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? ]

Reading the Comics, June 27, 2014: Pretty Easy Edition


I don’t mean to complain, because it really is a lot of fun to do these comic strip roundups, but Comic Strip Master Command has been sending a flood of comics my way. I hope it’s not overwhelming readers, or me. The downside of the great number of mathematics-themed comics this past week has been that they aren’t very deep examples, but, what the heck. Many of them are interesting anyway. As usual I’m including examples of the Comics Kingdom and the Creators.com comics because I’m not yet confident how long those links remain visible to non-subscribers.

A caveman invents the 'science fictiony' number of eleven.

Mike Peters’s _Mother Goose and Grimm_ for the 23rd of June, 2014

Mike Peters’ Mother Goose and Grimm (June 23) presents the cavemen-inventing-stuff pattern and the invention of a “science-fictiony” number. This is amusing, sure, but the dynamic is historically valid: it does seem like the counting numbers (1, 2, 3, and so on) were more or less intuitive, but negative numbers? Rationals? Irrationals? Zero? They required development and some fairly sophisticated reasoning to think of. You get a hint of the suspicion with which the newly-realized numbers were viewed when you think of the connotations of terms like “complex” numbers, or “imaginary” numbers, or even “negative” numbers. For that matter, Arabic numerals required some time for Europeans — who were comfortable with Roman numerals — to feel comfortable with; histories of mathematics will mention how Arabic numerals were viewed with suspicion and sometimes banned as being too easy for merchants or bankers to use to defraud customers who didn’t know what the symbols meant or how to use them.

Thom Bluemel’s Birdbrains (June 23) also takes us to the dawn of time and the invention of the calendar. Calendars are deeply intwined with mathematics, as they typically try to reconcile several things that aren’t quite perfectly reconcilable: the changes of the season, the cycles of the moon, the position of the sun in the sky, the length of the day. But the attempt to do as well as possible, using rules easy enough for normal human beings to understand, is productive.

Mark Pett’s Lucky Cow (June 23, rerun) lets Neil do some accounting the modern old-fashioned way. I trust there are abacus applications out there; somewhere in my pile of links I had a Javascript-based slide rule simulator, after all. I never quite got abacus use myself.

Mark Parisi’s Off The Mark (June 23) shows off one of those little hazards of skywriting and mathematical symbols. I admit the context threw me; I had to look again to read the birds as the less-than sign.

Dilton Doiley and Jughead find wonder in the vast and the numerous.

Henry Scarpelli and Craig Boldman’s _Archie_ rerun on the 24th of June, 2014

Henry Scarpelli and Craig Boldman’s Archie (June 24) has resident nerd Dilton Doiley pondering the vastness of the sky and the number of stars and feel the sense of wonder that inspires. The mind being filled with ever-increasing wonder and awe isn’t a unique sentiment, and thinking hard of very large, very numerous things is one of the paths to that sensation. Jughead has a similar feeling, evidently.

The orangutan who 'probably never had a thought in his life' works out a pretty involved derivative using the product and chain rules.

Mort Walker (Addison)’s _Boner’s Ark_ rerun the 26th of June, 2014 (originally run the 31st of July, 1968).

Mort Walker (“Addison”)’s Boner’s Ark (June 26, originally run July 31, 1968) features once again the motif of “a bit of calculus proves someone is really smart”. The orangutan’s working out of a derivative starts out well, too, using the product rule correctly through the first three lines, a point at which the chain rule and the derivative of the arccotangent function conspire to make things look really complicated. I admit I’m impressed Walker went to the effort to get things right that far in and wonder where he got the derivative worked out. It’s not one of the standard formulas you’d find in every calculus textbook, although you might find it as one of the more involved homework problem for Calculus I.

Mark Pett’s Lucky Cow comes up again (June 26, rerun) sees Neil a little gloomy at the results of a test coming back “negative”, a joke I remember encountering on The Office (US) too. It brings up the question of why, given the connotations of the words, a “positive” test result is usually a bad thing and a “negative” one a good, and it back to the language of statistics. Normally a test — medical, engineering, or otherwise — is really checking to see how often some phenomenon occurs within a given sample. But the phenomenon will normally happen a little bit anyway, even if nothing untoward is happening. It also won’t normally happen at exactly the same rate, even if there’s nothing to worry about. What statistics asks, then, is, “is this phenomenon happening so much in this sample space that it’s not plausible for it to just be coincidence?” And in that context, yeah, everything being normal is the negative result. What happens isn’t suspicious. Of course, Neil has other issues, here.

Chip Dunham’s Overboard (June 26) plays on the fact that “half” does have a real proper meaning, but will get used pretty casually when people aren’t being careful. Or when dinner’s involved.

