Reading the Comics, July 28, 2014: Homework in an Amusement Park Edition


I don’t think my standards for mathematics content in comic strips are seriously lowering, but the strips do seem to be coming pretty often for the summer break. I admit I’m including one of these strips just because it lets me talk about something I saw at an amusement park, though. I have my weaknesses.

Harley Schwadron’s 9 to 5 (July 25) builds its joke around the ambiguity of saying a salary is six (or some other number) of figures, if you don’t specify what side of the decimal they’re on. That’s an ordinary enough gag, although the size of a number can itself be an interesting thing to know. The number of digits it takes to write a number down corresponds, roughly, with the logarithm of a number, and in the olden days a lot of computations depended on logarithms: multiplying two numbers is equivalent to adding their logarithms; dividing two numbers, subtracting their logarithms. And addition and subtraction are normally easier than multiplication and division. Similarly, raising one number to a power becomes multiplying one number by the logarithm of another, and multiplication is easier than exponentiation. So counting the number of digits in a number might be something anyway.

Steve Breen and Mike Thompson’s Grand Avenue (July 25) has the kids mention something as being “like going to an amusement park to do math homework”, which gives me a chance to share this incident. Last year my love and I were in the Cedar Point amusement park (in Sandusky, Ohio), and went to the coffee shop. We saw one guy sitting at a counter, with his laptop and a bunch of papers sprawled out, looking pretty much like we do when we’re grading papers, and we thought initially that it was so very sad that someone would be so busy at work that (we presumed) he couldn’t even really participate in the family expedition to the amusement park.

And then we remembered: not everybody lives a couple hours away from an amusement park. If we lived, say, fifteen minutes from a park we had season passes to, we’d certainly at least sometimes take our grading work to the park, so we could get it done in an environment we liked and reward ourselves for getting done with a couple roller coasters and maybe the Cedar Downs carousel (which is worth an entry around these parts anyway). To grade, anyway; I’d never have the courage to bring my laptop to the coffee shop. So I guess all I’m saying is, I have a context in which yes, I could imagine going to an amusement park to grade math homework at least.

Wulff and Morgenthaler Truth Facts (July 25) makes a Venn diagram joke in service of asserting that only people who don’t understand statistics would play the lottery. This is an understandable attitude of Wulff and Morgenthaler, and of many, many people who make the same claim. The expectation value — the amount you expect to win some amount, times the probability you will win that amount, minus the cost of the ticket — is negative for all but the most extremely oversized lottery payouts, and the most extremely oversized lottery payouts still give you odds of winning so tiny that you really aren’t hurting your chances by not buying a ticket. However, the smugness behind the attitude bothers me — I’m generally bothered by smugness — and jokes like this one contain the assumption that the only sensible way to live is a ruthless profit-and-loss calculation to life that even Jeremy Bentham might say is a bit much. For the typical person, buying a lottery ticket is a bit of a lark, a couple dollars of disposable income spent because, what the heck, it’s about what you’d spend on one and a third sodas and you aren’t that thirsty. Lottery pools with coworkers or friends make it a small but fun social activity, too. That something is a net loss of money does not mean it is necessarily foolish. (This isn’t to say it’s wise, either, but I’d generally like a little more sympathy for people’s minor bits of recreational foolishness.)

Marc Anderson’s Andertoons (July 27) does a spot of wordplay about the meaning of “aftermath”. I can’t think of much to say about this, so let me just mention that Florian Cajori’s A History of Mathematical Notations reports (section 201) that the + symbol for addition appears to trace from writing “et”, meaning and, a good deal and the letters merging together and simplifying from that. This seems plausible enough on its face, but it does cause me to reflect that the & symbol also is credited as a symbol born from writing “et” a lot. (Here, picture writing Et and letting the middle and lower horizontal strokes of the E merge with the cross bar and the lowest point of the t.)

Berkeley Breathed’s Bloom County (July 27, rerun from, I believe, July of 1988) is one of the earliest appearances I can remember of the Grand Unification appearing in popular culture, certainly in comic strips. Unifications have a long and grand history in mathematics and physics in explaining things which look very different by the same principles, with the first to really draw attention probably being Descartes showing that algebra and geometry could be understood as a single thing, and problems difficult in one field could be easy in the other. In physics, the most thrilling unification was probably the explaining of electricity, magnetism, and light as the same thing in the 19th century; being able to explain many varied phenomena with some simple principles is just so compelling. General relativity shows that we can interpret accelerations and gravitation as the same thing; and in the late 20th century, physicists found that it’s possible to use a single framework to explain both electromagnetism and the forces that hold subatomic particles together and that break them apart.

It’s not yet known how to explain gravity and quantum mechanics in the same, coherent, frame. It’s generally assumed they can be reconciled, although I suppose there’s no logical reason they have to be. Finding a unification — or a proof they can’t be unified — would certainly be one of the great moments of mathematical physics.

