Stuff I Should Read: about p-Adics


This is mostly a post for myself, so that I remember the existence of something I mean to read. I have tried downloading and putting into scattered files stuff I mean to read. I’ve also tried stuffing links of stuff I mean to read into Yojimbo. Maybe putting it here will at least let someone read the things.

Anyway, this is a short essay by Joel Abraham that’s on arxiv.org. It’s Introduction to the p-adic Space. p-adics are a method of thinking about what the real numbers are. Why we need ways to think about what the real numbers are turn up when you think carefully about where our idea of them comes from.

It’s easy to see where the counting numbers like ‘1’ and ‘2’ and ‘3’ come from. They’re part of our evolutionary heritage, the part of mathematics that we know is understood also by apes and crows and raccoons. We understand some of it before we even have language.

With some thinking, and many people helping, we can go from these counting numbers to the idea of ‘0’. And even into the negative counting numbers like ‘-4’. And by thinking about multiplication, and how to reverse multiplication, we get fractions. Rational numbers. Positive and negative, given the chance. But then what are the irrational numbers? We can work out easily there have to be irrational numbers. We can name some of them. But how to give a clear definition of the whole mass of them? It should be more than just “also the other numbers”.

The p-adic numbers are one of ways to go about this. They start with thinking what we mean for two numbers to be “close to” one another. And thinking hard about how to write numbers. and this gets to interesting insights I don’t know as well as I’d like.

For this deficiency I blame Usenet. I first noticed p-adics in the voluminous and not particularly wise rantings of a crank poster to sci.math, back in the day. I forget what point, if any, he was trying to prove. But to first notice a subject as someone’s apparently idiosyncratic scheme of rewriting numbers so that everything we were already used to was useless, and in the service of some clearly nonsense goal (I think he was maybe trying to show how the number meant by 0.99999… was somehow different from the number meant by 1), is to badly hobble it. And I followed strongly mathematical-physics classes as an undergraduate and a graduate student. It’s easy to just miss problems of number representation. (This although p-adics could offer some advantages in numerical computing. They could make more numerically stable representations of irrational numbers.)

As I say, I want to fix that, and a friend linked to this arxiv post. And now that I’ve said stuff about it in public maybe it’ll coax me into going back and reading and understanding it all. We’ll see.

Advertisements

Reading the Comics, July 22, 2017: Counter-mudgeon Edition


I’m not sure there is an overarching theme to the past week’s gifts from Comic Strip Master Command. If there is, it’s that I feel like some strips are making cranky points and I want to argue against their cases. I’m not sure what the opposite of a curmudgeon is. So I shall dub myself, pending a better idea, a counter-mudgeon. This won’t last, as it’s not really a good name, but there must be a better one somewhere. We’ll see it, now that I’ve said I don’t know what it is.

Rabbits at a chalkboard. 'The result is not at all what we expected, Von Thump. According to our calculations, parallel universes may exist, and we may also be able to link them with our own by wormholes that, in strictly mathematical terms, end up in a black top hat.'
Niklas Eriksson’s Carpe Diem for the 17th of July, 2017. First, if anyone isn’t thinking of that Pixar short then I’m not sure we can really understand each other. Second, ‘von Thump’ is a fine name for a bunny scientist and if it wasn’t ever used in the rich lore of Usenet group alt.devilbunnies I shall be disappointed. Third, Eriksson made an understandable but unfortunate mistake in composing this panel. While both rabbits are wearing glasses, they’re facing away from the viewer. It’s always correct to draw animals wearing eyeglasses, or to photograph them so. But we should get to see them in full eyeglass pelage. You’d think they would teach that in Cartoonist School or something.

