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Monday, August 29, 2016

Ford Circles

(via WikimediaCommons)

Awhile back I mentioned Alfred Posamentier’s latest volume “The Circle,” and around now it should be showing up in bookstores -- another great little geometry offering from Dr. Posamentier (and Robert Geretschlager). One of so many interesting tidbits in it is about “Ford circles”:

Imagine you have two tangent circles sitting atop a number line, one tangent to that line at “0” and the other tangent at “1.” Now in the space between these circles draw another circle tangent to both the “parent” circles and to the number line as well — it will touch the number line at the 1/2 position. You can keep iteratively drawing such circles (to infinity) in the space created with each new (smaller) circle. Now, quoting from the book:
“Of course, the circles get very small very quickly. As it turns out points of tangency of all these infinitely many circles with the number line have a quite unexpected property. The points of tangency are precisely the rational numbers in the interval between 0 and 1. No circle created by this process touches the number line at an irrational point, and every rational number is the point of tangency for some circle created in this manner.”
Pretty amazing, and a nice demonstration of one area of mathematics, plane geometry, connecting to other areas of infinity and number theory. Further, these circles relate back to Farey sequences.
Here’s one of several treatments of Ford circles on the Web:

Sunday, August 28, 2016

Of Dogs and Financial Crashes

Recently finished (and very much enjoyed) Charles Wheelan's latest book, "Naked Money," and, though not very mathematical, will draw this Sunday reflection from it:
"...my dog was offered a preapproved Visa card with a five-digit credit limit sometime around 2005. (I subscribed to The New Yorker in his name, W. Buster Wheelan, and some credit card issuer obviously bought the list.) This would suggest that the credit markets were out of control. The subsequent real estate bust would not have been nearly as catastrophic if financial institutions had not been lending to dogs, literally and figuratively."
[...makes me think, that since you as a consumer are given a FICO score, perhaps financial institutions ought be assigned FIDO scores ;-) ]

Wednesday, August 24, 2016

18 and Counting

Not sure how he finds the time, but John Cook puts out 18 math-related Twitter accounts for our delectation (in addition to his blog):


If you’re on Twitter and not following at least a couple of John’s feeds you probably want to check them out.  Most of these accounts tweet about once-a-day, always with some pithy bit of interesting information (or link), ranging over topics from general science to more technical math or computer-science niches.

While I don’t doubt that there are scamsters/trolls with more than 18 Twitter accounts, I’m guessing there may be no serious mathematician/scientist in John’s league of multiplicity. Or am I wrong? Anyone know of someone who surpasses Dr. Cook's prolific accounts-output?
Of course by the time I post this John may have added another 1 or 2 accounts… ;-) 

Also, just realized, that an old (now defunct?) account I enjoyed, @TautologyFacts, almost has the look/feel (in a parody sort-of-way) of a Cook account, though I don't believe it had any connection to John?:

Tuesday, August 23, 2016

Erdös Trivia...

Just a line I recently found surprising in the Wikipedia entry for Paul Erdös....

Though Erdös collaborated with over 500 different individuals in his lifetime, publishing more than 1500 mathematical articles, he not only never won the Fields Medal himself, "nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes." That struck me as remarkable (and I assume it's accurate, unless someone knows a counter-example?). I'm not quite certain what the last phrase ("a pattern that extends to...") even means -- does it mean he never collaborated with the winner of any prize he had not himself won??? Anyway, apparently there have never been any Fields Medalists with "Erdös numbers" of "1" -- wouldn't have guessed that!

Monday, August 22, 2016

Life of Gardner...

Another YouTube video today, this time from Martin Gardner's official biographer, Dana Richards... over an hour long, but worth it if you're a Martin Gardner fan; lots of good stuff and interesting tidbits in addition to fairly well-known information about Martin:

(p.s... yesterday's 'Sunday reflection' came from Gardner)

Sunday, August 21, 2016

Dissolving Into Pure Mathematics

Sunday reflection, via Martin Gardner:
"In light of today's physics the entire universe has dissolved into pure mathematics. The cosmos is made of molecules, in turn made of atoms, in turn made of particles which in turn may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathematical structures. If a photon or quark or superstring isn't made of mathematics, pray tell me what it is made of?"

Thursday, August 18, 2016

Zhang and the Twin Primes...

