November 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Marden's/Siebeck's Theorem
by Dan Asimov 15 Nov '10

15 Nov '10

Another way to see the non-uniqueness of the triangle is to take any triangle containing the ellipse, and move its sides towards the ellipse until each one is first tangent to it. --Dan Rich wrote: << answering one question ... > Given an ellipse, is the triangle that the ellipse is inscribed > within unique? No. Start with an equilateral triangle, and the inscribed circle. Choose any direction, and stretch the diagram in that direction, while leaving perpendicular distances alone. This operation preserves some geometric properties, and mangles others. Tangency and midpoints are preserved. The inscribed circle becomes an inscribed ellipse. Foci are not preserved. We can get an ellipse of prescribed shape, in any direction. Working backwards, knowing an ellipse alone doesn't determine the orientation of the circumscribed "midpoint triangle". >> Those who sleep faster get more rest.

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Re: [math-fun] Perfect Squared Squares to order 28 are complete
by N. J. A. Sloane 15 Nov '10

15 Nov '10

Stuart, Ed: Nice work! I hope you will update the entries in the OEIS and create any new ones as appropriate. There's a new version of the OEIS as of two days ago. I'll post an official announcement here soon. The new URL for the OEIS is http://oeis.org [To contribute, you first need to register with the wiki part of the oeis: go to oeis.org/wiki, register (enter your full name in the top two windows, etc., then you get a password). Then go to the main database, oeis.org, and login with same name and password.] Best regards Neil

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[math-fun] Perfect Squared Squares to order 28 are complete
by Stuart Anderson 14 Nov '10

14 Nov '10

Hi All, Ed Pegg and I have finished processing all the 3-connected planar graphs (for simples) and 2-connected planar graphs (min degree 3) up to and including 29 edges for Mrs Perkin's Quilts and Perfect Squares. Many new Mrs Perkin's Quilts have been discovered, and there is much data yet to be worked through in Quilts. With the squared square results, there is less data and we have complete counts. As a result, we have obtained all simple squared squares in orders 21,22,23,24,25,26,27,28 and compound squares in orders 24,25,26,27,28. We can confirm the existing counts (orders 21-27) in simples squared squares are correct and complete. This confirms the work of Duijvestijn (spss orders 21,22,23,24,25,26) and Skinner (spss order 27) We prove the number of simple squared squares in order 28 is 3001 (and find 29 new simples). We can confirm Willcocks order 24 square is the lowest order compound and the only one of this order, We prove the 2 known order 25 compounds complete order 25 compounds We prove the number of compound squares in order 26 is 16, and find 1 new order 26 compound We prove the number of compound squares in order 27 is 46, and find 7 new order 27 compounds We prove the number of compound squares in order 28 is 142, and find 41 new order 28 compounds http://www.squaring.net/sq/ss/spss/spss.html http://www.squaring.net/sq/ss/cpss/cpss.html http://www.squaring.net/downloads/downloads.html I now intend to write a paper with Ed Pegg, substantiating the above claims, I intend to publish on the Archivx, not having published anything I assume that's all that's needed for publication these days I have already given some data to Neil Sloanes OEIS, but need to update the order 28 term in simples and compounds. We can also create the long overdue number sequence for Perfect Squares; 1,8,12,27,162,457,1198,3143 Couldnt be done before now, because we didnt have the compound sequence, we didnt have that because we couldnt confirm the 4 terms required to initiate the sequence. McKay/Brinkmann's plantri performed perfectly as a generator of the required graph classes. The efficiency of plantri in fast run time and its production of complete graph classes of non-isomorphic graphs allowed the required computations for exhaustive enumeration to be performed in a reasonable time frame. cheers, Stuart ps I may do order 29 as well, but that will depend very much on how many people and computers I can gain the assistance of. After this effort, I need a break and have no intention of starting a big run any time in the near future Some of the identifier labels for the squares may need changing, and no doubt other issues, I say there are only 44 isomers of 1015B , if you insist on canonical bouwkampcode, 4 have to be flipped about the diagonal, creating 4 duplicate bouwkampcodes, reducing the total from 48 to 44

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[math-fun] John Selfridge
by rcs＠xmission.com 14 Nov '10

14 Nov '10

Bob Baillie reports the death of number theorist John Selfridge. --Rich ---- http://www.ams.org/news/in-memory/in-memory John Selfridge (1927-2010) Friday November 12th 2010 Selfridge, who was known for his work in number theory, died October 31 at the age of 83. He earned his PhD from UCLA in 1958 and taught at various universities before joining the faculty at Northern Illinois University in 1971. Selfridge helped expand the department while serving as chair from 1972-75 and from 1986-90. For eight of the years in the interim, 1978-86, he was executive editor of Mathematical Reviews. Selfridge retired from Northern Illinois in 1991. He was a founder of the Number Theory Foundation, and co-authored 14 papers with Paul Erdos. Selfridge was an AMS member since 1953. Read more about his life in an obituary in the Daily Chronicle (DeKalb,IL). http://www.legacy.com/obituaries/daily-chronicle/ obituary.aspx?n=john-selfridge&pid=146547346&fhid=6514

