A couple months ago I wrote about the problem of counting the number of goldfish in the backyard pond. For those who’d missed it:

How To Count Fish, which presented a way to estimate a population by simply doing two samplings of the population.

How To Re-Count Fish, which described some of the numerical problems in estimation-based population samples.

How Not To Count Fish, which threatened to collapse the entire project under fiddly practical problems.

Spring finally arrived, and about a month ago we finally stopped having nights that touched freezing. So we moved the goldfish which had been wintering over in the basement out to the backyard. This also let us count just how many goldfish we’d caught, and I thought folks might like to know what the population did look like.

The counting didn’t require probabilistic methods this time. Instead we took the fish from the traps and set up a correspondence between them and an ordered subset of positive whole numbers. This is the way you describe “just counting” so that it sounds either ferociously difficult or like a game. Whether it’s difficult or a game depends on whether you were a parent or a student back when the New Math was a thing. My love and I were students.

Altogether then there were fifty goldfish that had wintered over in the stock tank in the basement: eight adults and 42 baby fish. (Possibly nine and 41; one of the darker goldfish is small for an adult, but large for a baby.) Over the spring I identified at least three baby fish that had wintered over outdoors successfully. It was a less harsh winter than the one before. So there are now at least 53 goldfish in the pond. There are surely more on the way, but we haven’t seen any new babies yet.

Also this spring we finally actually measured the pond. We’d previously estimated it to be about ten feet in diameter and two feet deep, implying a carrying capacity of about 60 goldfish if some other assumptions are made. Now we’ve learned it’s nearer twelve feet in diameter and twenty inches deep. Call that two meters radius and half a meter height. That’s a volume of about 6.3 cubic meters, or 6300 liters, or enough volume of water for about 80 goldfish. We’ll see what next fall brings.

I’d discussed a probability/sampling-based method to estimate the number of fish that might be in our pond out back, and then some of the errors that have to be handled if you want to have a reliable result. Now, I want to get into why the method doesn’t work, at least not without much greater insight into goldfish behavior than simply catching a couple and releasing them will do.

Catching a sample, re-releasing it, and counting how many of that sample we re-catch later on is a logically valid method, provided certain assumptions the method requires are accurately — or at least accurately enough — close to the way the actual thing works. Here are some of the ways goldfish fall short of the ideal.

First faulty assumption: Goldfish are perfectly identical. In this goldfish-trapped we make the assumption that there is some, fixed, constant probability of a goldfish being caught in the net. We have to assume that this is the same number for every goldfish, and that it doesn’t change as goldfish go through the experience of getting caught and then released. But goldfish have personality, as you learn if you have a bunch in a nice setting and do things like try feeding them koi treats or introduce something new like a wire-mesh trap to their environment. Some are adventurous and will explore the unfamiliar thing; some are shy and will let everyone else go first and then maybe not bother going at all. I empathize with both positions.

If there are enough goldfish, the variation between personalities is probably not going to matter much. There’ll be some that are easy to catch, and they’ll probably be roughly as common as the ones who can’t be coaxed into the trap at all. It won’t be exactly balanced unless we’re very lucky, but this would probably only throw off our calculations a little bit.

Whether the goldfish learn, and become more, or less, likely to be trapped in time is harder. Goldfish do learn, certainly, although it’s not obvious to me that the trapping and releasing experience would be one they draw much of a lesson from. It’s only a little inconvenience, really, and not at all harmful; what should they learn? Other than that there’s maybe an easy bit of food to be had here so why not go in? So this might change their behavior and it’s hard to predict how.

(I note that animal capture studies get quite frustrated when the animals start working out how to game the folks studying them. Bil Gilbert’s early-70s study of coatis — Latin American raccoons, written up in the lovely popularization Chulo: A Year Among The Coatimundis — was plagued by some coatis who figured out going into the trap was an easy, safe meal they’d be released from without harm, and wouldn’t go back about their business and leave room for other specimens.)

Second faulty assumption: Goldfish are not perfectly identical. This is the biggest challenge to counting goldfish population by re-catching a sample of them. How do you know if you caught a goldfish before? When they grow to adulthood, it’s not so bad, since they grow fairly distinctive patterns of orange and white and black and such, and they’ll usually settle into different sizes. (That said, we do have two adult fish who were very distinct when we first got them, but who’ve grown into near-twins.)

But baby goldfish? They’re basically all tiny black things, meant to hide into the mud at the bottom of ponds and rivers — their preferred habitat — and pretty near indistinguishable. As they get larger they get distinguishable, a bit, and start to grow patterns, but for the vast number of baby fish there’s just no telling one from another.

When we were trying to work out whether some mice we found in the house were ones we had previously caught and put out in the garage, we were able to mark them by squiring some food dye at their heads as they were released. The mice would rub the food dye from their heads onto their whole bodies and it would take a while before the dye would completely fade out. (We didn’t re-catch any mice, although it’s hard to dye a wild mouse efficiently because they will take off like bullets. Also one time when we thought we’d captured one there were actually three in the humane trap and you try squiring the food dye bottle at two more mice than you thought were there, fleeing.) But you can see how the food dye wouldn’t work here. Animal researchers with a budget might go on to attach collars or somehow otherwise mark animals, but if there’s a way to mark and track goldfish with ordinary household items I can’t think of it.

