that 21.6 should be about 21.60157 . Dan

9 11

Re: [math-fun] universe as doughnut
by Henry Baker 13 Mar '03

13 Mar '03

I thought that the article was interesting enough to post in its entirety, but in ASCII, not HTML. http://www.nytimes.com/2003/03/11/science/space/11COSM.html?pagewanted=prin… -------------------------------------------------------------------------------- March 11, 2003 Universe as Doughnut: New Data, New Debate By DENNIS OVERBYE Long ago in the dawn of the computer age, college students often whiled away the nights playing a computer game called Spacewar. It consisted of two rocket ships attempting to blast each other out of the sky with torpedoes while trying to avoid falling into a star at the center of the screen. Although cartoonish in appearance, the game was amazingly faithful to the laws of physics, complete with a gravitational field that affected both the torpedoes and the rockets. Only one feature seemed outlandish: a ship that drifted off the edge of the screen would reappear on the opposite side. Real space couldn't work that way. Or could it? Imagine that the Spacewar screen is wrapped around to form a cylinder or a section of a doughnut so that the two edges meet. That is the picture of space, some cosmologists say, that has been suggested by a new detailed map of the early universe. Their analysis of this map has now provided a series of hints though only hints that the universe may have a more complicated shape than astronomers presumed. Rather than being infinite in all directions, as the most fashionable theory suggests, the universe could be radically smaller in one direction than the others. As a result it may be even be shaped like a doughnut. "There's a hint in the data that if you traveled far and fast in the direction of the constellation Virgo, you'd return to Earth from the opposite direction," said Dr. Max Tegmark, a cosmologist at the University of Pennsylvania. The new data have generated both buzz and skepticism among cosmologists in recent weeks. Dr. Tegmark and other astronomers agree that the measurements are far from conclusive, or even persuasive about the shape of the universe. But if true, the doughnut universe would force cosmologists to reconsider their theories about what happened in the earliest moments after the universe was born in the Big Bang; those theories predict an infinite cosmos. The new findings have brought to center stage the hope that astronomers may be able to test speculations about the shape, or topology, of the universe that until recently have been relegated to the abstract mathematical margins of cosmology. The results are part of the bounty of data produced by a NASA satellite known as the Wilkinson Microwave Anisotropy Probe, built and operated by an international collaboration led by Dr. Charles L. Bennett of the Goddard Space Flight Center in Greenbelt, Md. The satellite recorded the pattern of heat, in the form of faint microwave radiation, that fills the sky. This radiation is believed to be the afterglow of the Big Bang itself, and thus constitutes a portrait of the universe when it was only 380,000 years old. As the COBE satellite first confirmed in 1992, the microwave cloud is laced with ripples and splotches lumps in the cosmic gravy from which galaxies and other cosmic structures would ultimately form. According to theory, these lumps are born as microscopic fluctuations during the first instant of time and then amplified into sound waves as the universe expands and matter and energy slosh around. Now the new satellite has illuminated the findings of COBE (pronounced KOE-bee, for Cosmic Background Explorer) in exquisite detail. By analyzing these waves cosmologists can determine many of the characteristics of the universe, which scientists have long debated, like its age and density. To their delight, the first results from the Wilkinson satellite, released last month, confirmed many of the strange ideas that cosmologists entertained in the last decade, including the notion that most of the universe consists of something called dark energy, which is pushing space apart at an accelerating rate. "Cosmologists have built a house of cards and it stands," said Dr. James Peebles, a cosmologist at Princeton. But to their even greater delight, perhaps, as they dig into the trove released last month, cosmologists are finding hints of even more strangeness. In principle, in an infinite universe, the waves in the cosmic fireball should appear randomly around the sky at all sizes. But, according to the new map, there seems to be a limit to the size of the waves, with none extending more than 60 degrees across the sky. The effect was first noted as a puzzle in the COBE data, according to Dr. Gary Hinshaw, an astronomer at the Goddard Space Flight Center and a member of the Wilkinson probe team, and now seems confirmed. If the universe were a guitar string, it would be missing its deepest notes, the ones with the longest wavelengths, perhaps because it is not big enough to sustain them. "The fact that there appears to be an angular cutoff hints at a special distance scale in the universe," Dr. Hinshaw said. Another analysis of the new map suggests that there is a special direction, as well as a special scale in the universe. While reanalyzing the Wilkinson data to eliminate radio noise from stars and our own galaxy, Dr. Tegmark, Dr. Angélica de Oliveira-Costa, also at Pennsylvania and married to Dr. Tegmark, and Dr. Andrew J. S. Hamilton of the University of Colorado have discovered that the universe appears lumpier in one direction through space than it does in another. When they combed finer variations out of the map, the remaining large-scale variations formed a line across the sky. It could be a chance alignment, a statistical fluke, Dr. Tegmark said, or contamination from radio noise from the galaxy. But in a paper posted on the physics Web site (at arXiv.org/pdf /astro-ph/0302496) late last month, the three cosmologists wrote that it was "difficult not to be intrigued" that their results bore all the earmarks of what are variously called small, compact, finite or periodic universes. If the universe is finite in one dimension, like a cylinder or a doughnut, Dr. Tegmark said in an interview, there is a limit to the size of clumps that can fit in that direction. They couldn't be bigger than the universe in that direction, just as a guitar string can only play a note so low, depending on its length. So the biggest blobs would have to squish out in a plane in other directions. The way home around the doughnut would be perpendicular to that plane. Nobody is yet claiming that this is a revolution. The notion of a special direction is on less firm ground than the discovery of a cutoff of large structures. "More detailed work in needed to clarify what's going on," Dr. Tegmark said. Dr. Martin Rees, a cosmologist at Cambridge University," said he didn't think there was evidence for "anything crazy" in the data. Even aficionados of finite universes are guarded. Dr. David Spergel, a Princeton cosmologist and Wilkinson satellite team member, called the results "intriguing," but cautioned that they could also be due to chance. Dr. Hinshaw called the findings of Dr. Tegmark's team "surprisingly robust," but added, "I'm not sure it says something profound about the universe." Dr. Alexei Starobinski, a theorist at the Landau Institute in Moscow, proposed in 1984 with his mentor, Dr. Yakov B. Zeldovich, that the universe could have been born as a doughnut. Dr. Starobinski emphasized that an infinite universe with ordinary Euclidean geometry was the most natural universe and still favored by theory. "However, theory is theory, but observations might tell us something different," he said in an e-mail