Wednesday, March 14, 2012

162: The Mathematician Who Wasn't There

Audio Link

Nicholas Bourbaki was an influential 20th-century mathematician who published a series of texts begnning in the 1930s, trying to rigourously describe the major core areas of mathematics based on the foundations of set theory. Many of his books became standard references in their fields. Among the Bourbaki contributions that have become familiar to modern math students are the use of the slashed zero to represent an empty set, and the terms injective, surjective, and bijective. The ill-fated New Math movement in education, which I discussed back in podcast 145 (http://mathmutation.blogspot.com/2011/12/145-why-johnny-couldnt-add.html), was also largely inspired by Bourbaki ideas. But if you try to find out about the life of Bourbaki himself, you will be in for a bit of a surprise: Nicholas Bourbaki did not actually exist.

Actually, to be more precise, Bourbaki was not an individual, but a secretive society of mathematicians established by Andre Weil, Henri Cartan, and other young mathematicians in Paris in the 1930s. They originally got together because they believed there was a serious gap in the available mathematics texts of the time, a problem magnified by the fact that a generation of potential leaders in the field had been wiped out by World War I. They hoped to re-establish the rigorous foundations of mathematics, while at the same time to provide a standard series of reference works. While secret societies tend to sound sinister in concept, their secrecy actually stemmed from rather noble motives: they wanted to ensure that any works they produced would be judged on the basis of their content, not on factors related to personal egos. So they created the ficticious persona of Nicholas Bourbaki, member of the Royal Academy of Poldavia, and agreed that he would be credited as the author of all books they produced.

　
The members of the Bourbaki group enjoyed having fun with the semi-secret nature of their small club. To anyone who asked, they would claim that Bourbaki was a real person of their acquaintance, and they even printed up a set of mathematical-pun-laden wedding invitations from Bourbaki's daughter, to show to anyone who doubted his authenticity. According to the invitation, "The trivial isomorphism (aka the sacrament of matrimony) will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.". You wouldn't think that would fool too many people-- but this prank backfired horribly during World War II, when Andre Weil (who had fled to Finland) was arrested on suspicion of spying. This apparently encoded letter from a strange foreign contact was considered a key piece of evidence. Weil was sentenced to death, though at the last minute a friend with government contacts managed to intervene and get him pardoned.

You would think they would be done with practical jokes after that, but the Bourbaki group tried to continue the whimsical spirit, perhaps a welcome break from the incredbly serious work they were attempting. In the late 1940s, American mathematician Ralph Boas was contacted by the Encyclopedia Britannica to assist with an article on modern mathematics, and since he knew Weil, he mentioned in his article that Bourbaki was actually a pseudonym for a collaboration rather than an actual person. Boas and the Encylopedia then received a letter, claiming to have been written by Bourbaki from an ashram in the Himalayas, asking "You miserable worm, how dare you say that I do not exist?" A series of letters went back and forth on the topic, though the encyclopedia editors were eventually convinced of the truth. But then Bourbaki members began to spread a new rumor, that Ralph Boas did not actually exist, and was actually a collective pseudonym for a group of American mathematicians!

The Bourbaki group lasted for several generations, producing a series of comprehensive texts on the foundations of modern mathematics. While they made many lasting contributions, they were also criticized for focusing too narrowly on foundations, omitting areas relevant to widely applicable topics such as logic and mathematical physics. They also had a strange aversion to pictures and illustrations, which meant that although some of their books were very useful as references, they ironically were not very good as textbooks. The group gradually declined after the 1970s, with their last major text, "Spectral Theory", being published in 1983. The decline may have been hastened by a long legal battle with their publishing company over royalties and translation rights: Pierre Cartier, one of the members during the final productive period, described the result as "both parties lost and the lawyer got rich."

In the show notes, you can find a link to a 1997 interview with Cartier, who shares some fascinating thoughts on the rise and fall of the Bourbaki group. Aside from the legal issues, one of the other major reasons he gave for Bourbaki's decline was that they had successfully achieved their objectives. As Cartier describes it, "In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution."

And this has been your math mutation for today.

　

References:


　

　
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Sunday, February 19, 2012

161: The Numbers Of Love

Math Mutation 161:  The Numbers Of Love

Before we get started, I'd like to thank listeners Foxy McLovin and The Devonian Kid, who recently posted nice reviews on iTunes.  Remember, you too can experience the thrill of having your name, or bizarre iTunes nickname, mentioned on Math Mutation by posting a review.  Or you can send a donation to your favorite charity in honor of Math Mutation, and email me to tell me about it.  Anyway, on to today's topic.