The idea of the grand unification theory as an explanation for everything is … well, fair enough. A grand unification theory should be able to explain what particles in the universe exist, and what forces they use to interact, and from there it would seem like the rest of reality is details. Perhaps so, but it’s a long way to go from a simple starting point to explaining something as complicated as a penguin. I guess what I’m saying is I doubt Oliver would notice the non-existence of Opus in the first couple pages of his work.

Thom Bluemel’s Birdbrains (July 28) takes us back to the origin of numbers. It also makes me realize I don’t know what’s the first number that we know of people discovering. What I mean is, it seems likely that humans are just able to recognize a handful of numbers, like one and two and maybe up to six or so, based on how babies and animals can recognize something funny if the counts of small numbers of things don’t make sense. And larger numbers were certainly known to antiquity; probably the fact that numbers keep going on forever was known to antiquity. And some special numbers with interesting or difficult properties, like pi or the square root of two, were known so long ago we can’t say who discovered them. But then there are numbers like the Euler-Mascheroni constant, which are known and recognized as important things, and we can say reasonably well who discovered them. So what is the first number with a known discoverer?

Lewis Carroll and my Playing With Universes


I wanted to explain what’s going on that my little toy universes with three kinds of elements changing to one another keep settling down to steady and unchanging distributions of stuff. I can’t figure a way to do that other than to introduce some actual mathematics notation, and I’m aware that people often find that sort of thing off-putting, or terrifying, or at the very least unnerving.

There’s fair reason to: the entire point of notation is to write down a lot of information in a way that’s compact or easy to manipulate. Using it at all assumes that the writer, and the reader, are familiar with enough of the background that they don’t have to have it explained at each reference. To someone who isn’t familiar with the topic, then, the notation looks like symbols written down without context and without explanation. It’s much like wandering into an Internet forum where all the local acronyms are unfamiliar, the in-jokes are heavy on the ground, and for some reason nobody actually spells out Dave Barry’s name in full.

Let me start by looking at the descriptions of my toy universe: it’s made up of a certain amount of hydrogen, a certain amount of iron, and a certain amount of uranium. Since I’m not trying to describe, like, where these elements are or how they assemble into toy stars or anything like that, I can describe everything that I find interesting about this universe with three numbers. I had written those out as “40% hydrogen, 35% iron, 25% uranium”, for example, or “10% hydrogen, 60% iron, 30% uranium”, or whatever the combination happens to be. If I write the elements in the same order each time, though, I don’t really need to add “hydrogen” and “iron” and “uranium” after the numbers, and if I’m always looking at percentages I don’t even need to add the percent symbol. I can just list the numbers and let the “percent hydrogen” or “percent iron” or “percent uranium” be implicit: “40, 35, 25”, for one universe’s distribution, or “10, 60, 30” for another.

Letting the position of where a number is written carry information is a neat and easy way to save effort, and when you notice what’s happening you realize it’s done all the time: it’s how writing the date as “7/27/14” makes any sense, or how a sports scoreboard might compactly describe the course of the game:

0 1 0   1 2 0   0 0 4   8 13 1
2 0 0   4 0 0   0 0 1   7 15 0

To use the notation you need to understand how the position encodes information. “7/27/14” doesn’t make sense unless you know the first number is the month, the second the day within the month, and the third the year in the current century, and that there’s an equally strong convention putting the day within the month first and the month in the year second presents hazards when the information is ambiguous. Reading the box score requires knowing the top row reflects the performance of the visitor’s team, the bottom row the home team, and the first nine columns count the runs by each team in each inning, while the last three columns are the total count of runs, hits, and errors by that row’s team.

When you put together the numbers describing something into a rectangular grid, that’s termed a matrix of numbers. The box score for that imaginary baseball game is obviously one, but it’s also a matrix if I just write the numbers describing my toy universe in a row, or a column:

40
35
25

or

10
60
30

If a matrix has just the one column, it’s often called a vector. If a matrix has the same number of rows as it has columns, it’s called a square matrix. Matrices and vectors are also usually written with either straight brackets or curled parentheses around them, left and right, but that’s annoying to do in HTML so please just pretend.

The matrix as mathematicians know it today got put into a logically rigorous form around 1850 largely by the work of James Joseph Sylvester and Arthur Cayley, leading British mathematicians who also spent time teaching in the United States. Both are fascinating people, Sylvester for his love of poetry and language and for an alleged incident while briefly teaching at the University of Virginia which the MacTutor archive of mathematician biographies, citing L S Feuer, describes so: “A student who had been reading a newspaper in one of Sylvester’s lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one os his elder brothers was living.” MacTutor goes on to give reasons why this story may be somewhat distorted, although it does suggest one solution to the problem of students watching their phones in class.

