Prime Progressions… (by 544,680,710)

Following up on yesterday's words from Richard Elwes I'll toss out another bit from his wonderful volume "Math In 100 Key Breakthroughs" (pg. 386). We're all familiar with "twin primes" like 11 & 13, and even triplet primes like 3, 5, 7. One can also signify longer sequences of primes that are separated by equal gaps: 11, 17, 23, 29, for example have spacing 6-apart (there are other primes, 13, 19, interspersed, but we're ignoring them).

Richard Elwes asks, "How long can sequences like this be?" and replies further, "The search quickly becomes hard, as the individual numbers involved become very large, too. The longest currently known arithmetic progression of primes consists of 26, beginning with 43,142,746,595,714,191 and then increasing in steps of 544,680,710. It has long been conjectured that there should be arithmetic progressions of primes of every possible length. This idea dates back at least to 1770, to the work of Edward Waring and Joseph Louis LaGrange. But the conjecture resisted all attempts at proof until 2004, when Ben Green and Terence Tao collaborated to prove their stunning theorem.
"If you want a list of 100 primes, each exactly the same distance from the last, the Green-Tao theorem guarantees there will be such a list somewhere. It does not, however, provide much useful information about where to start looking!"

...mind… blown… yet… again. . . .

…and, as long as we're speaking about primes, I hope most of you saw Web cartoonist xkcd's recent effort on the Goldbach conjecture(s):  http://xkcd.com/1310/

Finally, a safe, happy... and mathy NEW YEAR to one-and-all!!!

Meditating... on Chaos

Food for thought… an interesting passage from chapter 85 (on chaos theory, also known as "bifurcation theory") of Richard Elwes' "Math In 100 Key Breakthroughs" (pgs. 345-7):

"How can one produce a random number? In the late 1940's, John von Neumann proposed a very strange answer to that question.  He suggested that applying a simple algebraic rule a few times should do the job. The rule is to begin with some number, call it x, and then multiply x by (1 - x), and multiply the result by 4. That is to say: x --> 4 X x X (1 - x).
"There does not seem to be anything especially 'random' about this bit of algebra. Once the initial number is chosen, say x = 0.1, the result of applying the rule is then completely predetermined. But a little experimentation reveals von Neumann's insight. The sequence produced by this rule runs: 0.1, 0.36, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, and so on (each number given to 4 decimal places). There does not seem to be much of a pattern here, and in fact that is no illusion. You can extend the sequence for as long as you like and in fact no pattern will emerge. Someone who did not know the rule being used would find it virtually impossible to distinguish between this sequence and one produced by a genuinely random physical process such as radioactive decay."…

"Today, von Neumann's rule is known as the logistic map, and it is one of the simplest examples of mathematical chaos, a phenomenon which has been recognized in many different situations…."

"In von Neumann's pseudorandom number generator, everything rests on the number 4, known as the parameter. Changing that value completely alters the behavior of the system. If one replaces 4 with a new parameter of 2, the logistic map ceases to be chaotic. Instead, for any starting value, the sequence will quickly home in on a fixed value of 0.5. This is known as an attracting point of the system.
"Increase the parameter from 2 to 3.4, and something new occurs. After a while, the sequence will endlessly flicker back and forth between two values around 0.84 and 0.45. This is known as an attracting 2-cycle. Raise the parameter a little higher to 3.5, and this is replaced with an attracting 4-cycle, and then at 3.55, an attracting 8-cycle, and so on. As the parameter increases, the length of the attracting cycle keeps doubling 15, 32, 64, and so on. This behavior is what chaos theorists call a sequence of bifurcations."

He goes on to explain that the bifurcations end once the parameter hits a certain threshold value known as the Feigenbaum point (named after chaos theorist Mitchell Feigenbaum). Beyond that point (like "4" in the example) the produced sequence will act chaotically forever, producing the famous "butterfly effect" whereby two sequences beginning at only slightly different starting values "end up entirely unrecognizable from each other."