During this Valentine's week, it occurred to me that there was not yet a Math Mutation episode focusing on amicable numbers, said to be numbers that represent friendship and love.  Yes, there are such things.   And yes, math geeks do occasionally experience such emotions.  So what exactly are amicable numbers?

To start with, let's review the concept of a perfect number.  A perfect number, like 6 or 28, is precisely equal to the sum of its factors.  So 6, for example, is perfect because its factors are 1, 2, and 3, and 1 + 2 + 3 = 6.   The concept of amicable numbers is related to perfect numbers, except that they are pairs of numbers, such that each is the sum of the other's factors.  The smallest pair of amicable numbers is 220 and 284.  The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284.  And the factors of 284 are 1, 2, 4, 71 and 142, which add up to 220.   If you stretch your sense of analogy a but, you can kind of see how the way in which two numbers are unexpectedly intertwined might suggest love and friendship.

The pair 220 and 284 was known to the ancient Greeks.  But as you can probably guess, amicable numbers are relatively rare, and can be tricky to find.  In 850 A.D, Arabic mathematician Thabit ibn Kurrah discovered if n > 1 and each of p = 3*2^n-1-1, q = 3*2^n-1, and r = 9*2^2n-1-1 are prime, then (2^n)pq and (2^n)r are amicable numbers.  This was a nice advance, but finding numbers that meet the preconditions of this formula wasn't very easy-- several centuries later, it led to discovery of the pairs 17,296 and 18,416, resulting from n=4, and 9,363,584 and 9,437,056, which you get from n=7.  Euler later published a list of 64 new amicable pairs, but he actually made some mistakes, and the list had 2 bogus entries, which was not discovered until the 20th century.   But most surprisingly, the second-lowest pair of amicable numbers, 1184 and 1210, was not discovered until 1866, by a 16-year old Italian boy named Nicolo Paganini.   With the help of modern computers, millions of amicable pairs have been discovered, though the conjecture that there are an inifnite number of them has not yet been proven.

Amicable numbers have been thought to symbolize love and friendship since ancient times.   It is said that when Pythagoras was asked to define a friend, his definition was "one who is the other I, such as are 220 and 284."   In Genesis verse 32:14, Jacob gave his brother 220 sheep when he feared he was planning to kill him, and some Torah scholars interpret this as due to its status as an amicable number, to reinforce their brotherly love.  I think this last interpretation is a bit of a stretch though-- he also gave him 30 camels, 40 cows + 10 bulls, and 30 donkeys, which don't seem to have anything to do with amicable numbers.  And to get the number 284 to appear somehow, scholars have interpreted one partial word in the verse as a Hebrew encoding of a number, which really seems to be reaching.

According to one online site called "Renassance Astrology", you can make a magical love talisman by the following method:  "Make two images, and put one of the Fortunes at the ascendant and the Moon in Taurus conjunct Venus. Write on one image a number ... for 220 in the proper place, and on the other image write the same kind of figure for 284 in the proper place. Join the two figures together in an embrace, and then there will be perfect and lasting love between the two."  They go on to supply further mystical powers of these numbers: "If the aforesaid numbers were carved in wood, and bread or anything else edible was sealed with them and you gave it to someone to eat, he will delight in you with a great love. If those numbers are written on your clothing, your garments cannot be taken away from you; and if you write them on banners that are put in the street to draw business, they will draw business to you."   I'm thinking of writing some of these numbers on my socks, to stop them from disappearing in the dryer.

And at least some modern geeks still believe that amicable numbers represent love:  with a quick online search, I was able to find a pair of half-heart pendants, one labeled 220 and the other labeled 284, for sale at a site called mathsgear.com .    But math geekiness can only get you so far in a marriage-- when it comes to Valentine's day, I made the safe choice and got my wife some non-math-related flowers.  Perhaps you will be more courageous on your next romantic occasion.
And this has been your math mutation for today.