Cayley, meanwhile, competes with Leonhard Euler for prolific range in a mathematician. MacTutor cites him as having at least nine hundred published papers, covering pretty much all of modern mathematics, including work that would underlie quantum mechanics and non-Euclidean geometry. He wrote about 250 papers in the fourteen years he was working as a lawyer, which would by itself have made him a prolific mathematician. If you need to bluff your way through a mathematical conversation, saying “Cayley” and following it with any random noun will probably allow you to pass.

MathWorld mentions, to my delight, that Lewis Carroll, in his secret guise as Charles Dodgson, came in to the world of matrices in 1867 with an objection to the very word. In writing about them, Dodgson said, “”I am aware that the word `Matrix’ is already in use to express the very meaning for which I use the word `Block'; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. He’s got a fair point, really, but there wasn’t much to be done in 1867 to change the word, and it’s only gotten more entrenched since then.

Reading the Comics, July 24, 2014: Math Is Just Hard Stuff, Right? Edition


Maybe there is no pattern to how Comic Strip Master Command directs the making of mathematics-themed comic strips. It hasn’t quite been a week since I had enough to gather up again. But it’s clearly the summertime anyway; the most common theme this time seems to be just that mathematics is some hard stuff, without digging much into particular subjects. I can work with that.

Pab Sungenis’s The New Adventures of Queen Victoria (July 19) brings in Erwin Schrödinger and his in-strip cat Barfly for a knock-knock joke about proof, with Andrew Wiles’s name dropped probably because he’s the only person who’s gotten to be famous for a mathematical proof. Wiles certainly deserves fame for proving Fermat’s Last Theorem and opening up what I understand to be a useful new field for mathematical research (Fermat’s Last Theorem by itself is nice but unimportant; the tools developed to prove it, though, that’s worthwhile), but remembering only Wiles does slight Richard Taylor, whose help Wiles needed to close a flaw in his proof.

Incidentally I don’t know why the cat is named Barfly. It has the feel to me of a name that was a punchline for one strip and then Sungenis felt stuck with it. As Thomas Dye of the web comic Newshounds said, “Joke names’ll kill you”. (I’m inclined to think that funny names can work, as the Marx Brotehrs, Fred Allen, and Vic and Sade did well with them, but they have to be a less demanding kind of funny.)

John Deering’s Strange Brew (July 19) uses a panel full of mathematical symbols scrawled out as the representation of “this is something really hard being worked out”. I suppose this one could also be filed under “rocket science themed comics”, but it comes from almost the first problem of mathematical physics: if you shoot something straight up, how long will it take to fall back down? The faster the thing starts up, the longer it takes to fall back, until at some speed — the escape velocity — it never comes back. This is because the size of the gravitational attraction between two things decreases as they get farther apart. At or above the escape velocity, the thing has enough speed that all the pulling of gravity, from the planet or moon or whatever you’re escaping from, will not suffice to slow the thing down to a stop and make it fall back down.

The escape velocity depends on the size of the planet or moon or sun or galaxy or whatever you’re escaping from, of course, and how close to the surface (or center) you start from. It also assumes you’re talking about the speed when the thing starts flying away, that is, that the thing doesn’t fire rockets or get a speed boost by flying past another planet or anything like that. And things don’t have to reach the escape velocity to be useful. Nothing that’s in earth orbit has reached the earth’s escape velocity, for example. I suppose that last case is akin to how you can still get some stuff done without getting out of the recliner.

Mel Henze’s Gentle Creatures (July 21) uses mathematics as the standard for proving intelligence exists. I’ve got a vested interest in supporting that proposition, but I can’t bring myself to say more than that it shows a particular kind of intelligence exists. I appreciate the equation of the final panel, though, as it can be pretty well generalized.

To disguise a sports venue it's labelled ``Math Arena'', with ``lectures on the actual odds of beating the casino''.

Bill Holbrook’s _Safe Havens_ for the 22nd of July, 2014.

Bill Holbrook’s Safe Havens (July 22) plays on mathematics’ reputation of being not very much a crowd-pleasing activity. That’s all right, although I think Holbrook makes a mistake by having the arena claim to offer a “lecture on the actual odds of beating the casino”, since the mathematics of gambling is just the sort of mathematics I think would draw a crowd. Probability enjoys a particular sweet spot for popular treatment: many problems don’t require great amounts of background to understand, and have results that are surprising, but which have reasons that are easy to follow and don’t require sophisticated arguments, and are about problems that are easy to imagine or easy to find interesting: cards being drawn, dice being rolled, coincidences being found, or secrets being revealed. I understand Holbrook’s editorial cartoon-type point behind the lecture notice he put up, but the venue would have better scared off audiences if it offered a lecture on, say, “Chromatic polynomials for rigidly achiral graphs: new work on Yamada’s invariant”. I’m not sure I could even explain that title in 1200 words.

Missy Meyer’s Holiday Doodles (July 22) revelas to me that apparently the 22nd of July was “Casual Pi Day”. Yeah, I suppose that passes. I didn’t see much about it in my Twitter feed, but maybe I need some more acquaintances who don’t write dates American-fashion.