If you've never seen this 2015 documentary (uploaded to YouTube about a month ago) on Yitang Zhang and his Twin Prime Conjecture work it's worth a watch:

It's almost an hour long (starting off with fairly basic information) and very enjoyable, both for the remarkable storyline and for the appearances of lots of key people.

Tuesday, August 16, 2016

Watch 'em

Yesterday I linked to Tadashi Tokieda's videos over at Numberphile because... they are just so wonderful:

...and it occurs to me there are several YouTube video math presenters I really enjoy and ought be sure readers are aware of:

The general Numberphile site is of course almost unsurpassed in their quality, consistency, and entertaining (as well as instructional) value:

James Grime and Matt Parker are wonderful and often appear on Numberphile, but also have their own channels:



also entertaining, but perhaps less well-known is Mathologer:

Presh Talwalkar has many very-nicely-done videos from his wide-ranging blog “Mind Your Decisions”:

and finally, 3Blue1Brown is another less-well known site of good math videos:

If you're not familiar with some of these, check 'em out... but be forewarned, they can be addictive.

Sunday, August 14, 2016

Conversations Across Millennia

Sunday reflection:
"Ask mathematicians about their experience of the craft, and most will talk about an intense feeling of intellectual camaraderie. 'A very central part of any mathematician's life is this sense of connection to other minds, alive today and going back to Pythagoras,' said Steven Strogatz, a professor of mathematics at Cornell University. 'We are having this conversation with each other going over millennia.'"
 -- Gareth Cook in the NY Times (2015)

Thursday, August 11, 2016

Of Drugs and Canaries

Saying There’s a real problem in the way that clinical trials report their results,” reform-minded Ben Goldacre argues that “Statins are the canary in the cage for problems in modern medicine.”
This isn’t strictly math, but is an important read on how clinical research gets “fiddled” with:

From the piece:
"[clinical] Outcome priorities are changed; negative results are omitted; trials are foreshortened or extended to better massage the data. In the book [Big Pharma], Goldacre terms these tactics 'a quiet and diffuse scandal.'"

…and to think there was a day, not so long ago, when Big Pharma was trying to convince folks that statins (“the most commonly prescribed medication in the developed world”) ought be added routinely to municipal water supplies. 

Tuesday, August 9, 2016


Meanwhile, that whirring sound you hear in the distance is the sound of Lincoln, Ike, and Reagan spinning in their graves:
“The leader of the American Nazi Party has said the election of Donald Trump as president would present ‘a real opportunity for people like white nationalists’ to start ‘acting intelligently’, with the aim of building a mainstream political presence….” 

Sunday, August 7, 2016

...the beautiful

A simple, succinct Sunday reflection today:
 "I always try to combine the true with the beautiful, but when I have to choose one or the other, I usually choose the beautiful."
 -- Hermann Weyl (mathematician/physicist)


Thursday, August 4, 2016

Knock 'em, Sock 'em, Rock 'em...

OK, a just-for-fun post today… My favorite show on TV (I mean after Seinfeld re-runs of course) is “Battlebots” which, I was thrilled to find on the ABC network, since I obstinately refuse to pay for cable. The show features MIT-like dissertation creations ;-) doing battle for mechanical supremacy and honors. And hey, somewhere in all this imaginative, intricate engineering a whole lot of math has to be buried! 

Anyway, the show airs on Thursday nights at 8pm. Eastern time (tonight), and here’s a sampling of an older bout, in case you're unfamiliar:

Wednesday, August 3, 2016

Playing With Primes...

James Maynard continues his fascinating work with prime numbers:
Maynard has shown that there are an infinite number of primes that include no number “7” digit. As the article states:
" ‘The vast majority of big numbers have lots and lots of 7s in them, so having no 7s is a rare property for any whole number to have.’ The fact that, despite this rarity, Maynard was able to prove that there are infinitely many such primes counts as an impressive feat. There's nothing special about the number 7, incidentally. Maynard's proof works equally well for any other number: so we now know that there are infinitely many primes without 1 as a digit, or 2 as a digit, or 3, or 4, or 5, and so on.”
The article proceeds with some interesting exposition about primes, music, Fourier analysis, “waves,” and cuboids.
But returning to the digit analysis I can’t help but wonder if there are also an infinite number of primes missing any two digits, say “7” and “3” for example, or “1” and “2”… or of course then three digits… or…. (what are the limiting cases here, and what, if anything, would they mean?). Or, starting at the other end, are there an infinite number of primes composed of say, nothing but 1s or 7s  (DOHH!!, that wouldn't work).
I don’t see any indication in the piece how much of this has already been looked at? HEY, Mike Lawler, a weekend project! ;-)