References:

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Thursday, December 29, 2011

156: The Ultimate Answer

I celebrate my 42nd birthday this week.  This is an important milestone, since in his classic Hitchhikers series of science fiction novels, the late Douglas Adams described 42 as the ultimate answer to life, the universe, and everything.  This was figured out by a massive computer, Deep Thought, after 7.5 million years of calculation.  Once they heard the answer, people realized that maybe they should have figured out what the question was first.  But we can ask a more pertinent question in real life:  why did Adams settle on 42 as the answer in his books?  It somehow seems to have a nice ring to it, but what makes 42 a good "ultimate answer"?

Intrinsically, 42 does have a few things going for it.  One fun fact is that you can build a 3x3x3 magic cube with each number from 1 to 27 appearing exactly once, and every row, column, and diagonal summing up to 42.  42 is also the 5th Catalan Number, which means that 42 is the number of triangulations of a heptagon, or the number of ways you can cut a heptagon into triangles using straight lines.  (You may recall that a heptagon is a 7-sided regular polygon.)  It's a Harshad number, or an integer divisible by the sum of its digits.   There are also many more obscure properties of this number, too numerous to list here. 

Science provides some more cool applications of this number.  In 1268, English philosopher Roger Bacon calculated the geometric properties of rainbows, and discovered that the summit of a rainbow cannot appear more than 42 degrees above the horizon.  More recently, in 1966, mathematician Paul Cooper calculated that if you bore a frictionless hole all the way through the earth and try to travel to China by just jumping in and using gravity, the trip would take you exactly 42 minutes.  Surprisingly, this calculation works even if your hole doesn't pass through the center of the earth:  the reduction in gravitational force and distance traveled exactly balance each other out.

The number 42 also appears in various places in religion and culture.  It's unlucky in Japan, since the name of it sounds like the words for "unto death".  In ancient Egypt, there were 42 principles of Ma'at, or religious laws.  In the Christian book of Revelations, the Beast will hold dominion over the Earth for 42 months.  The Jewish Talmud references the "42-lettered name of god", and perhaps most relevant, Kabbalistic tradition makes the related claim that this number was somehow used by God to create the universe. 

The Wikipedia article also points out that Lewis Carroll, popular 19th-century author of lighthearted math-influenced works, used the number 42 numerous times:  42 illustrations in "Alice in Wonderland", the rule 42 in the same book (requiring all persons more than a mile high to leave the court), 42 boxes in The Hunting of the Snark, and a few other places. 

One more intriguing possiblity is that a semi-serious guide book from the 1970s and source of Adams's title, "The Hitchhiker's Guide to Europe", mentioned that travelers to the UK looking for family roots were likely to find the answers disappointing.  And that comment was on page 42.

So, were one of these the reasoning as to why Adams chose 42 as the ultimate answer?  These possibilities, mostly included on the wikipedia page linked in the show notes, all have some level of plausibility.   That page also includes various other references to 42, too numerous to include here.  But these just scratch the surface-- while doing online research for this podcast, I found that author Peter Gill wrote a whole book on the topic, which I haven't yet read. 

On the other hand, most of these references to the number are what I would describe as, well, very miscellaneous at best.   There's no overarching theme that really convinces me that 42 is a good candidate for the answer to the universe.  I would bet that with a little research, I could find an equal number of properties and references to any other two-digit number you might select.  Think about it:  in any instance where a number less than 100 appears in history, religion, or even some mathematical grouping, there's roughly a 1% chance that it's the number you want!  

So my inclination is to guess that Adams was playing a massive joke on the public, and really did choose the number randomly, just to see what interpretations his readers would come up with.  Since the author is no longer with us, we never will know for sure.

And this has been your math mutation for today.

  • Review of Gill book
  • 42 at Wikipedia
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    153: Cracking Pythagoras


    A few days ago I was browsing the infotainment site cracked.com, and saw an amusing article.  It was titled "6 Famous Firsts That You Learned In History Class (And Are Total BS)."  One of these 'famous firsts' was Pythogras's discovery of the Pythagorean Theorem.  As you probably recall, this is the theorem that if the legs of a right triangle are A and B, and the hypotenuse is C, A squared plus B squared equals C squared.  Pythagoras is known for having first proven this theorem around 550 BC.  Is it really the case that this is total BS?

    Actually, I wasn't too surprised to see this theorem on such a list.  This theorem is critical for measuring distances and amounts of materials in construction, real estate boundaries, and many other real cases where life doesn't always provide you with straight lines.  Because of this, it was discovered much earlier than Pythagoras-- ancient Babylonians had been using it for at least 1000 years by Pythagoras's time, and there are also references to it in ancient India, China, and Egypt.  Cracked.com even included a photo of an ancient Babylonian stone tablet inscribed with Pythagorean triples, sets of integers such that A squared plus B squared equals C squared.