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[math-fun] Mac "Grapher" utility
by Dan Asimov 14 Nov '10

14 Nov '10

Does any other math-funnik use the Mac Grapher utility? It can do so many things to create a 2D or 3D image . . . BUT it is buggy as hell. At least the version that came with my Mac -- among the very last with a Motorola chip -- plus all Apple updates to my software. It was, I think, created in a story that is now legendary among programmers: < http://www.pacifict.com/Story/ >. But I sure wish they would debug the *&^%$@ thing. Case in point: Just now I wanted to draw a family of plane curves with a simple parametric equation depending on one additional parameter . . . for six values of that parameter. Grapher supposedly has a syntax for doing this: just stick the additional parameter in curly brackets, and either explicitly write all parameter values, or the first two and the last one. In my case, according to the examples and the (very sparse) Help file, that should appear as either {0.5, 0.6, 0.7, 0.8, 0.9, 1.0} or {0.5, 0.6...1.0}, respectively. But neither one works. Boiled down: Using either {1.0} or {0.5} my tiny Grapher program draws the correct curve. But using {0.5, 1.0} instead, it draws two curves that are both wrong! Has anyone figured how to get around stupid bugs like this? --Dan Those who sleep faster get more rest.

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[math-fun] new web site : http://www.plouffe.fr/simon/
by Simon Plouffe 14 Nov '10

14 Nov '10

Hello everybody, I just opened my new website at http://www.plouffe.fr/simon/ also redirected the old one at that same address. I intend to re-open the inverter with the new format at 64 digits, it will be called either plouffe inverter or inverse symbolic calculator. In the meantime, all the data of the old inverter is available at http://www.plouffe.fr/simon/plouffe_inverter_tables/ The new tables are all here ; http://www.plouffe.fr/simon/ip/ About sequences, all of the conjectures are here (most of them are the real ones), http://www.ploufffe.fr/simon/OEIS/conjectures/OEIS_conjectured_formulas.txt from which I computed about 148 billion digits of sequences, which are all here also : http://www.plouffe.fr/simon/seqc/ each file is formated this way, for example the file c000045.txt located at http://www.plouffe.fr/simon/seqc/000/c000045.txt looks like this sequence index value A000045 0 0 A000045 1 1 A000045 2 1 A000045 3 2 A000045 4 3 A000045 5 5 A000045 6 8 A000045 7 13 A000045 8 21 A000045 9 34 A000045 10 55 A000045 11 89 A000045 12 144 A000045 13 233 A000045 14 377 A000045 15 610 A000045 16 987 A000045 17 1597 I will try now to fix all <404> errors. have a nice day, simon plouffe

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[math-fun] A peculiar numerical coincidence
by Victor Miller 12 Nov '10

12 Nov '10

I thought that this was rather neat: http://sbseminar.wordpress.com/2010/10/06/a-peculiar-numerical-coincidence/…

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[math-fun] DNA-tape Turing Machines ?
by Henry Baker 10 Nov '10

10 Nov '10

FYI -- One step closer to DNA-tape Turing Machines with a 1nm tape-reading head. This is really cool stuff! I wonder if there is a mechanism to _write_ a DNA tape quickly? http://www.gizmag.com/graphene-rapid-dna-sequencing/16360/ RESEARCH WATCH Graphene could speed up DNA sequencing By Ben Coxworth 21:00 September 13, 2010 Graphene is pretty amazing stuff. Just a couple of months ago, we heard about how the one-atom thick sheets of bonded carbon atoms had been used to create the strongest pseudo-electric magnetic fields ever sustained in a lab Â and that was just the latest use that had been discovered for it. Now, word comes from Harvard University and MIT that graphene could be used to rapidly sequence DNA. Researchers at Harvard drilled a few-nanometers wide hole, known as a nanopore, in a sheet of graphene. That sheet was stretched over a silicon-based frame, and inserted between two liquid reservoirs. When those reservoirs were filled with water, the graphene sheet was the only barrier between the two bodies of water. At less than one nanometer in width, graphene is the thinnest material known to be able to separate two liquid compartments from one another. ThatÂs actually not too surprising, as itÂs also the strongest known material. Graphene is also electrically-conducive, so when a current was applied to the sheet, it attracted ions in the water toward it. As the ions passed through the nanopore, they registered as a continuous electrical signal. When long DNA chains that had been added to the water went through the opening one-by-one, they blocked the flow of ions, resulting in a change in the electrical signal. The exact nature of those changes allowed the researchers to determine the size of each DNA molecule, and to identify their individual nucleobases, which are the letters of their genetic code. There are still some hurdles to overcome, such as controlling the speed at which the DNA strands pass through the nanopore, but the scientists believe it could lead to rapid, low-cost DNA sequencing. ÂWe were the first to demonstrate DNA translocation through a truly atomically thin membrane. The unique thickness of the graphene might bring the dream of truly inexpensive sequencing closer to reality,Â said HarvardÂs Prof. Daniel Branton. ÂThe research to come will be very exciting.Â The Harvard/MIT findings were recently published in Nature.