Tuesday, August 2, 2016

'...the End of Days'

If you’re already having a bad day, maybe don’t go read this right away!:

Keith Devlin reviews and comments on a new documentary titled “Zero Days” about the “Stuxnet” computer virus hatched by the U.S. and Israel to destroy centrifuges involved in the Iranian nuclear production program. Devlin calls the film “arguably the most important movie of the present century” and titles his piece “Mathematics and the End of Days,” which may give you a hint as to its thrust (Variety calls it "a white knuckle thriller"). But in case you want stronger indication, here are some sentences near the end of Keith's piece:
“The weapon is, after all, just a mathematical structure; a piece of code. Designing it is a mathematical problem. Unlike a nuclear bomb, the mathematician does not have to hand over her results to a large, well-funded organization to build the weapon. She can create it herself at a keyboard.
"That raw power has been the nature of mathematics since our ancestors first began to develop the subject several thousand years ago. Those of us in the mathematics profession have always known that. It seems we have now arrived at a point where that power has reached a new level, certainly no less awesome than nuclear weapons.”
I hadn’t even heard of this film prior to Dr. Devlin's piece, but now am certainly anxious to see it. An old Murphy-like aphorism says that ‘anything that can happen will eventually happen’ -- in this day of computer malware and cyber-warfare that’s never been a scarier thought.
Here is the official trailer for this must-see film:

Monday, August 1, 2016

Just Sayin’…

Judge Philip Forman:  “Now, Mr. Gödel, where do you come from?” 
Kurt Gödel “Where I come from? Austria.”
Judge:  “What kind of government did you have in Austria?” 
Gödel:  “It was a republic, but the constitution was such that it finally was changed into a dictatorship.” 
Judge “Oh! This is very bad. This could not happen in this country.” 
Gödel:  “Oh, yes, I can prove it.” 

With the intensity of the political conventions passed I thought I could drag myself away from politics… but, well, not quite yet!:

Kurt Gödel was no slouch of a thinker or logician, and many or most of you will know the legendary story of his trip to attain U.S. citizenship in 1947. 
The above conversation snippet is part of the purported discussion between Kurt Gödel and the presiding examiner, Judge Philip Forman, when Gödel went (with his friends Albert Einstein and Oskar Morgenstern as character witnesses) to a hearing applying for U.S. citizenship. Using pure reasoning, Gödel was certain he had found a flaw (a self-contradiction) in the U.S. Constitution that would permit a dictator to attain power in the country (in fact, according to some versions of the story I’ve seen, he believed it almost inevitable, given enough time). And, he thought he could prove it!

Einstein and Morgenstern knew Gödel was courting disaster (in applying for citizenship) if he insisted on pushing his view that the country was bound for possible dictatorship, so they helped maneuver Gödel nervously through the mundane procedure, and all ended well.

The story has been told widely, though never with complete clarity, nor with details as to what Gödel’s specific argument with the Constitution was. Many believe it had to do with Article V which spelled out the right to amend the Constitution… in Gödelian recursive thinking this would imply also the right, at some point, to amend the Constitution to say it may no longer be amended, following a despot gaining power. There are, however, likely several other spots in the Constitution that could harbor the seeds of eventual dictatorship. 

For a further long legalistic discussion of Gödel’s story legend see pdf link here:

Anyway, I find it interesting that now, almost 70 years since mathematician Gödel expressed his concerns to Einstein and Morgenstern, there is suddenly more talk of fascism and dictatorship in America than probably ever before in history.  As a recent tweet I saw on Twitter said, "When Fascism comes to America, it won't release its tax returns" ;-) 
One can almost imagine Kurt Gödel, somewhere in the Great Beyond, nodding knowingly...

Sunday, July 31, 2016

Clever Hardy

"Hardy used the Riemann hypothesis as insurance when crossing the North sea after his summer visit to his friend Harald Bohr in Denmark. Before leaving port he would send his friend a postcard with the claim that he had just proved the Riemann hypothesis. It was a clever each way bet. If the boat sank he would have the posthumous honour of solving the great problem. On the other hand, if God did exist he would not let an atheist like Hardy have that honour and would therefore prevent the boat from sinking."

-- Tony Crilly

[p.s... with the political conventions over, blogging here and at MathTango, will likely return to normal in the coming week]