July 2011 - mathfun@mailman.xmission.com

[math-fun] David's "27" packing
by Bill Gosper 28 Jul '11

28 Jul '11

My conjecture that 31 might be the first rotationally symmetric set of radii inexpressible in radicals is on the ropes. Corey and I worked out the equations for David's 27 (http://www2.stetson.edu/~efriedma/cirRcir/) but Reduce can't solve them algebraically anytime soon. Good old FindIntegerNullVector and FindRoot (999 digits) quickly give degree 20 (again?!) polynomials for the radii of all but the light orange disks, where PSLQ complains: FindIntegerNullVector::lowpr: FindIntegerNullVector cannot find an integer null vector because of insufficient precision of the input. >> Even with 9999 digits. LatticeReduce comes to the rescue, and says it's just a 10th degree over sqrt3, like the other 5. Here is the mysterious radius, in case you want to experiment: Root[4782969 - 1111774572 #1 + 44221126878 #1^2 - 307992419388 #1^3 - 8070585578919 #1^4 + 90167866460064 #1^5 + 150017012821404 #1^6 - 1381565181544056 #1^7 - 2541769632528930 #1^8 - 12793102409225160 #1^9 - 16223229604493784 #1^10 + 99440200315651104 #1^11 + 127666170124763526 #1^12 + 245161940234004576 #1^13 + 448033243426240932 #1^14 + 23246278469793432 #1^15 - 14112704124290727 #1^16 + 448087696965684 #1^17 + 19969942229946 #1^18 + 37004992764 #1^19 + 11771761 #1^20 &, 13] ~0.1760155767437583601645544130545779438560210712086956361545336164666108949916301221 Note that this is the only size which is surrounded by six different sizes. Finally, for David's Sum(radii) metric, Out[23]= s == Root[772690429687389147028310024964029054460615661037074186965170657489148640157424 - 1252150034158679892073592213738221544763982273185101716937850019400615469267216 #1 + 894101646306967418990714166469728854086811534040961919825626454389349568482607 #1^2 - 370535002186753148816065781791793841373361064007009093917305796349950654837894 #1^3 + 98585499109277440100892024783571503674461841813340963516397348934746393551831 #1^4 - 17537282099175471585324919382047103164067402676914160496887027103365237985296 #1^5 + 2095329496883146641591880827676789201936700111609182408312499347314971125404 #1^6 - 162581406984533745130162686637360629662639564698220123332118975994424295496 #1^7 + 7315174101074478959948591907336518346543812578462103252159725113209934156 #1^8 - 110179967256584058265325808692517286196365022221976714892569942238719952 #1^9 - 5210786844166514462554894258016492610911648226017109091287319543880958 #1^10 + 235504976723519352232932797704723384517491437953853656479644035104764 #1^11 - 822038136083999158156413079062172274783684008146700851353930742942 #1^12 - 78272319479518900556761239891321970600088764432072085939114898288 #1^13 + 560802701967280190031324433718254607649684741002636396646811692 #1^14 + 8012926781889727306623190565874256627453257915557081010606840 #1^15 - 22160201351067819572752983769650479439077975537076544188276 #1^16 - 77873705468844592325047782942025877646289806118814532768 #1^17 + 173017798526655483804206798601616162580746057680701703 #1^18 \ + 10267478177298250805153325223822351093391050035930 #1^19 + 118755914270972677050504275876817937004163071 #1^20 &, 8] In[24]:= N[%[[2]], 22] Out[24]= 4.762847452755912443159 This factors over sqrt3, but Root won't accept noninteger polynomials. rwg>May[be] he was seeing double after attempting a cholecystectomy? According to www.rhinoresourcecenter.com/ref_files/1178936542.pdf (p5?): black rhinos don't have gallbladders. (I wonder if the prosectorium was secure against Chinese herbalists.) The rhinos probably turn black when all that leaking bile collects in their skin. Somebody should cross a monotreme with a marsupial and name it Monomial monomial. --rwg

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Re: [math-fun] Mandelbrot's Conjecture (as mentioned in this video interview)
by Dan Asimov 28 Jul '11

28 Jul '11

I certainly give Mandelbrot credit for inventing the word "fractal" and popularizing the notion with at least 1 1/2 highly engaging books, to the point that it is now a household word. But he made no effort to dispel the widespread erroneous notion that he invented the concept of fractional-dimensional sets. That idea goes back at least to Hausdorff, who invented -- surprise! -- Hausdorff measure. (Does it go back further?) --Dan Fred wrote: << The one time I attended a talk by Mandelbrot, I came away with a very low opinion of his verbal skills: whatever might have been going on in his mind, when he attempted to express it he seemed incapable of anything more than high-flown waffling. The interview you quoted does little to dispel this impression. . . . >> Sometimes the brain has a mind of its own.