Niklas Eriksson’s Carpe Diem for the 17th features the blackboard full of equations as icon for serious, deep mathematical work. It also features rabbits, although probably not for their role in shaping mathematical thinking. Rabbits and their breeding were used in the simple toy model that gave us Fibonacci numbers, famously. And the population of Arctic hares gives those of us who’ve reached differential equations a great problem to do. The ecosystem in which Arctic hares live can be modelled very simply, as hares and a generic predator. We can model how the populations of both grow with simple equations that nevertheless give us surprises. In a rich, diverse ecosystem we see a lot of population stability: one year where an animal is a little more fecund than usual doesn’t matter much. In the sparse ecosystem of the Arctic, and the one we’re building worldwide, small changes can have matter enormously. We can even produce deterministic chaos, in which if we knew exactly how many hares and predators there were, and exactly how many of them would be born and exactly how many would die, we could predict future populations. But the tiny difference between our attainable estimate and the reality, even if it’s as small as one hare too many or too few in our model, makes our predictions worthless. It’s thrilling stuff.

Vic Lee’s Pardon My Planet for the 17th reads, to me, as a word problem joke. The talk about how much change Marian should get back from Blake could be any kind of minor hassle in the real world where one friend covers the cost of something for another but expects to be repaid. But counting how many more nickels one person has than another? That’s of interest to kids and to story-problem authors. Who else worries about that count?

Fortune teller: 'All of your money problems will soon be solved, including how many more nickels Beth has than Jonathan, and how much change Marian should get back from Blake.'
Vic Lee’s Pardon My Planet for the 17th of July, 2017. I am surprised she had no questions about how many dimes Jonathan must have, although perhaps that will follow obviously from knowing the Beth nickel situation.

Jef Mallet’s Frazz for the 17th straddles that triple point joining mathematics, philosophy, and economics. It seems sensible, in an age that embraces the idea that everything can be measured, to try to quantify happiness. And it seems sensible, in age that embraces the idea that we can model and extrapolate and act on reasonable projections, to try to see what might improve our happiness. This is so even if it’s as simple as identifying what we should or shouldn’t be happy about. Caulfield is circling around the discovery of utilitarianism. It’s a philosophy that (for my money) is better-suited to problems like how ought the city arrange its bus lines than matters too integral to life. But it, too, can bring comfort.

Corey Pandolph’s Barkeater Lake rerun for the 20th features some mischievous arithmetic. I’m amused. It turns out that people do have enough of a number sense that very few people would let “17 plus 79 is 4,178” pass without comment. People might not be able to say exactly what it is, on a glance. If you answered that 17 plus 79 was 95, or 102, most people would need to stop and think about whether either was right. But they’re likely to know without thinking that it can’t be, say, 56 or 206. This, I understand, is so even for people who aren’t good at arithmetic. There is something amazing that we can do this sort of arithmetic so well, considering that there’s little obvious in the natural world that would need the human animal to add 17 and 79. There are things about how animals understand numbers which we don’t know yet.

Alex Hallatt’s Human Cull for the 21st seems almost a direct response to the Barkeater Lake rerun. Somehow “making change” is treated as the highest calling of mathematics. I suppose it has a fair claim to the title of mathematics most often done. Still, I can’t get behind Hallatt’s crankiness here, and not just because Human Cull is one of the most needlessly curmudgeonly strips I regularly read. For one, store clerks don’t need to do mathematics. The cash registers do all the mathematics that clerks might need to do, and do it very well. The machines are cheap, fast, and reliable. Not using them is an affectation. I’ll grant it gives some charm to antiques shops and boutiques where they write your receipt out by hand, but that’s for atmosphere, not reliability. And it is useful the clerk having a rough idea what the change should be. But that’s just to avoid the risk of mistakes getting through. No matter how mathematically skilled the clerk is, there’ll sometimes be a price entered wrong, or the customer’s money counted wrong, or a one-dollar bill put in the five-dollar bill’s tray, or a clerk picking up two nickels when three would have been more appropriate. We should have empathy for the people doing this work.

Reading the Comics, December 10, 2016: E = mc^2 Edition


And now I can finish off last week’s mathematically-themed comic strips. There’s a strong theme to them, for a refreshing change. It would almost be what we’d call a Comics Synchronicity, on Usenet group rec.arts.comics.strips, had they all appeared the same day. Some folks claiming to be open-minded would allow a Synchronicity for strips appearing on subsequent days or close enough in publication, but I won’t have any of that unless it suits my needs at the time.