message. The Science of Shapes A Compact Universe Like Mirrored Halls The new work involves topology, the branch of mathematics that deals with shapes. Topologists are often accused of not knowing the difference between a coffee mug and a doughnut; because each object has one hole, the two can be deformed into each other and are thus topologically equivalent. In a similar vein, a figure 8 and a pair of eyeglass frames are also the same to a topologist. The more holes, the more complicated the topology. The simplest topology is just the infinite space of the Euclidean geometry taught in high school. But some cosmologists have a hard time calculating how an infinite universe could have appeared in that kind of space. Nature, they contend, might have had an easier time making a small "compact" universe than an infinite one, and they assume Nature would take the easy way out. "The basic idea is that God's on a budget," said Dr. George Smoot, a physicist at the University of California's Lawrence Berkeley Laboratory and a leader on the COBE team. The simplest of these compact universes is something called a 3-torus, a doughnut wrapped in three different dimensions. This object is essentially impossible to visualize: it is the equivalent, in a way, of a cube whose opposite sides are somehow glued together. In two dimensions it works just like the Spacewar screen. Living in such a universe would be like being inside a hall of mirrors, Dr. Tegmark said. Instead of seeing new stars deeper and deeper in space, you see the same things over and over again as light travels out one side of your cube and back in the other. This mirror game is not limited to cubes and doughnuts. Over the years mathematicians, particularly Dr. William Paul Thurston, now at the University of California at Davis, and Dr. Jeffrey Weeks, an independent mathematician, have speculated about universes composed of various polyhedrons glued together in various ways. In 1996 the French astronomer Dr. Jean-Pierre Luminet of the Paris Observatory and his colleagues Dr. Roland Lehoucq and Dr. Marc Lachieze-Rey, both of the Center for Astrophysical Studies in Saclay, France, developed a method called "cosmic crystallography," using galaxy statistics to detect and diagnose the repeating periodic patterns that would be created in the sky by light going around and around in differently shaped universe. Finite or Infinite? Problems Are Posed For Favored Theory Why would the universe want to do this to us? Partly to avoid the difficulties of the infinite, said Dr. Glenn Starkman, an astronomer at the Case Western Reserve University in Cleveland. Besides being difficult to create, an infinite universe is philosophically unattractive. In an infinite volume, he pointed out, anything that can happen will happen. "Somewhere there are two guys having this same conversation," Dr. Starkman said in a telephone interview, "except that one of them has a purple phone." Moreover, the idea that dimensions could be curled in loops occurs naturally in theories that try to unite gravity and particle physics, several physicists pointed out. For example, according to string theory, the leading candidate for a theory of everything, the universe actually has 10 dimensions 9 of space and 1 of time rather than the 4 we are familiar with. The extra dimensions are curled up into submicroscopic loops, like the threads in an uncut carpet pile, so that we don't notice them in ordinary life. "This is the same idea on a very large scale," Dr. Smoot said. Knowing that all nine of the spatial dimensions predicted by string theory are finite and thus on the same footing could help string theorists decide among the nearly endless possibilities allowed by the theory, scientists say. But a finite universe would create big problems for the reigning theory of the Big Bang, inflation theory. It posits that the universe underwent a burst of hyperexpansion in its earliest moments. Among other things, it implies that the observable universe today, a bubble 28 billion light-years in diameter, is only a speck on the surface of a vastly greater realm trillions upon trillions of light-years across. "There's no natural way yet proposed to get the inflation to stop and give a space that's big enough to house all the galaxies but small enough to see within the observable horizon," said Dr. Janna Levin, a Cambridge University cosmologist who wrote about finite universes in her 1992 book, "How the Universe Got Its Spots, Diary of a Finite Time in a Finite Space." Dr. Spergel added, "If the universe were finite, then this would rule out inflation and require something new." The Search for Patterns One Convincing Sign Of the Doughnut So far, sporadic searches for repeating patterns of quasars or distant galaxy clusters that would occur in a hall of mirrors universe have been unsuccessful. For finite universe aficionados, the first encouragement of note was COBE's discovery that the universe appeared to be deficient in large-scale fluctuations. There were no structures extending more than about 60 degrees across the sky. But the finding was subject to large statistical uncertainties, astronomers said. There are other possible explanations for the cutoff in fluctuation size, Dr. Starkman explained. According to inflation the biggest longest waves are created first, and thus the missing notes are the earliest ones that would have been strummed by inflation's guitar. Perhaps, he said, this is telling us something about the beginning of inflation. Dr. George Efstathiou of Cambridge University has pointed out in a recent paper submitted to the Monthly Notices of the Royal Astronomical Society that the Wilkinson satellite data are also marginally consistent with yet another finite shape, namely a sphere. In that case, fluctuations larger than the radius of the sphere might be dampened, he said, producing the observed cutoff. The most convincing sign of a doughnut universe, if it exists, astronomers say, could come from a search of the satellite data now being performed by Dr. Spergel, Dr. Starkman and Dr. Neil J. Cornish of Montana State University. "We're looking for circles in the sky," Dr. Starkman said. In a 1998 paper they point out that if the universe is small enough, part of the cosmic background radiation, which essentially fills the sky surrounding us, will hit the sides of the "box" or the space war screen we are in and appear on the other side. The result, in the simplest case, would be identical circles on opposite sides of the sky with the same patterns of hot and cold running around them. In the simplest case, the size of the circles would depend on the distance between the "walls" of the universe: the smaller the universe, the bigger the circles. Success or even a definitive failure is not guaranteed. "It would be fantastic if something like that was found," Dr. Hinshaw said of the circles. But success or even a definitive failure is not guaranteed. If the universe is finite but still much larger than today's observable universe 28 billion light-years in diameter the circles will not show. "Usually in science when we see an intriguing pattern that appears to contradict existing theory we do a better experiment," Dr. Spergel wrote in an e-mail message, but in this case, "Ultimately we will be limited by the fact that we can only observe the `visible' universe." Dr. Levin was doubtful, "I suspect every last one of us would be flabbergasted if the universe was so small," she said in an e-mail message. When she first heard about the new satellite data, she reported, "I tried on the idea that we were really and truly seeing the finite extent of space and I was filled with dread. "But I'm enjoying it too." Copyright 2003 The New York Times Company | Privacy Policy