    But there's a huge difference between having observed the theorem experimentally and actually knowing for sure that it will always be true.  The history of math and science is full of examples of supposedly true facts that were later overturned; just ask your doctor if he has any leeches in stock.  If you read your high school math textbook a little more carefully, you'll see that Pythagoras is credited not with discovering the theorem, but with *proving* it.  So if you really want to claim it is illegitimate to cite Pythagoras, you need to supply an earlier proof of the theorem.  The Cracked guys are on their toes though-- they have done this as well, pointing to an earlier Chinese text known as the Chou Pei Suan Ting.

    This book does indeed contain a diagram which seems to illustrate a common geometric proof of the theorem.  You can see the picture if you follow the links in the show notes.  The way it works is that you put four copies of the triangle together to form a large square, in such a way that each side contains one of each of the A and B legs of the original triangle.  This forms a large square whose sides are each of length (A+B), and in its center is a smaller square whose side is of length C.  The total area of the large square is (A+B), the quantity, squared, or A squared + 2AB + B squared.  The four triangles are each of area AB/2, using the standard fomula for area of a right triangle, so together their area adds up to 2AB.  But the remaining square in the middle has sides each of length C, with total area C squared.   So we have A squared + 2AB + B squared equals C squared + 2AB--- or A squared + B squared equals C squared!   Does this illustration thus establish that the Chinese proved the theorem before Pythagoras?  Cracked seems to think so.

    But not so fast-- there are a few problems with the Chinese illustration.  Most glaringly, it has grid markings that show it is addressing a particular case, of 3-4-5 right triangles, with no clear evidence that it was thought of as a general proof.  There is even some dispute as to whether the diagram was actually part of the original book, or transcribed by later commentators.  While the book was begun as early as 1046 BC, annotations and additions continued until 220 A.D.  And the book is not part of any kind of general document created to supply axiomatic proofs of mathematical discoveries; it's a collection of 246 specific problems encountered by the Duke of Zhou and his astrologer.

    But most importantly, we need to keep in mind that the Pythagorean Theorem is not taught as a standalone discovery, a single isolated contribution of the Greeks.  It was one of the beginnings of a systematic approach to mathematics, of not just being satisfied with observations, but of proving hypotheses based on fundamental assumptions, that culminated in Euclid's Elements.  Whether earlier societies knew some of the facts discussed by Greek mathematicians, or even came up with a few isolated proofs, is beside the point.  And if you want to unseat the Greeks as founders of modern mathematics, you need to point out systematic efforts to prove theorems on the basis of fundamental axioms, not grab a few random factoids out of other societies' texts.   I love cracked.com, but I think they truly are cracked on this one.

    And this has been your math mutation for today.

  • Cracked article
  • Chou Pei Suan Ching at Wikipedia
  • Pythagoream Theorem at Wikipedia
  • Another Pythagorean Theorem article
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    150: A Podcast About Nothing

    Before we start, I thought might be nice to highlight a few ways you can show your support for Math Mutation, since I received a recent query about donations.   I don't actually accept money, as I like the idea of total independence.  But if you find the power of the podcast to be so overwhelming as to loosen your wallet, please donate some money to your favorite charity in honor of Math Mutation, and send me an email about it.  I also love hearing directly from fans, either in the form of an email to erik(e r i k)@mathmutation.com, or by posting a review to iTunes.  And don't forget to sign up as a fan on Facebook.

    Anyway, you may recall that in the last episode, I asked for topic suggestions for Episode 150.  While I received a few unrelated emails from listeners, nobody was brave enough to actually sugget a topic.  So, being in a glass-half-full kind of mood, I decided to treat the lack of suggestions as a suggestion in itself, and do a topic that has been on my back burner for a while, the history of the number zero.

    To start with, we should clarify a few things about zero.  In one sense, it is a simple representation of nothing.  But perhaps even more importantly, it plays the critical role of a placeholder in our positional number system.  For example, how do you know you are listening to episode 150 and not episode 15?  It's because of that 0 in the ones place, which pushes the 5 into the tens place and the 1 into the hundreds place.  This seems like an obvious idea now, but that wasn't always the case.  You may recall that the Roman number system was basically non-positional:  with a few minor exceptions, you essentially wrote a bunch of symbols down in any order, added their values, and got a total.  Systems like the Roman numerals quickly grow cumbersome in the face of large numbers, and led to the confusing situation of many possible representations for the same number.