Ernie Bushmiller’s for the 6th would fit thematically better as a Cameo Edition comic. It mentions arithmetic but only because it’s the sort of thing a student might need a cheat sheet on. I can’t fault Sluggo needing help on adding eight or multiplying by six; they’re hard. Not remembering 4 x 2 is unusual. But everybody has their own hangups. The strip originally ran the 6th of December, 1949.

People contorted to look like a 4, a 2, and a 7 bounce past Dethany's desk. She ponders: 'Performance review time ... when the company reduces people to numbers.' Wendy, previous star of the strip, tells Dethany 'You're next.' Wendy's hair is curled into an 8.
Bill holbrook’s On The Fastrack for the 7th of December, 2016. Don’t worry about the people in the first three panels; they’re just temps, and weren’t going to appear in the comic again.

Bill holbrook’s On The Fastrack for the 7th seems like it should be the anthropomorphic numerals joke for this essay. It doesn’t seem to quite fit the definition, but, what the heck.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers on the 7th starts off the run of E = mc2 jokes for this essay. This one reminds me of Gary Larson’s Far Side classic with the cleaning woman giving Einstein just that little last bit of inspiration about squaring things away. It shouldn’t surprise anyone that E equalling m times c squared isn’t a matter of what makes an attractive-looking formula. There’s good reasons when one thinks what energy and mass are to realize they’re connected like that. Einstein’s famous, deservedly, for recognizing that link and making it clear.

Mark Pett’s Lucky Cow rerun for the 7th has Claire try to use Einstein’s famous quote to look like a genius. The mathematical content is accidental. It could be anything profound yet easy to express, and it’s hard to beat the economy of “E = mc2” for both. I’d agree that it suggests Claire doesn’t know statistics well to suppose she could get a MacArthur “Genius” Grant by being overheard by a grant nominator. On the other hand, does anybody have a better idea how to get their attention?

Harley Schwadron’s 9 to 5 for the 8th completes the “E = mc2” triptych. Calling a tie with the equation on it a power tie elevates the gag for me. I don’t think of “E = mc2” as something that uses powers, even though it literally does. I suppose what gets me is that “c” is a constant number. It’s the speed of light in a vacuum. So “c2” is also a constant number. In form the equation isn’t different from “E = m times seven”, and nobody thinks of seven as a power.

Morrie Turner’s Wee Pals rerun for the 8th is a bit of mathematics wordplay. It’s also got that weird Morrie Turner thing going on where it feels unquestionably earnest and well-intentioned but prejudiced in that way smart 60s comedies would be.

Sarge demands to know who left this algebra book on his desk; Zero says not him. Sarge ignores him and asks 'Who's been figuring all over my desk pad?' Zero also unnecessarily denies it. 'Come on, whose is it?!' Zero reflects, 'Gee, he *never* picks on *me*!'
Mort Walker’s vintage Beetle Bailey for the 18th of May, 1960. Rerun the 9th of December, 2016. For me the really fascinating thing about ancient Beetle Bailey strips is that they could run today with almost no changes and yet they feel like they’re from almost a different cartoon universe from the contemporary comic. I don’t know how that is, or why it is.

Mort Walker’s Beetle Bailey for the 18th of May, 1960 was reprinted on the 9th. It mentions mathematics — algebra specifically — as the sort of thing intelligent people do. I’m going to take a leap and suppose it’s the sort of algebra done in high school about finding values of ‘x’ rather than the mathematics-major sort of algebra, done with groups and rings and fields. I wonder when holding a mop became the signifier of not just low intelligence but low ambition. It’s subverted in Jef Mallet’s Frazz, the title character of which works as a janitor to support his exercise and music habits. But it is a standard prop to signal something.

The End 2016 Mathematics A To Z: Normal Numbers


Today’s A To Z term is another of gaurish’s requests. It’s also a fun one so I’m glad to have reason to write about it.