Move Over, Clifford Pickover

Richard Elwes has a new volume out, "Math In 100 Key Breakthroughs," that I'd add to the Holiday math book shopping list I've already posted. It's a bit reminiscent of Cliff Pickover's "The Math Book" -- I like a lot of Pickover's stuff, and he was kind enough to do an interview for me here, but I was never greatly enamored of that particular volume from Cliff, despite its wide success and popularity -- I do however like Elwes' effort to combine math text and gorgeous graphics in a delicious way, that flows along nicely.

Elwes' book runs essentially in chronological order and while the first third didn't grab my interest that much, covering earlier math history, it gets more interesting with coverage of more modern mathematics (say starting with Newton onward). The text is again (like Cliff's book) on the pithy side, but a bit more substantive than the latter; and I always find Elwes to be one of the very best, clearest, current explicators of mathematical ideas for a lay audience… all the more reason I wish he had gone just a tad more deeply into some of the subjects addressed here.
Still, the volume represents, I think, a splendid introduction to the variety and range of mathematics, especially for a young person with such inclinations. It is already 400 pages long (perhaps at least 1/3rd of that from graphics/pictures), so maybe further, pedagogic text would've added too much. While organized into 100 chapters or "breakthroughs," each chapter covers multiple specific topics, so there's a lot more than 100 topics touched upon here, and more, I think, than is covered in Pickover's choppy volume of 250 "milestones."

My one beef with the book is that there is no bibliography included (Cliff's book has one at the end)… or even better yet, would have been a "for further study" listing following each chapter, referencing sources to further the reader's interest/knowledge if so inclined. One might argue that because anyone can Google any subject these days and find copious additional material, such bibliographic references are no longer needed… but it is exactly because Google returns such copious, ill-prioritized suggestions, that a honed list of excellent selections from the author would be valuable.

Anyway, I highly recommend this beautiful book, especially if you liked Pickover's more coffee-table-like version… OR, even moreso if you didn't find Pickover's volume satisfying, but still fancy the concept of combining wide-ranging, informative mathematical text with beautiful illustrations.

The Unassuming Rock Star

This will likely be a month of a lot of Martin Gardner reminiscences on blogs, with the combination of his birthday approaching (10/21) and his new autobiography making the rounds (I've reviewed the autobio. twice over at MathTango: short review  &  long review ).

The most recent NY Times "Wordplay" puzzle column is dedicated to an old Gardner classic problem (Monkey and the Coconuts), but what I enjoy most about the column is hearing from Martin's son James who relays some brief memories of his famous dad:

http://wordplay.blogs.nytimes.com/2013/10/07/gardner-2/

James remarks at one point, in words that I've heard echoed by others:

“The thing I find fascinating — in some realms Dad was a rock star. He had groupies. People were excited to meet him. In other realms … if you were not enmeshed in his writing, you had no idea who he was."

I too always found this true. Over the years when I mentioned to friends that I was a Martin Gardner fan, the response was either along the lines of, 'Oh yeah, isn't that guy great!' or alternatively, 'Whooooooooo???' …or, on still other occasions, 'isn't he the guy that wrote those pages at the back of Scientific American for awhile?' …to which I always wanted to reply, 'Uhhh, yeah, sorta like that Einstein guy, who fiddled around with light for awhile, I guess'….


Also worth noting that The Aperiodical recently ran a podcast talking to Colm Mulcahy specifically about Gardner here:

http://aperiodical.com/2013/10/all-squared-number-8-martin-gardner-colm-mulcahy-part-2/

In other notes, but still speaking of math popularizers, one of my favorite current ones, Richard Elwes, was recently interviewed here, promoting his most recent books:

http://www.kevinhouston.net/blog/2013/10/interview-with-richard-elwes/

And finally, as a heads-up, looks like I'll be hosting the November "Carnival of Math" so be thinking of posts (yours or others) you'd like to send along for inclusion (I'll probably be putting out a reminder each week through end-of-month). The submission page is here:

http://tinyurl.com/krobqjp

A Couple of Popularizers

First, a quick note that Brit Richard Elwes, one of my favorite popular math writers, has a new volume out "Maths In 100 Key Breakthroughs":

http://richardelwes.co.uk/2013/09/12/maths-in-100-breakthroughs/

I'll note that Elwes' books, published in Britain, unfortunately are not always readily available in the U.S. right away, and sometimes show up at a later date, under a different title! (but worth keeping an eye out for)