    One of the earliest positional number systems was used by the Babylonians, well-established by the second millenium B.C.  Their system was base-60 instead of base-10, and we still hear echoes of it today when we measure time or angles.  Initially, they did not have a symbol for zero, which meant that written numbers were inherently ambiguous:  you could not tell whether you had written 61 or 3601, 60 squared plus 1, because there was no symbol marking any non-used powers of 60 in the number.  By the first millenium B.C., the system had been enhanced by several authors to use placeholder symbols such as a pair of wedges, but these were only used internally between two digits: trailing zeros in the lower places could only be identified by context.   There must have been a lot of arguments with the waiter over the check in ancient Babylonian resturants.

    Strangely, even when used by a few mathematicians, this placeholder concept did not take hold very quickly.   Despite all their advances in mathematics, the Greeks didn't develop a true positional number system.  This may have been partially due to the Euclidean emphasis on geometry, where numbers were respected mainly for their usefulness when talking about drawn figures.  A few Greek and Roman astronomers, such as Ptolemy in 130 A.D., used the Babylonian system enhanced by a placeholder zero when recording their observations, but this was still considered an esoteric usage. 

    We should also mention that the native Olmecs and Mayans of the Americas independently developed a mixed base-20 and base-18 positional system as early as the 1st century B.C., which also included a true zero placeholder symbol.  Their Long Count calendar basically counted the days from 3114 B.C, when Raised-up-Sky-Lord caused three stones to be set by associated gods at Lying-Down-Sky, First-Three-Stone-Place.  While their numbers were mostly base-20, the second digit from the end rolled over whenever it hit 18 rather than 20.  The mixed base is kind of strange, until you think about the fact that 20x18 is 360, very close to an actual year.

    Most sources seem to agree that the widespread use of a true zero originated in India between 500-700 A.D.  In Brahmagupta's treatise "The Opening Of The Universe", he laid out a number of rules for mathematical operations on numbers including zero.  Some of his rules are very familiar to us today, such as adding zero to a negative number gives you a negative, and adding zero to a positive number gives you a positive.  But oddly, he tried to define division by zero, claiming that zero over zero equals zero.  Today we see that is clearly wrong:  if a calculation results in 0/0, you need more context to figure out a reasonable interpretation.  For example, look at the function y = x/x.  You can see that this is 1 for all nonzero values-- so shouldn't it also be 1 for x=0, thus showing that 0/0 = 1?  But on the other hand, look at y=2x/x.  With the same reasoning, we find that 0/0 equals two! 

    The Hindu-Arabic number system, a positional base-10 system with zero, was originally brought into Spain in the 11th century by the Moors.  Apparently it had spread into common usage among merchants in that civilization.   It was popularized in Christian Europe by Leonardo of Pisa, or Fibonacci, in 1202.  Even then it did not exactly take the continent by storm:  while mathematicians embraced it, merchants continued using the non-positional Roman numeral system for several more centuries before it slowly died out in the face of a superior competitor.  Except, of course, for the critical tasks of expressing motion picture copyright dates, or naming Popes.

    As usual, I've barely scratched the surface here.  Whole books have been written about the number zero, such as Robert Kaplan's highly regarded "The Nothing That Is".  And you can also find some excellent online articles linked in the show notes. But hopefully this podcast has given you a non-zero number of things to think about.

    And this has been your math mutation for today.

  • Robert Kaplan's book
  • Zero at Wikipedia
  • Another online history of zero
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    149: Robot Planet

    Before we start, I've observed that we're rapidly approaching the episode 150, one of those big round numbers that is supposed to be significant or something.  Having trouble coming up with an appropriately weighty topic, I thought I'd throw the problem out to you listeners:  what topics do YOU think would be worthy of episode 150?  Please email erik (e r i k) @ mathmutation.com with your answer.  If you suggest the topic I use, you will win the honor of international fame and fortune as I mention you on the podcast.  Now on to today's topic.