Thom Bluemel’s Birdbrains (July 24) again uses mathematics — particularly, Calculus — as not just the marker for intelligence but also as The Thing which will decide whether a kid goes on to success in life. I think the dolphin (I guess it’s a dolphin?) parent is being particularly horrible here, as it’s not as if a “B+” is in any way a grade to be ashamed of, and telling kids it is either drives them to give up on caring about grades, or makes them send whiny e-mails to their instructors about how they need this grade and don’t understand why they can’t just do some make-up work for it. Anyway, it makes the kid miserable, it makes the kid’s teachers or professors miserable, and for crying out loud, it’s a B+.

(I’m also not sure whether a dolphin would consider a career at Sea World success in life, but that’s a separate and very sad issue.)

Next In A Continuing Series


For today’s entry in the popular “I suppose everybody heard about this already like five years ago but I just found out about it now”, there’s the Online Encyclopedia of Integer Sequences, which is a half-century-old database (!) of various commonly appearing sequences of integers. It started, apparently, when Neil J A Sloane (a graduate student at Cornell University) needed to know the next terms in a sequence describing a particular property of trees, and he couldn’t find a way to look it up and so we got what I imagine to be that wonderful blend of frustration (“it should be easy to find this”) and procrastination (“surely having this settled once and for all will speed my dissertation”) that produces great things.

It’s even got a search engine, so that if you have the start of a sequence — say, “1, 4, 5, 16, 17, 20, 21” — it can find whether there’s any noteworthy sequences which begin that way and even give you a formula for finding successive terms, programming code for the terms, places in the literature where it might have appeared, and other neat little bits.

This isn’t foolproof, of course. Deductive logic will tell you that just because you know the first (say) ten terms in a sequence you don’t actually know what the eleventh will be. There are literally infinitely many possible successors. However, we’re not looking for deductive inevitability with this sort of search engine. We’re supposing that our sequence starts off describing some pattern that can be described by some rule that looks simple and attractive to human eyes. (So maybe my example doesn’t quite qualify, though their name for it makes it sound pretty nice.) There’s bits of whimsy (see the first link I posted), and chances to discover stuff I never heard of before (eg, the Wilson Primes: the encyclopedia says it’s believed there are infinitely many of them, but only three are known — 5, 13, and 563, with the next term unknown but certainly larger than 20,000,000,000,000), and plenty of stuff about poker and calendars.

Anyway, it’s got that appeal of a good reference tome in that you can just wander around it all afternoon and keep finding stuff that makes you say “huh”. (There’s a thing called Canada Perfect Numbers, but there are only four of them.)


On the title: some may protest, correctly, that a sequence and a series are very different things. They are correct: mathematically, a sequence is just a string of numbers, while a series is the sum of the terms in a sequence, and so is a single number. It doesn’t matter. Titles obey a logic of their own.

Reading the Comics, July 18, 2014: Summer Doldrums Edition


Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.

Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.

Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.

Richard Thompson’s Richard’s Poor Almanac (July 7, rerun) brings back our favorite set of infinite monkeys, here, to discuss their ambitious book set at the Museum of Natural History.

Tom Thaves’s Frank and Ernest (July 16) builds on the (true) point that the ancient Greeks had no symbol for zero, and would probably have had a fair number of objections to the concept.

'The day Einstein got the wind knocked out of his sails': Einstein tells his wife he's discovered the theory of relativity.

Joe Martin’s _Mr Boffo_ strip for the 18th of July, 2014.

Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.

Charles Schulz’s Peanuts (July 18) — a rerun, of course, from the 21st of July, 1967 — mentions Sally going to Summer School and learning all about the astronomical details of summertime. Astronomy has always been one of the things driving mathematical discovery, but I admit, thinking mostly this would be a good chance to point out Dr Helmer Aslaksen’s page describing the relationship between the solstices and the times of earliest and latest sunrise (and sunset). It’s not quite as easy as finding when the days are longest and shortest. Dr Aslaksen has a number of fascinating astronomy- and calendar-based pages which I think worth reading, so, I hope you enjoy.

What’s Going On In The Old Universe


Last time in this infinitely-old universe puzzle, we found that by making a universe of only three kinds of atoms (hydrogen, iron, and uranium) which shifted to one another with fixed chances over the course of time, we’d end up with the same distribution of atoms regardless of what the distribution of hydrogen, iron, and uranium was to start with. That seems like it might require explanation.

(For people who want to join us late without re-reading: I got to wondering what the universe might look like if it just ran on forever, stars fusing lighter elements into heavier ones, heavier elements fissioning into lighter ones. So I looked at a toy model where there were three kinds of atoms, dubbed hydrogen for the lighter elements, iron for the middle, and uranium for the heaviest, and made up some numbers saying how likely hydrogen was to be turned into heavier atoms over the course of a billion years, how likely iron was to be turned into something heavier or lighter, and how likely uranium was to be turned into lighter atoms. And sure enough, if the rates of change stay constant, then the universe goes from any initial distribution of atoms to a single, unchanging-ever-after mix in surprisingly little time, considering it’s got a literal eternity to putter around.)