Percy Crosby’s Skippy (June 26, rerun) must have originally run in March sometime, and it does have Skippy and the other kid arguing about how many months it is until Christmas. Counting intervals like this does invite what’s termed a “fencepost error”, and the kids present it perfectly: do you count the month you’re in if you want to count how many months until something? There isn’t really an absolutely correct answer, though; you and the other party just have to agree on whether you mean, say, the pages on the calendar you’ll go through between today and Christmas, or whether you mean how many more times you’ll pass the 24th of the month until you get to Christmas. You will see this same dynamic in every argument about conventions ever. Two spaces after the end of the sentence.

Moose has a fair idea for how to get through the algebra book in time.

Henry Scarpelli and Craig Boldman’s _Archie_ rerun for the 27th of June, 2014.

In Henry Scarpelli and Craig Boldman’s Archie (June 27, rerun), Moose has a pretty good answer to how to get the whole algebra book read in time. It’d be nice if it quite worked that way.

Mel Henze’s Gentle Creatures (June 27) has the characters working out just what the calculations for a jump into hyperspace would be. I admit I’ve always wondered just what the calculations for that sort of thing are, but that’s a bit silly of me.

In A Really Big Universe


I’d got to thinking idly about Olbers’ Paradox, the classic question of why the night sky is dark. It’s named for Heinrich Wilhelm Olbers, 1758-1840, who of course was not the first person to pose the problem nor to give a convincing answer to it, but, that’s the way naming rights go.

It doesn’t sound like much of a question at first, after all, it’s night. But if we suppose the universe is infinitely large and is infinitely old, then, along the path of any direction you look in the sky, day or night, there’ll be a star. The star may be very far away, so that it’s very faint; but it takes up much less of the sky from being so far away. The result is that the star’s intensity, as a function of how much of the sky it takes up, is no smaller. And there’ll be stars shining just as intensely in directions that are near to that first star. The sky in an infinitely large, infinitely old universe should be a wall of stars.

Oh, some stars will be dimmer than average, and some brighter, but that doesn’t matter much. We can suppose the average star is of average brightness and average size for reasons that are right there in the name of the thing; it makes the reasoning a little simpler and doesn’t change the result.

The reason there is darkness is that our universe is neither infinitely large nor infinitely old. There aren’t enough stars to fill the sky and there’s not enough time for the light from all of them to get to us.

But we can still imagine standing on a planet in such an Olbers Universe (to save myself writing “infinitely large and infinitely old” too many times), with enough vastness and enough time to have a night sky that looks like a shell of starlight, and that’s what I was pondering. What might we see if you looked at the sky, in these conditions?

Well, light, obviously; we can imagine the sky looking as bright as the sun, but in all directions above the horizon. The sun takes up a very tiny piece of the sky — it’s about as wide across as your thumb, held at arm’s length, and try it if you don’t believe me (better, try it with the Moon, which is about the same size as the Sun and easier to look at) — so, multiply that brightness by the difference between your thumb and the sky and imagine the investment in sunglasses this requires.

It’s worse than that, though. Yes, in any direction you look there’ll be a star, but if you imagine going on in that direction there’ll be another star, eventually. And another one past that, and another past that yet. And the light — the energy — of those stars shining doesn’t disappear because there’s a star between it and the viewer. The heat will just go into warming up the stars in its path and get radiated through.

This is why interstellar dust, or planets, or other non-radiating bodies doesn’t answer why the sky could be dark in a vast enough universe. Anything that gets enough heat put into it will start to glow and start to shine from that light. The stars will slow down the waves of heat from the stars behind them, but given enough time, it will get through, and in an infinitely old universe, there is enough time.

The conclusion, then, is that our planet in an Olbers Universe would get an infinite amount of heat pouring onto it, at all times. It’s hard to see how life could possibly exist in the circumstance; water would boil away — rock would boil away — and the planet just would evaporate into dust.

Things get worse, though: it’s not just our planet that would get boiled away like this, but as far as I can tell, the stars too. Each star would be getting an infinite amount of heat pouring into it. It seems to me this requires the matter making up the stars to get so hot it would boil away, just as the atmosphere and water and surface of the imagined planet would, until the star — until all stars — disintegrate. At this point I have to think of the great super-science space-opera writers of the early 20th century, listening to the description of a wave of heat that boils away a star, and sniffing, “Amateurs. Come back when you can boil a galaxy instead”. Well, the galaxy would boil too, for the same reasons.

Even once the stars have managed to destroy themselves, though, the remaining atoms would still have a temperature, and would still radiate faint light. And that faint light, multiplied by the infinitely many atoms and all the time they have, would still accumulate to an infinitely great heat. I don’t know how hot you have to get to boil a proton into nothingness — or a quark — but if there is any temperature that does it, it’d be able to.

So the result, I had to conclude, is that an infinitely large, infinitely old universe could exist only if it didn’t have anything in it, or at least if it had nothing that wasn’t at absolute zero in it. This seems like a pretty dismal result and left me looking pretty moody for a while, even I was sure that EE “Doc” Smith would smile at me for working out the heat-death of quarks.