    In a recent hilarious episode of the TV cartoon Futurama, the robot Bender acquired an attachment that allowed him to make 2 duplicates of himself, each 60% of his size, consuming an equal mass of nearby matter in the process for their raw materials.  When the professor asked him to do some work, he was lazy, so activated this machine to make 2 smaller replicas of himself to do it.  But then these replicas each contained a smaller copy of the duplicating machine as well, and they were just as lazy as the original, so they could make 2 copies each of themselves.  You can see where this is going:  since all the Benders were equally lazy, they kept duplicating, potentially on to infinity.  The alarmed professor flashed an equation on the screen, and everyone gasped.  "The equation is divergent", he explained, which meant that soon replicating Benders would consume all the matter on Earth!  What was he talking about?

    Let's review the concept of convergent and divergent series.  Suppose you are adding together an infinite series of smaller and smaller numbers: say, 1/2 + 1/3 + 1/4 ...  .  There are two things that could happen.  Either the series is divergent, in which case the sum approaches infinity, or it's convergent, in which case the total is some finite number.  At first, it might seem counter-intuitive that the sum of an infinite series could ever do anything but diverge to infinity.  But here's a simple counterexample:  Look at the decimal number .99999..., with an infinite number of nines after the decimal point.  I think we would all agree at a minimum that this is a finite value, less than or equal to 1.  (It's actually precisely equal to 1, but we'll leave that nuance for another podcast.)  In any case, if you look at each digit of the number separately, you can see it is just an infinite series:  the first 9 after the decimal represents 9/10, the 2nd represents 9/100, and so on.  So .99999... is the same as the infinite series 9/10 + 9/100 + 9/1000..., and we know that despite the infinite number of terms, the sum never gets past 1.  Similarly, if the total mass of the infinite Benders converges to a small finite number, the Earth is not doomed after all.

    So, let's take another look at the problem of the replicating Benders.  Rather than kilograms, let's simplify our calculations and measure mass in Bender-masses, or Bs, where the mass of the original Bender is 1B.  What is the mass of the two mini-Benders he creates?  First we have to define what '60% of the size' means:  I think it would be logical to assume we are reducing the measure to .6 times the original in each of the 3 dimensions: length, width, height.  Assuming the mass is proportional to the volume, this means that the mass of each mini-Bender is .6*.6*.6 Bs, or .216 times the mass of the original Bender.  So, the total mass of the two half-Benders is 2*.216 = .432 Bs.   Similarly, the Nth generation of Benders should have mass of (.432)^N Bs, since its mass is .432 times that of the previous generation.  And the sum we're dealing with is (1 + .432 + .432^2 + ...).  Good news-- this series converges!  The easiest way to see that is to notice that .432 is less than 1/2, so the Nth term is always less than (1/2^N), a well-known convergent series that adds up to 2.  (A quick way to prove this is to look at the base-2 number .1111...)   So, no matter how many replicas there are, the total mass will be less than 2 Benders, and our planet is safe.

    Unfortunately, the plot of the episode is dependent on the fact that the Bender-masses will add up to a divergent series guaranteed to consume the earth.  On the cartoon, the sum the Professor flashes is a series where each term equalled a Bender mass times 2^n * (1 / ((2^n) * (n+1))).  You can see here that the 2^n terms cancel out in the top and bottom, making this effectively the sum of 1/(n+1), or 1/2 + 1/3 + 1/4..., a well-known divergent series.  If accurate, this would indeed show that all the mass in the universe would ultimately be consumed by the replicating Benders.  But where does this equation come from?  If the smaller Benders were 60% of the mass instead of 60% of the size in each dimension, then each generation would be 1.2x the mass of the previous, a constantly growing series which obviously diverges, but doesn't match the Professor's series either.  I have the feeling the writers are just messing with us, and came up with an arbitrary divergent series to advance the plot.  Or maybe the professor messed up the equation; after all, he is often portrayed as a bit senile.  On the other hand,  I could be the senile one-- tell me if you manage to find something I missed, and think of a good justification for the series in this episode.

    And this has been your math mutation for today.