3 2

Re: [math-fun] To Rich as math-fun moderator
by Henry Baker 13 Mar '03

13 Mar '03

Bernie, et al: For the record, I've been a subscriber to the NYTimes for many years, now. I assume that some fraction of this list are subscribers as well. I could have scanned the article and posted the image, but that would have been significantly larger and less convenient. Henry ----- At 07:22 AM 3/13/03 -0500, Bernie Cosell wrote: >On 12 Mar 2003 at 20:12, asimovd(a)aol.com wrote: > >> As for the article from the NYTimes I just sent to math-fun that's "being held >> for moderator approval" ... > >Does math-fun have a policy on sending out copyrighted materials (I assume >without permission)? > > /Bernie\ > >-- >Bernie Cosell Fantasy Farm Fibers >mailto:bernie@fantasyfarm.com Pearisburg, VA > --> Too many people, too few sheep <--

4 3

[math-fun] To Rich as math-fun moderator
by asimovd＠aol.com 13 Mar '03

13 Mar '03

Rich, As for the article from the NYTimes I just sent to math-fun that's "being held for moderator approval" (it"s 44kb, about 10% over the limit), I misspelled the word address in the first sentence, but worse I *forgot* to de-MIME it. Apologies -- there will be times when I forget. If you send it out could you please de-MIME it for me, or else let me know and I'll send it out that way, or else I could both de-MIME it and send it in two halves. --Dan

3 2

[math-fun] Swiss 10 Fr note
by N. J. A. Sloane 12 Mar '03

12 Mar '03

A correspondent just sent the following - any comments anyone? NJAS > The Swiss 10 Fr note has some graphs on it that are accompanied by what > appears to be a partial list of the zeroes: > > 43, 53, 70, 86, 113, 140, 183, 226, 296. > > There is reason to think from the graphs that these values are actually > integer parts of a sequence that goes to 0 in one direction and infinity > in the other. At any rate, your numerical sequences web site fanned on > this sequence and I was curious if you know anything about it. One point: > the sequence seems to be spliced together from two sequences (since the > odd numbered terms are in one color and the even numbered ones in another) > and in each of the sequences, the ratio of successive terms is > suspiciously close to the golden ratio. Actually, I just noticed that > each odd numbered term is the sum of the two preceding odd numbered terms, > but this is not quite true of the even numbered ones. > > Michael Barr >

1 0