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[math-fun] Linnaean "monomials"
by Bill Gosper 26 Jul '11

26 Jul '11

For reasons to emerge, Fred's rhinoceromania led me to ask Corey for examples of Linnaean binomials(!) where genus = species. He promptly found 24 animals at http://www.sporcle.com/games/KyleBarron/same_genus_species , causing me to kick myself for not being able to remember so many of them, including Genetta genetta, mentioned to me by Whit Diffie, who owned a Genetta tigrina. This list must be very incomplete, because I found Gulo gulo before seeing it. But some namesmiths seem to shun these "monomials" for sounding childish or something. E.g., Hydrochoerus hydrochaeris. And whoever named the black rhinoceros was just having fun: diceros bicornis! --rwg May he was seeing double after attempting a cholecystectomy?

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[math-fun] Lots of piles of bills
by Eric Angelini 25 Jul '11

25 Jul '11

For Simon Plouffe: http://www.wtfnoway.com/Best, Best, É.

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[math-fun] 1x pad: Codebook Shows an Encryption Form Dates Back to Telegraphs
by Henry Baker 25 Jul '11

25 Jul '11

FYI -- I'd be willing to bet that one-time pads were used in ancient times -- e.g., Greeks, Romans, Persians or Chinese. There was a relatively large amount of message traffic from spies, both personal & state-sponsored, that needed to be securely encrypted. I would also imagine that steganography was also pretty common in ancient times. Of course, it may be a little difficult to prove this w/o having the one-time pad! --- http://www.nytimes.com/2011/07/26/science/26code.html July 25, 2011 Codebook Shows an Encryption Form Dates Back to Telegraphs By JOHN MARKOFF If not for a computer scientistÂs hobby of collecting old telegraph codebooks, a crucial chapter in modern cryptography might have been lost to history. The collector is Steven M. Bellovin, a professor of computer science at the Columbia University School of Engineering and a former computer security researcher at AT&T Bell Laboratories. On a recent trip to Washington he found himself with a free afternoon and decided to spend it at the Library of Congress, looking for codebooks that werenÂt in his collection. In the 19th century codebooks were used not so much for secrecy as for compression, to bring down the prohibitive cost of telegraph communication. (The first trans-Atlantic cables cost $5 a word.) Designers devised lists of words to replace phrases and even sentences. But when Dr. Bellovin hunted though the card catalog, his interest was piqued by an 1882 codebook whose title included the word Âsecrecy.Â ÂI thought, ÂO.K., let me go see how they did it,Â Â he recalled. ÂWhen I read the two-page preface, my jaw dropped.Â He could plainly see that the document described a technique called the one-time pad fully 35 years before its supposed invention during World War I by Gilbert Vernam, an AT&T engineer, and Joseph Mauborgne, later chief of the Army Signal Corps. Although not widely used today because it is relatively difficult to work with, the one-time pad is still viewed as one of strongest ways to encrypt a communication. The technique is distinguished by the use of a random key, shared by both parties, to encode the message and decode it; the key must be used only once and then securely disposed of. It was the Soviet UnionÂs misuse of the technique Â code clerks were occasionally reusing the one-time pads instead of discarding them Â that led to the Venona project, the collaboration between the United States and British intelligence services that yielded code-cracking coups during World War II and the cold war. The 1882 monograph that Dr. Bellovin stumbled across in the Library of Congress was ÂTelegraphic Code to Insure Privacy and Secrecy in the Transmission of Telegrams,Â by Frank Miller, a successful banker in Sacramento who later became a trustee of Stanford University. In MillerÂs preface, the key points jumped off the page: ÂA banker in the West should prepare a list of irregular numbers to be called Âshift numbers,Â Â he wrote. ÂThe difference between such numbers must not be regular. When a shift-number has been applied, or used, it must be erased from the list and not be used again.Â That sent the astonished Dr. Bellovin to the Internet to try to find out whether Mr. MillerÂs innovation was known to the later inventors. The results of his largely online detective work can be found in the July issue of the journal Cryptologia. Born in Milwaukee in 1842, Mr. Miller attended Yale and then joined the Union Army, where he fought at Antietam and was wounded at the Second Battle of Bull Run. He was transferred to the Army inspector generalÂs office, where he became a member of a squad of detectives investigating LincolnÂs assassination Â perhaps his first contact with cryptanalysis, Dr. Bellovin speculates. He seems to have kept a diary that still belongs to his descendants, but Dr. Bellovin was unable to obtain it. According to several independent specialists in cryptography, Mr. Miller was undoubtedly the first to propose the concept of the one-time pad. ÂMiller probably invented the one-time pad, but without knowing why it was perfectly secure or even that it was,Â said David Kahn, the author of the definitive 1967 book ÂThe Codebreakers.Â ÂMoreover, unlike MauborgneÂs conscious invention, or the GermansÂ conscious adoption of the one-time pad to superencipher their Foreign Office codes, it had no echo, no use in cryptology. It sank without a trace Â until Steve found it by accident.Â Dr. Bellovin found no evidence that either Mr. Vernam or Mr. Mauborgne ever met Mr. Miller, but he did uncover one more tantalizing clue Â in the society pages of The San Francisco Chronicle, of all places. At a military ball at the Presidio in 1907, Mr. Miller met Parker Hitt, a cryptographer who was a student and colleague of Mr. MauborgneÂs. ÂIt is quite certain that if Hitt knew of MillerÂs system,Â Dr. Bellovin writes, Âhe would have shared that knowledge with Mauborgne when they were together at the Army Signal School in Fort Leavenworth.Â But as he acknowledges, that is still a big Âif.Â