The first question, it seems, is whether I happened to pick a freak set of numbers for the change of one kind of atom to another. It’d be a stroke of luck, but, these things happen. In my first model, I gave hydrogen a 25 percent chance of turning to iron, and no chance of turning to helium, in a billion years. Let’s change that so any given atom has a 20 percent chance of turning to iron and a 20 percent chance of turning to uranium. Similarly, instead of iron having no chance of turning to hydrogen and a 40 percent chance of turning to uranium, let’s try giving each iron atom a 25 percent chance of becoming hydrogen and a 25 percent chance of becoming uranium. Uranium, first time around, had a 40 percent chance of becoming hydrogen and a 40 percent chance of becoming iron. Let me change that to a 60 percent chance of becoming hydrogen and a 20 percent chance of becoming iron.

With these chances of changing, a universe that starts out purely hydrogen settles on being about 50 percent hydrogen, a little over 28 percent iron, and a little over 21 percent uranium in about ten billion years. If the universe starts out with equal amounts of hydrogen, iron, and uranium, however, it settles over the course of eight billion years to … 50 percent hydrogen, a little over 28 percent iron, and a little over 21 percent uranium. If it starts out with no hydrogen and the rest of matter evenly split between iron and uranium, then over the course of twelve billion years it gets to … 50 percent hydrogen, a litte over 28 percent iron, and a little over 21 percent uranium.

Perhaps the problem is that I’m picking these numbers, and I’m biased towards things that are pretty nice ones — halves and thirds and two-fifths and the like — and maybe that’s causing this state where the universe settles down very quickly and stops changing any. We should at least try that before supposing there’s necessarily something more than coincidence going on here.

So I set the random number generator to produce some element changes which can’t be susceptible to my bias for simple numbers. Give hydrogen a 44.5385 percent chance of staying hydrogen, a 10.4071 percent chance of becoming iron, and a 45.0544 percent chance of becoming uranium. Give iron a 25.2174 percent chance of becoming hydrogen, a 32.0355 percent chance of staying iron, and a 42.7471 percent chance of becoming uranium. Give uranium a 2.9792 percent chance of becoming hydrogen, a 48.9201 percent chance of becoming iron, and a 48.1007 percent chance of staying uranium. (Clearly, by the way, I’ve given up on picking numbers that might reflect some actual if simple version of nucleosynthesis and I’m just picking numbers for numbers’ sake. That’s all right; the question this essay is, are we stuck getting an unchanging yet infinitely old universe?)

And the same thing happens again: after nine billion years a universe starting from pure hydrogen will be about 18.7 percent hydrogen, about 35.7 percent iron, and about 45.6 percent uranium. Starting from no hydrogen, 50 percent iron, and 50 percent uranium, we get to the same distribution in again about nine billion years. A universe beginning with equal amounts hydrogen, iron, and uranium under these rules gets to the same distribution after only seven billion years.

The conclusion is this settling down doesn’t seem to be caused by picking numbers that are too particularly nice-looking or obviously attractive; and the distributions don’t seem to have an obvious link to what the probabilities of changing are. There seems to be something happening here, though admittedly we haven’t proven that rigorously. To spoil a successor article in this thread: there is something here, and it’s a big thing.

(Also, no, we’re not stuck with an unchanging universe, and good on you if you can see ways to keep the universe changing without, like, having the probability of one atom changing to another itself vary in time.)

To Build A Universe


So I kept thinking about what the distribution of elements might be in an infinitely old universe. It’s a tough problem to consider, if you want to do it exactly right, since you have to consider how stars turn lighter atoms in a blistering array of possibilities. Besides the nearly hundred different elements — which represents the count of how many protons are in the nucleus — each element has multiple isotopes — representing how many neutrons are in the nucleus — and I don’t know how many there are to consider but it’s certainly at least several hundred to deal with. There’s probably a major work in the astrophysics literature describing all the ways atoms and their isotopes can get changed over the course of a star’s lifetime, either actually existing or waiting for an indefatigable writer to make it her life’s work.

But I can make a toy model, because I want to do mathematics, and I can see what I might learn from that. This is basically a test vehicle: I want to see whether building a more accurate model is likely to be worthwhile.

For my toy model of the universe I will pretend there are only three kinds of atoms in the universe: hydrogen, iron, and uranium. These represent the lighter elements — which can fuse together to release energy — and Iron-56 — which can’t release energy by fusing into heavier or by fissioning into lighter elements — and the heavier elements — which can fission apart to release energy and lighter elements. I can describe the entire state of the universe with three numbers, saying what fraction of the universe is hydrogen, what fraction is iron, and what fraction is uranium. So these are pretty powerful toys.