Speaking of favorite math popularizers, I've had occasion to think about Martin Gardner lately, and so will re-run one of his classic puzzles that I used here a couple years back -- am quoting it directly from his "The Jinn From Hyperspace" volume:

"Now for a final paradox. There is a certain event that I guarantee will or will not take place during the next ten minutes. You are absolutely  incapable of predicting correctly whether it will or won't occur. I don't mean that it's unlikely you can predict it. I mean it is logically impossible to predict it!

"You don't believe it? Then do the following. If you think the event will occur write 'Yes' inside the blank rectangle below. If you think it won't happen, write 'No' inside the rectangle.

"If you predicted correctly, I'll send you a million dollars.

"The event is: You will write 'No' inside the rectangle."  

A wonderful paradox/conundrum entangled with self-reference, causation/prediction, and human language and logic.
I should have a bit more to pass along about Martin in an upcoming post at MathTango, but for now you can read the current post  up there about "skepticism." 

Math In the Modern World via Richard Elwes

"Chaotic Fishponds and Mirror Universes" is the new volume from perhaps my favorite, little-known-in-the-US-but-altogether-worthwhile-knowing-about math popularizer, Richard Elwes. He hails from Britain, and unfortunately his books often don't achieve wide distribution or publicity over here.
Read about the new volume here (I haven't read it yet myself):

http://richardelwes.co.uk/2013/05/03/chaotic-fishponds-and-mirror-universes/

Richard's personal webpage is here:

http://richardelwes.co.uk/

And the rest of his works, through Amazon listed here:

http://tinyurl.com/co6x678

Or you can sample some of his writing at plus.maths.org here:

http://plus.maths.org/content/list-by-author/Richard%20Elwes

Richard would easily make it onto my list of 5 favorite current math popularizers (maybe even top three!)... check him out.

Richard Elwes... Again

I've been aware of Brit Richard Elwes for barely over a year now, but he's already vaulted to one of my favorite math expounders. An interview with him here from Q Blog (and great to see that he has a new volume out, "The Maths Handbook," and yet another on the way! ....his first two books are two of my favorites):

http://www.quercusbooks.co.uk/blog/2011/12/07/the-quercus-couch-richard-elwes/

Richard Elwes... Never Boring

Some time ago I stumbled across Richard Elwes' "Mathematics 1001" volume in a bookstore (having never heard of either Elwes or the book) and was quite delighted with that encyclopedic compendium of mathematical information. This weekend, another Elwes volume, "Mathematics, Without the Boring Bits," was my lucky, accidental find in a bookstore, and it too looks to delight. My sense is that Elwes' books, coming from Britain, don't get as wide a distribution and publicity as they deserve here (in US).

I'm barely into this volume but it looks to be another wonderful, what-I-call 'nugget' book -- even at 200 pages it serves up math in very palatable bite-size nuggets, often introducing some topic in a page or less... and it only brings up the sort of topics a non-professional math person will find fun or interesting. The selection is excellent, the format attractive, and the writing entertaining and engaging. Having said that, some of the topics are covered so briefly I'm not sure a mathematical novice will always get the point or fully appreciate the significance, and a professional mathematician, on-the-other-hand, may find little new here, he/she isn't quite familiar with. So the intended audience (I think) for the book may be those who already have some background and inclination toward math, but not enough academic training to make these particular topics old hat. In any event, nice to see this array of mind-bending topics brought together succinctly in the pages of a single breezy volume.

Elwes also has a blog here:

http://richardelwes.co.uk/blog/

...and here's a fun post he did a bit ago on one of my favorite topics, self-reference/recursion:

http://richardelwes.co.uk/2011/06/06/an-idiotic-paradox/