Normal Numbers

A normal number is any real number you never heard of.

Yeah, that’s not what we say a normal number is. But that’s what a normal number is. If we could imagine the real numbers to be a stream, and that we could reach into it and pluck out a water-drop that was a single number, we know what we would likely pick. It would be an irrational number. It would be a transcendental number. And it would be a normal number.

We know normal numbers — or we would, anyway — by looking at their representation in digits. For example, π is a number that starts out 3.1415926535897931159979634685441851615905 and so on forever. Look at those digits. Some of them are 1’s. How many? How many are 2’s? How many are 3’s? Are there more than you would expect? Are there fewer? What would you expect?

Expect. That’s the key. What should we expect in the digits of any number? The numbers we work with don’t offer much help. A whole number, like 2? That has a decimal representation of a single ‘2’ and infinitely many zeroes past the decimal point. Two and a half? A single ‘2, a single ‘5’, and then infinitely many zeroes past the decimal point. One-seventh? Well, we get infinitely many 1’s, 4’s, 2’s, 8’s, 5’s, and 7’s. Never any 3’s, nor any 0’s, nor 6’s or 9’s. This doesn’t tell us anything about how often we would expect ‘8’ to appear in the digits of π.

In an normal number we get all the decimal digits. And we get each of them about one-tenth of the time. If all we had was a chart of how often digits turn up we couldn’t tell the summary of one normal number from the summary of any other normal number. Nor could we tell either from the summary of a perfectly uniform randomly drawn number.

It goes beyond single digits, though. Look at pairs of digits. How often does ’14’ turn up in the digits of a normal number? … Well, something like once for every hundred pairs of digits you draw from the random number. Look at triplets of digits. ‘141’ should turn up about once in every thousand sets of three digits. ‘1415’ should turn up about once in every ten thousand sets of four digits. Any finite string of digits should turn up, and exactly as often as any other finite string of digits.

That’s in the full representation. If you look at all the infinitely many digits the normal number has to offer. If all you have is a slice then some digits are going to be more common and some less common. That’s similar to how if you fairly toss a coin (say) forty times, there’s a good chance you’ll get tails something other than exactly twenty times. Look at the first 35 or so digits of π there’s not a zero to be found. But as you survey more digits you get closer and closer to the expected average frequency. It’s the same way coin flips get closer and closer to 50 percent tails. Zero is a rarity in the first 35 digits. It’s about one-tenth of the first 3500 digits.

The digits of a specific number are not random, not if we know what the number is. But we can be presented with a subset of its digits and have no good way of guessing what the next digit might be. That is getting into the same strange territory in which we can speak about the “chance” of a month having a Friday the 13th even though the appearances of Fridays the 13th have absolutely no randomness to them.

This has staggering implications. Some of them inspire an argument in science fiction Usenet newsgroup rec.arts.sf.written every two years or so. Probably it does so in other venues; Usenet is just my first home and love for this. In a minor point in Carl Sagan’s novel Cosmos possibly-imaginary aliens reveal there’s a pattern hidden in the digits of π. (It’s not in the movie version, which is a shame. But to include it would require people watching a computer. So that could not make for a good movie scene, we now know.) Look far enough into π, says the book, and there’s suddenly a string of digits that are nearly all zeroes, interrupted with a few ones. Arrange the zeroes and ones into a rectangle and it draws a pixel-art circle. And the aliens don’t know how something astounding like that could be.

Nonsense, respond the kind of science fiction reader that likes to identify what the nonsense in science fiction stories is. (Spoiler: it’s the science. In this case, the mathematics too.) In a normal number every finite string of digits appears. It would be truly astounding if there weren’t an encoded circle in the digits of π. Indeed, it would be impossible for there not to be infinitely many circles of every possible size encoded in every possible way in the digits of π. If the aliens are amazed by that they would be amazed to find how every triangle has three corners.