Over time the stars in this universe will change some of their hydrogen into iron, and some of their iron into uranium. The uranium will change some of itself into hydrogen and into iron. How much? I’m going to make up some nice simple numbers and say that over the course of a billion years, one-quarter of all the hydrogen in the universe will be changed into iron; three-quarters of the hydrogen will remain hydrogen. Over that same time, let’s say two-fifths of all the iron in the universe will be changed to uranium, while the remaining three-fifths will remain iron. And the uranium? Well, that decays; let’s say that two-fifths of the uranium will become hydrogen, two-fifths will become iron, and the remaining one-fifth will stay uranium. If I had more elements in the universe I could make a more detailed, subtle model, and if I didn’t feel quite so lazy I might look up more researched figures for this, but, again: toy model.

I’m by the way assuming this change of elements is constant for all time and that it doesn’t depend on the current state of the universe. There are sound logical reasons behind this: to have the rate of nucleosynthesis vary in time would require me to do more work. As above: toy model.

So what happens? This depends on what we start with, sure. Let’s imagine the universe starts out made of nothing but hydrogen, so that the composition of the universe is 100% hydrogen, 0% iron, 0% uranium. After the first billion years, some of the hydrogen will be changed to iron, but there wasn’t any iron so there’s no uranium now. The universe’s composition would be 75% hydrogen, 25% iron, 0% uranium. After the next billion years three-quarters of the hydrogen becomes iron and two-fifths of the iron becomes uranium, so we’ll be at 56.25% hydrogen, 33.75% iron, 10% uranium. Another billion years passes, and once again three-quarters of the hydrogen becomes iron, two-fifths of the iron becomes uranium, and two-fifths of the uranium becomes hydrogen and another two-fifths becomes iron. This is a lot of arithmetic but the results are easy enough to find: 46.188% hydrogen, 38.313% iron, 15.5% uranium. After some more time we have 40.841% hydrogen, 40.734% iron, 18.425% uranium. It’s maybe a fair question whether the universe is going to run itself all the way down to have nothing but iron, but, the next couple billions of years show things settling down. Let me put all this in a neat little table.

Composition of the Toy Universe
Age
(billion years)
Hydrogen Iron Uranium
0 100% 0% 0%
1 75% 25% 0%
2 56.25% 33.75% 10%
3 46.188% 38.313% 15.5%
4 40.841% 40.734% 18.425%
5 38% 42.021% 19.979%
6 36.492% 42.704% 20.804%
7 35.691% 43.067% 21.242%
8 35.265% 43.260% 21.475%
9 35.039% 43.362% 21.599%
10 34.919% 43.417% 21.665%
11 34.855% 43.446% 21.700%
12 34.821% 43.461% 21.718%
13 34.803% 43.469% 21.728%
14 34.793% 43.473% 21.733%
15 34.788% 43.476% 21.736%
16 34.786% 43.477% 21.737%
17 34.784% 43.478% 21.738%
18 34.783% 43.478% 21.739%
19 34.783% 43.478% 21.739%
20 34.783% 43.478% 21.739%

We could carry on but there’s really no point: the numbers aren’t going to change again. Well, probably they’re changing a little bit, four or more places past the decimal point, but this universe has settled down to a point where just as much hydrogen is being lost to fusion as is being created by fission, and just as much uranium is created by fusion as is lost by fission, and just as much iron is being made as is being turned into uranium. There’s a balance in the universe.

At least, that’s the balance if we start out with a universe made of nothing but hydrogen. What if it started out with a different breakdown, for example, a universe that started as one-third hydrogen, one-third iron, and one-third uranium? In that case, as the universe ages, the distribution of elements goes like this:

Composition of the Toy Universe
Age
(billion years)
Hydrogen Iron Uranium
0 33.333% 33.333% 33.333%
1 38.333% 41.667% 20%
2 36.75% 42.583% 20.667%
3 35.829% 43.004% 21.167%
4 35.339% 43.226% 21.435%
5 35.078% 43.345% 21.578%
10 34.795% 43.473% 21.732%
15 34.783% 43.478% 21.739%

We’ve gotten to the same distribution, only a tiny bit faster. (It doesn’t quite get there after fourteen billion years.) I hope it won’t shock you if I say that we’d see the same thing if we started with a universe made of nothing but iron, or of nothing but uranium, or of other distributions. Some take longer to settle down than others, but, they all seem to converge on the same long-term fate for the universe.

Obviously there’s something special about this toy universe, with three kinds of atoms changing into one another at these rates, which causes it to end up at the same distribution of atoms.

In A Really Old Universe


So, my thinking about an “Olbers Universe” infinitely old and infinitely large in extent brought me to a depressing conclusion that such a universe has to be empty, or at least just about perfectly empty. But we can still ponder variations on the idea and see if that turns up anything. For example, what if we have a universe that’s infinitely old, but not infinite in extent, either because space is limited or because all the matter in the universe is in one neighborhood?