Of course, there’s no reason that a universe has to, or even should, be pleasing to imagine. And there is a little thread of hope for life, or at least existence, in a Olbers Universe.

All the destruction-of-everything comes about from the infinitely large number of stars, or other radiating bodies, in the universe. If there’s only finitely much matter in the universe, then, their total energy doesn’t have to add up to the point of self-destruction. This means giving up an assumption that was slipped into my Olbers Universe without anyone noticing: the idea that it’s about uniformly distributed. If you compare any two volumes of equal size, from any time, they have about the same number of stars in them. This is known in cosmology as “isotropy”.

Our universe seems to have this isotropy. Oh, there are spots where you can find many stars (like the center of a galaxy) and spots where there are few (like, the space in-between galaxies), but the galaxies themselves seem to be pretty uniformly distributed.

But an imagined universe doesn’t have to have this property. If we suppose an Olbers Universe without then we can have stars and planets and maybe even life. It could even have many times the mass, the number of stars and everything, that our universe has, spread across something much bigger than our universe. But it does mean that this infinitely large, infinitely old universe will have all its matter clumped together into some section, and nearly all the space — in a universe with an incredible amount of space — will be empty.

I suppose that’s better than a universe with nothing at all, but somehow only a little better. Even though it could be a universe with more stars and more space occupied than our universe has, that infinitely vast emptiness still haunts me.

(I’d like to note, by the way, that all this universe-building and reasoning hasn’t required any equations or anything like that. One could argue this has diverted from mathematics and cosmology into philosophy, and I wouldn’t dispute that, but can imagine philosophers might.)

Reading the Comics, June 22, 2014: Name-Dropping Stuff Edition


Comic Strip Master Command apparently really is ordering strips to finish their mathematics jokes before the summer vacation sets in, based on how many we’ve gotten in the past week. I confess this set doesn’t give me so much to write about; it’s more a set of mathematics things getting name-dropped. And there’s always something, isn’t there?

Tom Thaves’s Frank and Ernest (June 17) showcases a particularly severe form of math anxiety. I’m sympathetic to people who’re afraid of mathematics, naturally; it’s rotten being denied a big and wonderful and beautiful part of human ingenuity. I don’t know where math anxiety comes from, although I’d imagine a lot of it comes from that mix of doing something you aren’t quite sure you’re doing correctly and being hit too severely with a sense of rejection in the case that you did it wrong. I’d like to think that recreational mathematics puzzles would help overcome that, but I have no evidence that it does, just my hunch that getting to play with numbers and pictures and logic puzzles is good for you.

Russell Myers’ Broom Hilda (June 18) taunts the schoolkid Nerwin with the way we “used to do math with our brains instead of calculators”. One hesitates to know too much about the continuity of Broom Hilda, but I believe she’s over a thousand years old and so when she was Nerwin’s age they didn’t even have Arabic numerals just yet. I’ll assume there’s some way she’d have been in school then. (Also, given how long Broom Hilda‘s been running Nerwin did used to be in classes that did mathematics without calculators.)

Hagar can't count how many beers he had, and so proposes getting a math tutor.

Chris Browne’s Hagar the Horrible for the 19th of June, 2014.

Chris Brown’s Hagar the Horrible (June 19) tries to get itself cut out and put up on the walls of math tutors’ offices. Good luck.

Crankshaft does arithmetic by counting on his fingers, including long division.

Tom Batiuk and Chuck Ayers’s Crankshaft for the 20th of June, 2014.

Tom Batiuk and Chuck Ayers’ Crankshaft (June 20) spent a couple days this week explaining how he just counts on fingers to do his arithmetic. It’s a curious echo of the storyline several years ago revealing Crankshaft suffered from Backstory Illiteracy, in which we suddenly learned he had gone all his life without knowing how to read. I hesitate to agree with him but, yeah, there’s no shame in counting on your fingers if that does all the mathematics you need to do and you get the answers you want reliably. I don’t know what his long division thing is; if it weren’t for Tom Batiuk writing the comic strip I’d call it whimsy.

Keith Knight’s The Knight Life carried on with the story of the personal statistician this week. I think the entry from the 20th is most representative. It’s fine, and fun, to gather all kinds of data about whatever you encounter, but if you aren’t going to study the data and then act on its advice you’re wasting your time. The personal statistician ends up quitting the job.

Steve McGarry’s kid-activity feature KidTown (June 22) promotes the idea of numbers as a thing to notice in the newspapers, and includes a couple of activities, one featuring a maze to be navigated by way of multiples of seven. It also has one of those math tricks where you let someone else pick a number, give him a set of mathematical operations to do, and then you can tell them what the result is. It seems to me working out why that scheme works is a good bit of practice for someone learning algebra, and developing your own mathematics trick that works along this line is further good practice.