I’m a more forgiving reader. And I’ll give Sagan this amazingness. I have two reasons. The first reason is on the grounds of discoverability. Yes, the digits of a normal number will have in them every possible finite “message” encoded every possible way. (I put the quotes around “message” because it feels like an abuse to call something a message if it has no sender. But it’s hard to not see as a “message” something that seems to mean something, since we live in an era that accepts the Death of the Author as a concept at least.) Pick your classic cypher `1 = A, 2 = B, 3 = C’ and so on, and take any normal number. If you look far enough into its digits you will find every message you might ever wish to send, every book you could read. Every normal number holds Jorge Luis Borges’s Library of Babel, and almost every real number is a normal number.

But. The key there is if you look far enough. Look above; the first 35 or so digits of π have no 0’s, when you would expect three or four of them. There’s no 22’s, even though that number has as much right to appear as does 15, which gets in at least twice that I see. And we will only ever know finitely many digits of π. It may be staggeringly many digits, sure. It already is. But it will never be enough to be confident that a circle, or any other long enough “message”, must appear. It is staggering that a detectable “message” that long should be in the tiny slice of digits that we might ever get to see.

And it’s harder than that. Sagan’s book says the circle appears in whatever base π gets represented in. So not only does the aliens’ circle pop up in base ten, but also in base two and base sixteen and all the other, even less important bases. The circle happening to appear in the accessible digits of π might be an imaginable coincidence in some base. There’s infinitely many bases, one of them has to be lucky, right? But to appear in the accessible digits of π in every one of them? That’s staggeringly impossible. I say the aliens are correct to be amazed.

Now to my second reason to side with the book. It’s true that any normal number will have any “message” contained in it. So who says that π is a normal number?

We think it is. It looks like a normal number. We have figured out many, many digits of π and they’re distributed the way we would expect from a normal number. And we know that nearly all real numbers are normal numbers. If I had to put money on it I would bet π is normal. It’s the clearly safe bet. But nobody has ever proved that it is, nor that it isn’t. Whether π is normal or not is a fit subject for conjecture. A writer of science fiction may suppose anything she likes about its normality without current knowledge saying she’s wrong.

It’s easy to imagine numbers that aren’t normal. Rational numbers aren’t, for example. If you followed my instructions and made your own transcendental number then you made a non-normal number. It’s possible that π should be non-normal. The first thirty million digits or so look good, though, if you think normal is good. But what’s thirty million against infinitely many possible counterexamples? For all we know, there comes a time when π runs out of interesting-looking digits and turns into an unpredictable little fluttering between 6 and 8.

It’s hard to prove that any numbers we’d like to know about are normal. We don’t know about π. We don’t know about e, the base of the natural logarithm. We don’t know about the natural logarithm of 2. There is a proof that the square root of two (and other non-square whole numbers, like 3 or 5) is normal in base two. But my understanding is it’s a nonstandard approach that isn’t quite satisfactory to experts in the field. I’m not expert so I can’t say why it isn’t quite satisfactory. If the proof’s authors or grad students wish to quarrel with my characterization I’m happy to give space for their rebuttal.

It’s much the way transcendental numbers were in the 19th century. We understand there to be this class of numbers that comprises nearly every number. We just don’t have many examples. But we’re still short on examples of transcendental numbers. Maybe we’re not that badly off with normal numbers.

We can construct normal numbers. For example, there’s the Champernowne Constant. It’s the number you would make if you wanted to show you could make a normal number. It’s 0.12345678910111213141516171819202122232425 and I bet you can imagine how that develops from that point. (David Gawen Champernowne proved it was normal, which is the hard part.) There’s other ways to build normal numbers too, if you like. But those numbers aren’t of any interest except that we know them to be normal.

Mere normality is tied to a base. A number might be normal in base ten (the way normal people write numbers) but not in base two or base sixteen (which computers and people working on computers use). It might be normal in base twelve, used by nobody except mathematics popularizers of the 1960s explaining bases, but not normal in base ten. There can be numbers normal in every base. They’re called “absolutely normal”. Nearly all real numbers are absolutely normal. Wacław Sierpiński constructed the first known absolutely normal number in 1917. If you got in on the fractals boom of the 80s and 90s you know his name, although without the Polish spelling. He did stuff with gaskets and curves and carpets you wouldn’t believe. I’ve never seen Sierpiński’s construction of an absolutely normal number. From my references I’m not sure if we know how to construct any other absolutely normal numbers.