Suppose we have stars. Stars do many things. One of them is they turn elements into other elements, mostly heavier ones. For the lightest of atoms — hydrogen and helium, for example — stars create heavier elements by fusing them together. Making the heavier atoms from these lighter ones, in the net, releases energy, which is why fusion is constantly thought of as a potential power source. And that’s good for making elements up to as heavy as iron. After that point, fusion becomes a net energy drain. But heavier elements still get made as the star dies: when it can’t produce energy by fusion anymore the great mass of the star collapses on itself and that shoves together atom nucleuses regardless of the fact this soaks up more energy. (Warning: the previous description of nucleosynthesis, as it’s called, was very simplified and a little anthropomorphized, and wasn’t seriously cross-checked against developments in the field since I was a physics-and-astronomy undergraduate. Do not use it to attempt to pass your astrophysics qualifier. It’s good enough for everyday use, what with how little responsibility most of us have for stars.)

The important thing to me is that a star begins as a ball of dust, produced by earlier stars (and in our, finite universe, from the Big Bang, which produced a lot of hydrogen and helium and some of the other lightest elements), that condenses into a star, turns many of the elements into it into other elements, and then returns to a cloud of dust that mixes with other dust clouds and forms new stars.

Now. Over time, over the generations of stars, we tend to get heavier elements out of the mix. That’s pretty near straightforward mathematics: if you have nothing but hydrogen and helium — atoms that have one or two protons in the nucleus — it’s quite a trick to fuse them together into something with more than two, three, or four protons in the nucleus. If you have hydrogen, helium, lithium, and beryllium to work with — one, two, three, and four protons in the nucleus — it’s easier to get products of from two up to eight protons in the nucleus. And so on. The tendency is for each generation of stars to have relatively less hydrogen and helium and relatively more of the heavier atoms in its makeup.

So what happens if you have infinitely many generations? The first guess would be, well, stars will keep gathering together and fusing together as long as there are any elements lighter than iron, so that eventually there’d be a time when there were no (or at least no significant) amounts of elements lighter than iron, at which point the stars cease to shine. There’s nothing more to fuse together to release energy and we have a universe of iron- and heavier-metal ex-stars. I’m not sure if this is an even more depressing infinite universe than the infinitely large, infinitely old one which couldn’t have anything at all in it.

Except that this isn’t the whole story. Heavier elements than iron can release energy by fission, splitting into two or more lighter elements. Uranium and radium and a couple other elements are famous for them, but I believe every element has at least some radioactive isotopes. Popular forms of fission will produce alpha particles, which is what they named this particular type of radioactive product before they realized it was just the nucleus of a helium atom. Other types of radioactive decay will produce neutrons, which, if they’re not in the nucleus of an atom, will last an average of about fifteen minutes before decaying into a proton — a hydrogen nucleus — and some other stuff. Some more exotic forms of radioactive decay can produce protons by themselves, too. I haven’t checked the entire set of possible fission byproducts but I wouldn’t be surprised if most of the lighter elements can be formed by something’s breaking down.

In short, even if we fused the entire contents of the universe into atoms heavier than iron, we would still get out a certain amount of hydrogen and of helium, and also of other lighter elements. In short, stars turn hydrogen and helium, eventually, into very heavy elements; but the very heavy elements turn at least part of themselves back into hydrogen and helium.

So, it seems plausible, at least qualitatively, that given enough time to work there’d be a stable condition: hydrogen and helium being turned into heavier atoms at the same rate that heavier atoms are producing hydrogen and helium in their radioactive decay. And an infinitely old universe has enough time for anything.

And that’s, to me, anyway, an interesting question: what would the distribution of elements look like in an infinitely old universe?

(I should point out here that I don’t know. I would be surprised if someone in the astrophysics community has worked it out, at least in a rough form, for an as-authentic-as-possible set of assumptions about how nucleosynthesis works. But I am so ignorant of the literature I’m not sure how to even find the answer they’ve posed. I can think of it as a mathematical puzzle at least, though.)

Reading the Comics, July 3, 2014: Wulff and Morgenthaler Edition


Sorry to bring you another page of mathematics comics so soon after the last one, but, I don’t control Comic Strip Master Command. I’m not sure who does, but it’s obviously someone who isn’t paying very close attention to Mary Worth because the current psychic-child/angel-warning-about-pool-safety storyline is really going off the rails. But I can’t think of a way to get that back to mathematical topics, so let me go to safer territories instead.

Mickey's nephew figures this is rocket science.

The Disney Corporation’s _Mickey Mouse_ comic strip rerun on the 28th of June, 2014.

The Disney Corporation’s Mickey Mouse (June 28, rerun) uses the familiar old setup of mathematics stuff — here crossbred with rocket science — as establishment that someone is just way smarter than the rest of the room.