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[math-fun] Large cardinals: maths shaken by the 'unprovable'
by Henry Baker 09 Nov '10

09 Nov '10

FYI -- Does anyone here know what this is all about? http://www.telegraph.co.uk/science/8118823/Large-cardinals-maths-shaken-by-… Large cardinals: maths shaken by the 'unprovable' A shocking discovery has unsettled the world of numbers, says Richard Elwes. By Richard Elwes Published: 7:50AM GMT 09 Nov 2010 55 Comments In the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities. To understand them, we must understand what makes mathematics different from other sciences. The difference is proof. Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction. But proofs have to start somewhere. For all its sophistication, mathematics is not alchemy: we cannot conjure facts from thin air. Every proof must be based on some underlying assumptions, or axioms. And there we reach a thorny question. Even today, we do not fully understand the ordinary whole numbers 1,2,3,4,5Â or the age-old ways to combine them: addition and multiplication. Over the centuries, mathematicians have arrived at basic axioms which numbers must obey. Mostly these are simple, such as "a+b=b+a for any two numbers a and b". But when the Austrian logician Kurt GÃ¶del turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His "incompleteness theorems" showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms. GÃ¶del's theorems showed that maths meant that mathematicians could not hope to prove every true statement: there would always be "unprovable theorems", which cannot be deduced from the usual axioms. Most known examples, it's true, will not change how you add up your shopping bill. For practical purposes, the laws of arithmetic seemed good enough. However, as revealed in his forthcoming book, Boolean Relation Theory and Concrete Incompleteness, Harvey Friedman has discovered facts about numbers which are far more unsettling. Like GÃ¶del's unprovable statements, they fall through the gaps between axioms. The difference is that these are no longer artificial curiosities. Friedman's theorems are "concrete", meaning they contain genuinely interesting information concerning patterns among the numbers, which must always appear once certain conditions are met. Yet, Friedman has shown, the fact that such patterns always appear does not follow from the usual laws of arithmetic. These patterns are not yet affecting physicists or engineers, but mathematicians are having to take unprovability seriously. In the past, they just had to show whether an idea is true or false. Now, results such as Friedman's raise the awkward possibility that the standard laws of mathematics may not provide an answer. There is one easy way to make an unprovable theorem provable: adding more axioms. But which axioms do we need? The new axioms require a hard look at one of the most contentious issues in mathematics: infinity. For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3Â On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever. This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not. At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants. Dr Richard Elwes is the author of 'Maths 1001: Absolutely Everything That Matters in Mathematics' (Quercus Publishing)

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Re: [math-fun] pi cf record
by Bill Gosper 05 Nov '10

05 Nov '10

Bill Gosper: > Has anyone listed the largest terms in Hans's data? Hans Havermann: >I have. :) >http://chesswanks.com/seq/cfpi/ For his 13th birthday, funster Neil Bickford has just computed and verified 485M partial quotients (http://www.neilbickford.com/picf.htm) The longest previous CF we can find remains Hans Havermann's 180M, wherein Neil found two glitches--a "q" instead of a "1" at position 4,311,037 (a picked 2^6 bit) and a semicolon following term 100,000,000. The probably biggest news is the survival of 878783625 as the largest known partial quotient, seeming to signal the end of a pattern of "premature" high water marks, assuming the lg 1+t distribution for the tails of almost all reals. If pi is to maintain its reputation, the next high should be a doozy. (I dimly recall deriving that lg 1+t implies that, in an n term burst, the largest should be about e*n. Can anyone help here?) Neil also reports "Taking 9,701 seconds (161 minutes), the longest string of second difference zeroes is of length 18, starting at 173,533,907 and ending at 173,533,924." I.e., he has significantly increased the numerators and denominators necessary for any rational linear relationship of pi to e. Expect details shortly at http://nbickford.wordpress.com/ . Is there reason to expect the longest burst of vanishing second differences should have a difference interval of 1? Or did Neil neglect to mention that it was >1? --rwg

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