So that is the strange state of things. Nearly every real number is normal. Nearly every number is absolutely normal. We know a couple normal numbers. We know at least one absolutely normal number. But we haven’t (to my knowledge) proved any number that’s otherwise interesting is also a normal number. This is why I say: a normal number is any real number you never heard of.

Reading the Comics, June 25, 2016: Busy Week Edition


I had meant to cut the Reading The Comics posts back to a reasonable one a week. Then came the 23rd, which had something like six hundred mathematically-themed comic strips. So I could post another impossibly long article on Sunday or split what I have. And splitting works better for my posting count, so, here we are.

Charles Brubaker’s Ask A Cat for the 19th is a soap-bubbles strip. As ever happens with comic strips, the cat blows bubbles that can’t happen without wireframes and skillful bubble-blowing artistry. It happens that a few days ago I linked to a couple essays showing off some magnificent surfaces that the right wireframe boundary might inspire. The mathematics describing how a soap bubbles’s shape should be made aren’t hard; I’m confident I could’ve understood the equations as an undergraduate. Finding exact solutions … I’m not sure I could have done. (I’d still want someone looking over my work if I did them today.) But numerical solutions, that I’d be confident in doing. And the real thing is available when you’re ready to get your hands … dirty … with soapy water.

Rick Stromoski’s Soup To Nutz for the 19th Shows RoyBoy on the brink of understanding symmetry. To lose at rock-paper-scissors is indeed just as hard as winning is. Suppose we replaced the names of the things thrown with letters. Suppose we replace ‘beats’ and ‘loses to’ with nonsense words. Then we could describe the game: A flobs B. B flobs C. C flobs A. A dostks C. C dostks B. B dostks A. There’s no way to tell, from this, whether A is rock or paper or scissors, or whether ‘flob’ or ‘dostk’ is a win.

Bill Whitehead’s Free Range for the 20th is the old joke about tipping being the hardest kind of mathematics to do. Proof? There’s an enormous blackboard full of symbols and the three guys in lab coats are still having trouble with it. I have long wondered why tips are used as the model of impossibly difficult things to compute that aren’t taxes. I suppose the idea of taking “fifteen percent” (or twenty, or whatever) of something suggests a need for precision. And it’ll be fifteen percent of a number chosen without any interest in making the calculation neat. So it looks like the worst possible kind of arithmetic problem. But the secret, of course, is that you don’t have to have “the” right answer. You just have to land anywhere in an acceptable range. You can work out a fraction — a sixth, a fifth, or so — of a number that’s close to the tab and you’ll be right. So, as ever, it’s important to know how to tell whether you have a correct answer before worrying about calculating it.

Allison Barrows’s Preeteena rerun for the 20th is your cheerleading geometry joke for this week.

'I refuse to change my personality just for a stupid speech.' 'Fi, you wouldn't have to! In fact, make it an asset! Brand yourself as The Math Curmudgeon! ... The Grouchy Grapher ... The Sour Cosine ... The Number Grump ... The Count of Carping ... The Kvetching Quotient' 'I GET IT!'
Bill Holbrook’s On The Fastrack for the 22nd of June, 2016. There are so many bloggers wondering if Holbrook is talking about them.

I am sure Bill Holbrook’s On The Fastrack for the 22nd is not aimed at me. He hangs around Usenet group rec.arts.comics.strips some, as I do, and we’ve communicated a bit that way. But I can’t imagine he thinks of me much or even at all once he’s done with racs for the day. Anyway, Dethany does point out how a clear identity helps one communicate mathematics well. (Fi is to talk with elementary school girls about mathematics careers.) And bitterness is always a well-received pose. Me, I’m aware that my pop-mathematics brand identity is best described as “I guess he writes a couple things a week, doesn’t he?” and I could probably use some stronger hook, somewhere. I just don’t feel curmudgeonly most of the time.