Wulff and Morgenthaler’s Truth Facts — a new strip from the people who do that WuMu which is replacing the strangely endless reruns of Get Fuzzy in your local newspaper (no, I don’t know why Get Fuzzy has been rerunning daily strips since November, and neither do its editors, so far as they’re admitting) — shows a little newspaper sidebar each day. The premise is sure to include a number of mathematics/statistics type jokes and on June 28th they went ahead with the joke that delivers statistics about statistics, so that’s out of the way.

Dave Whamond’s Reality Check (June 29) brings out two of the songs that prominently mention numbers.

Mel Henze’s Gentle Creatures (June 30) drops in a bit of mathematics technobabble for the sake of sounding all serious and science-y and all that. But “apply the standard Lagrangian model” is a better one than average since Joseph-Louis Lagrange was an astoundingly talented and omnipresent mathematician and physicist. Probably his most useful work is a recasting of Newton’s laws of physics in a form in which you don’t have to worry so much about forces at every moment and can instead look at the kinetic and potential energy of a system. This generally reduces the number of equations one has to work with to describe what’s going on, and that usually means it’s easier to understand them. That said I don’t know a specific “Lagrangian model” that would necessarily be relevant. The most popular “Lagrangian model” I can find talks about the flow of particles in a larger fluid and is popular in studying atmospheric pollutants, though the couple of medical citations stuggest it’s also useful for studying how things get transported by the bloodstream. Anyway, it’s nice to hear somebody besides Einstein get used as a science name.

Mary Beth figures if she works her apple-dividing and giving right she can get al the apple.

John Rose’s _Barney Google and Snuffy Smith_ for the 1st of July, 2014, featuring neither Barney Google nor Snuffy Smith.

John Rose’s Barney Google and Snuffy Smith (July 1) plays with division word problems and percentages and the way people can subvert the intentions of a problem given any chance.

Bill Watterson’s Calvin and Hobbes (July 1, rerun) lets Calvin’s Dad gently blow Calvin’s mind by pointing out that rotational motion means that different spots on the same object are moving at different speeds yet the object stays in one piece. When you think hard enough about it rotation is a very strange phenomenon (I suppose you could say that about any subject, though), and the difference in speeds within a single object is just part of it. Sometime we must talk about the spinning pail of water.

Wulff and Morgenthaler’s WuMo (July 1) — I named this edition after them for some reason, after all — returns to the potential for mischief in how loosely one uses the word “half”.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (July 3) dips into the well of mathematics puns. I admit I had to reread the caption before noticing where the joke was. It’s been a busy week.

June 2014 In Mathematics Blogging


And with the start of July I look over how well the mathematics blog did in June and see what I can learn from that. It seems more people are willing to read when I post stuff, which is worth knowing, I guess. After May’s near-record of 751 views and 315 visitors I expected a fall, and, yeah, it came. The number of pages viewed dropped to 492, which is … well, the fourth-highest this year at least? And the number of unique visitors fell to 194, which is actually the lowest of this year. The silver lining is this means the views per visitor, 2.54, was the second-highest since WordPress started sharing those statistics with me, so, people who come around find themselves interested. I start the month at 16,174 views total and won’t cross 17,000 at that rate come July, but we’ll see what I can do. And between WordPress and Twitter I’m (as of this writing) at exactly 400 followers, which isn’t worldshaking but is a nice big round number. I admit thinking how cool it would be if that were 400 million but I’d probably get stage fright if it were.

If one thing defined June it was “good grief but there’s a lot of mathematics comics”, which I attributed to Comic Strip Master Command ordering cartoonists to clear the subject out before summer vacation. It does mean the top five posts for June are almost comically lopsided, though:

Now, that really is something neat about triangles in the post linked above so please do read it. What I’m not clear about is why the June 16th comics post was so extremely popular; it’s nearly twice as viewed as the runner-up. If I were sure what keyword is making it so popular I’d do more with that.

Now on to the international portion of this contest: what countries are sending me the most visitors? Of course the United States comes in first, at 336 views. Denmark finished second with 17, and there was a three-way tie for third as Australia, Austria, and the United Kingdom sent sixteen each. (Singapore and Canada came in next with nine each.) I had a pretty nice crop of single-reader countries this month: Argentina, Bosnia and Herzegovina, Cambodia, Egypt, Ghana, Hong Kong, Indonesia, Japan, Paraguay, Saudi Arabia, Switzerland, and Thailand. Hong Kong, Japan, and Switzerland are repeats from last month and nobody’s got a three-month streak going.

Among the interesting search terms to bring people to me:

  • names for big numbers octillion [ happy to help? ]
  • everything to need to know about trapezoids [ I'm going to be the world's authority on trapezoids! ]
  • what does the fact that two trapezoids make a parallelogram say about tth midline [ I have some ideas but don't want to commit to anything particular ]
  • latching onto you 80 version [ I ... think I'm being asked for lyrics? ]
  • planet nebus [ I feel vaguely snarked upon, somehow ]
  • origin is the gateway to your entire gaming universe [ ... thank you? ]
  • nebus student job for uae [ Um ... I guess I can figure out a consulting fee or something if you ask? ]