  • The Basel Problem at Wikipedia
  • Convergent Series at Wikipedia
  • Online discussion including screenshot of the Professor's equation
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    147: Elemental Obsession


    Recently I was reading Theodore Gray's colorful coffee-table book/ebook "The Elements", where he shows an illustration of each element and a few trivia bits about their physical and chemical properties.  I was surprised to see a few passing references to the ease or difficulty of acquiring samples for "collectors".  Are there really people who collect chemical elements as a hobby?  A bit of quick web searching showed that this was no joke-- there is a noticeable element-collecting community out there, who take their hobby quite seriously.  The goal of an element collector is to acquire samples of every element in the Periodic Table.
    I know what you're probably thinking now.  "Sounds interesting, but this is a math podcast, not a chemistry podcast!"  True, but if you think about it, I believe this interest is a kind of mathematical obsession.  After all, the beauty of the Periodic Table is the way that a simple mathematical pattern has be used to understand and predict the occurrence of the basic elements that make up our universe. 

    Let's review the basic concepts of the periodic table.  As you may recall, each row of the table represents an electron shell, which you can think of as a layer of orbiting electrons around the nucleus of an atom.  Each column represents a configuration of 'valence electrons', the outermost electrons that are most important in determining chemical reactions.  Once a shell is full of electrons, the next row begins filling in a new shell with a similar pattern of electrons-- thus the pattern of valence electrons repeats in the next row, which is what makes the periodic table periodic.  The electrons in each shell must fit into a finite set of known orbit patterns, but the inner shells have fewer orbits than outer ones, which is why there are the noticeable gaps in the upper rows, representing the inner shells.  Once the basic pattern is established, the elements go in the table in order of increasing number of protons, or atomic number, and the families with similar sets of valence electrons fall neatly into the columns.  There are a few other complications which you can read about in detail on the wikipedia page in the show notes, and modern quantum mechanics shows that the solar-system-like vision of simple orbiting particles isn't quite right.  But for this podcast, the important point is the mathematical elegance of having a table filled in using simple patterns, and being able to connect this to reality by neatly checking off each box as you acquire samples of each of the elements.

    But for people who actually want to take up element collecting, there are a few minor issues they have to deal with.  An obvious one is simple chemical safety:  numerous elements are poisonous in their pure form.  It sure looks tempting to open that vial of mercury and actually feel the bizarre sensation of a liquid metal rolling around your hand-- but that is not medically advisable, to say the least, as mercury is extremely poisionous and can be absorbed through the skin.  Many other elements are relatively safe in solid chunks but can be extemely dangerous in powdered form, which is unfortunately often the easiest to acquire.  One Russian company used to sell cheap samples of many elements by mail order, including a glass vial of powered beryllium, which could be fatal if it broke during shipping and someone inhaled the scattered powder.  Any element collector has to think carefully about the form and storage requirements for each element.

    And then there are the radioactive elements.  Aside from the obvious dangers of radiation poisoning and the need for careful storage, there is the question of how to acquire them in the first place.  Some that have real-life applications are available in tiny amounts in commercial products:  for example, in the 1990's, teenager David Hahn, nicknamed the "radioactive boy scout", ordered decommissioned smoke detectors by the dozens to extract the tiny clumps of americium.  There are also some historical sources for some radioactive elements that were discovered before their dangers were known: in the early 20th century, radium paint was used to create glow-in-the-dark clocks and watches.  Hahn got his nice radium sample when he found an old clock in an antique shop with a vial of spare paint to refresh the dials.   Hahn's success in acquiring radioactive elements eventually led to a national security incident, and the need for a federal cleanup of his backyard.  And if you look at the scary online photo of Hahn today, covered in open sores from the effects of radiation, your admiration for his element-collecting skills might be dampened somewhat.

    Another issue is that such a collection can never be truly complete.  As you get to the heaviest elements, they are not only highly radioactive, but decay almost instantaneously when formed.  The highest-numbered element actually synthesized so far is number 118, ununoctium, but only four atoms of this element have ever been known to exist.  All were synthesized in scientific labs and detected only indirectly.  Good luck collecting that!

    Anyway, if you're a fan of this podcast, you probably realize that you can appreciate the periodic table from a poster or chart with or without actually holding the elements.  In the show notes you'll find a link to Theodore Gray's book and the beautiful associated poster, which I highly recommend.  The existence of physical objects is just a minor corollary of this mathematically elegant table, and the underlying quantum mechanical laws that make it possible; why sully it by obsessively acquiring a bunch of rocks, liquids, and powders that could potentially kill you anyway? 
      
    And this has been your Math Mutation for today.
  • Element Collecting at Wikipedia
  • Another Element Collecting Page
  • Page showing how to collect americium from smoke detectors
  • Periodic Table at Wikipedia
  • David Hahn at Wikipedia
  • Theodore Gray's book
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