Darby Conley’s Get Fuzzy rerun for the 22nd is about arithmetic as a way to be obscure. We’ve all been there. I had, at first, read Bucky’s rating as “out of 178 1/3 π” and thought, well, that’s not too bad since one-third of π is pretty close to 1. But then, Conley being a normal person, probably meant “one-hundred seventy-eight and a third”, and π times that is a mess. Well, it’s somewhere around 550 or so. Octave tells me it’s more like 560.251 and so on.

Roller Coaster Immortality Update!


Several years ago I had the chance to go to Lakemont Park, in Altoona, Pennsylvania. It’s a lovely and very old amusement park, featuring the oldest operating roller coaster, Leap The Dips. As roller coasters go it’s not very large and not very fast, but it’s a great ride. It does literally and without exaggeration leap off the track, though not far enough to be dangerous. I recommend the park and the ride to people who have cause to be in the middle of Pennsylvania.

I wondered whether any boards in it might date from the original construction in 1902 by the E Joy Morris company. If we make some assumptions we can turn this into a probability problem. It’s a problem of a type that always seems to be answered 1/e. (The problem is “what is the probability that any particular piece of wood has lasted 100 years, if a piece of wood has a one percent chance of needing replacement every year?”) That’s a probability of about 37 percent. But I doubted this answer meant anything. My skepticism came from wondering why every piece of wood should be equally likely to survive every year. Different pieces serve different structural roles, and will be exposed to the elements differently. How can I be sure that the probability one piece needs replacement is independent of the probability some other piece needs replacement? But if they’re not independent then my calculation doesn’t give a relevant answer.

The Leap-The-Dips roller coaster at Lakemont Park, Altoona, Pennsylvania.
The Leap-The-Dips roller coaster at Lakemont Park, Altoona, Pennsylvania.

A recent post on the Usenet roller coaster enthusiast newsgroup rec.roller-coaster, in a discussion titled “Age a coaster should be preserved”, suggests I was right in my skepticism. Derek Gee writes:

According to the video documentary the park produced around
1999, all of the original upright lumber was found to be in excellent shape.
The E. Joy Morris company had waterproofed it by sealing it in ten coats of
paint and it was old-growth hardwood. All the horizontal lumber was
replaced as I recall.

I am aware this is not an academically rigorous answer to the question of how much of the roller coaster’s original construction is still in place. But it is a lead. It suggests that quite a bit of the antique ride is as antique as could be.

Who Was Karl Pearson?


An offhanded joke in the Usenet newsgroup alt.fan.cecil-adams — a great spot for offhanded jokes, as the audience is reasonably demanding — about baseball being a game of statistics but this is ridiculous prompted me to say I hoped the Pearson Chi-Squared Test had a good season since it was at the core of my baseball statistics fantasy team. One respondent asked if this was connected to Pearson Publishing, which has recently screwed up its composition of standardized tests for New York State quite severely, including giving as a reading comprehension assignment a bit of nonsense composed to have no meaning, and twenty mistakes in the non-English translation of a math exam. There’s no connection of which I’m aware; but, why not take a couple paragraphs to talk about Karl Pearson?

Continue reading “Who Was Karl Pearson?”

Early April’s Math Comics


I had started to think the mathematics references in the comics pages were fading out and I might not have an installment to offer anytime soon. Then, on April 3, Pab Sugenis’s The New Adventures of Queen Victoria — a clip art comic strip which supposes the reader will recognize an illustration of King Edward VI — skipped its planned strip for the day (Sugenis’s choice, he says) and ran a Fuzzy Bunny Time strip calling on pretty much the expected rabbits and mathematics comic strip. (Some people in the Usenet group alt.fan.cecil-adams, which I read reliably and write to occasionally, say Sugenis was briefly a regular; perhaps so, but I don’t remember.) This would start a bumper crop of math strips for the week.

Continue reading “Early April’s Math Comics”