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Re: [math-fun] Geometry question
by Dan Asimov 25 Jul '11

25 Jul '11

Nice one. Interestingly, a paper published only this year proves that if an NxN Hadamard matrices is also a circulant -- like Rich's matrix -- then N is 1 or 4. See < http://blms.oxfordjournals.org/content/early/2011/01/24/blms.bdq112.full.pdf >. --Dan ----- Rich wrote: << How about recasting as - + + + + - + + + + - + + + + - for "obvious" symmetry? To go from your reverse-quaternion-sign matrix to mine, negate the first column and the last three rows; then swap rows 2 & 4. >> Sometimes the brain has a mind of its own.

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[math-fun] The Code
by Adam P. Goucher 25 Jul '11

25 Jul '11

Greetings! It has come to my attention that Marcus du Sautoy is presenting an exciting tripartite TV programme about mathematics, starting from this Wednesday (27th July): http://www.bbc.co.uk/tv/features/code/ There's also a code challenge running parallel to the series, for those cryptanalysts amongst you. Indeed, I hear that a group of you had fascinating sessions in the Birmingham camp, such as one about the RSA cipher. If you want to try another RSA cipher yourself, that would be 2429528048. (n = 4294967297, e = 17. This time, I have actually encrypted the message! : ) Have fun! Sincerely, Adam P. Goucher

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Re: [math-fun] Geometry question
by Dan Asimov 24 Jul '11

24 Jul '11

Suppose we define a map L:R^n -> C = R^2 by sending the kth standard basis vector e_k |-> t^k, where t := exp(2k*pi/n), and extending linearly. L takes the origin O in the n-cube [0,1]^n to the origin in C, and the vertices edge-adjacent to O to the nth roots of unity. Then it's fairly straightforward to check that the vectors C := (cos(2k*pi/n)) (k = 1,...,n) and S := (sin(2k*pi/n)) (k = 1,...,n) define ker(L) via: ker(L) = (v in R^n | v*C = v*S = 0}, and so C and S form a basis for the orthogonal complement Perp(L) of ker(L). And since C*C = S*S (= n/2), L takes Perp(L) to C via a uniform scaling (of factor sqrt(n/2)), which ensures that L is an orthogonal projection R^n -> C. --Dan David W. wrote: << I had the following conjecture, probably nothing more than a corollary to projective geometers: Given an n-dimensional hypercube H, it is possible to orthogonally project H to some suitably coordinated complex plane P so that some vertex V of H maps to 0 and the vertices H-edge-adjacent to V map onto the n-th roots of unity. I leave it to the geometers to state this correctly, and if true, I imagine it is a fairly elementary theorem of projective geometry. >> Sometimes the brain has a mind of its own.

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Re: [math-fun] Dupin cyclides --- another rhinoceros' pancreas?
by Dan Asimov 24 Jul '11

24 Jul '11

I think the complex singularities can be brought into the real domain by an appropriate transformation, yielding a surface like this: < http://jalape.no/math/ dupin_s.gif >. --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] Geometry question
by N. J. A. Sloane 24 Jul '11

24 Jul '11

Fred, Did you see A Note on Projecting the Cubic Lattice on my home page, about 6 items from the top? Also arXiv: 1004.3072 We show that given any (n-1)-dimensional lattice L, there is a vector v in Z^n such that the projection of Z^n onto v^perp is arbitrarily close to L. Not quite your problem, but in the same spirit. Best regards Neil Home page: http://www.research.att.com/~njas/

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