(No, we will not be taking the bits of americium in our smoke detectors out and injecting them into trapped goldfish; among the objections, I don’t have a radioactivity detector.)

Third faulty assumption: Goldfish are independent entities. The first two faulty assumptions are ones that could be kind of worked around. If there’s enough goldfish then the distribution of how likely any one is to get caught will probably be near enough normal that we can pretend there’s an identical chance of catching each, and if we really thought about it we could probably find some way of marking goldfish to tell if we re-caught any. Independence, though; this is the point on which so many probability-based schemes fall.

Independence, in the language of probability, is the principle that one thing’s happening does not affect the likelihood of another thing happening. For our problem, it’s the assumption that one goldfish being caught does not make it any more or less likely that another goldfish will be caught. We like independence, in studying probability. It makes so many problems easier to study, or even possible to study, and it often seems like a reasonable supposition.

A good number of interesting scientific discoveries amount to finding evidence that two things are not actually independent, and that one thing happening makes it more (or less) likely the other will. Sometimes these turn out to be vapor — there was a 19th-century notion suggesting a link between sunspot activity and economic depressions (because sunspots correlate to solar activity, which could affect agriculture, and up to 1893 the economy and agriculture were pretty much the same thing) — but when there is a link the results can be profound, as see the smoking-and-cancer link, or for something promising but still (to my understanding) under debate, the link between leaded gasoline and crime rates.

How this applies to the goldfish population problem, though, is that goldfish are social creatures. They school, loosely, forming and re-forming groups, and would much rather be around another goldfish than not. Even as babies they form these adorable tiny little schools; that may be in the hopes that someone else will get eaten by a bigger fish, but they keep hanging around other fish their own size through their whole lives. If there’s a goldfish inside the trap, it is hard to believe that other goldfish are not going to follow it just to be with the company.

Indeed, the first day we set out the trap for the winter, we pulled in all but one of the adult fish, all of whom apparently followed the others into the enclosure. I’m sorry I couldn’t photograph that because it was both adorable and funny to see so many fish just station-keeping beside one another — they were even all looking in the same direction — and waiting for whatever might happen next. Throughout the months we were able to spend bringing in fish, the best bait we could find was to have one fish already in the trap, and a couple days we did leave one fish in a few more hours or another night so that it would be joined by several companions the next time we checked.

So that’s something which foils the catch and re-catch scheme: goldfish are not independent entities. They’re happy to follow one another into trap. I would think the catch and re-catch scheme should be salvageable, if it were adapted to the way goldfish actually behave. But that requires a mathematician admitting that he can’t just blunder into a field with an obvious, simple scheme to solve a problem, and instead requires the specialized knowledge and experience of people who are experts in the field, and that of course can’t be done. (For example, I don’t actually know that goldfish behavior is sufficiently non-independent as to make an important difference in a population estimate of this kind. But someone who knew goldfish or carp well could tell me, or tell me how to find out.)

For those curious how the goldfish worked out, though, we were able to spend about two and a half months catching fish before the pond froze over for the winter, though the number we caught each week dropped off as the temperature dropped. We have them floating about in a stock tank in the basement, waiting for the coming of spring and the time the pond will be warm enough for them to re-occupy it. We also know that at least some of the goldfish we didn’t catch made it to, well, about a month ago. I’d seen one of the five orange baby fish who refused to go into the trap through a hole in the ice then. It was holding close to the bottom but seemed to be in good shape.

This coming year should be an exciting one for our fish population.

I’m still getting back to normal after the Christmas and New Year’s disruption of, well, everything, which is why I’m taking it easy and just doing another comics review. I have to suppose Comic Strip Master Command was also taking it easy over the holidays since most of the subjects are routine genres — word answer problems, mathematics-connected puns, and the like — with the Bloom County reruns the cartoons that give me most to write about. It’s all part of the wondrous cycle of nature; I’m sure there’ll be a really meaty collection of topics along soon.

Gordon Bess’s Redeye (January 8, originally run August 21, 1968) is an example of the student giving a mischievous answer to a word problem. I feel like I should have a catchy name for this genre, given how much it turns up, but I haven’t got anything good that comes to mind. (I don’t tend to talk about the drawing much in these strips — most of the time it isn’t that important, and comic strips have been growing surprisingly indifferent to drawing — but I did notice while uploading this that Pokey’s stance and expression in the first panel is really quite good. You should be able to open the image in a new tab and see it at its fullest-available 1440-by-431 pixel size and that shows off well the crafting that went into the figure.)

I had assumed that at some point the good folks at Comics Kingdom would let any of their cartoonists do a panel that’s got mathematical content relevant enough for me to chat about, but apparently that’s just not happening. So for a third time in a row here’s a set of Gocomics-only comic strips, with reasonably stable links and images I don’t feel the need to include. Enjoy, please.

Fred Wagner’s Animal Crackers (October 26) presents an old joke — counting the number of animals by counting the number of legs and dividing by four — although it’s only silly because it’s hard to imagine a case where it’s easier to count the legs on a bunch of animals than it is to count the animals themselves. But if it’s the case that every animal has exactly four legs, then, there’s what’s called a one-to-one relationship between the set of animals and the set of animal legs: if you have some number of animals you have exactly four times that number of animal legs, and if you have some number of animal legs you have exactly one-fourth that number of animals, and you can count whatever’s the more convenient for you and use that to get what you’re really interested in. Showing such a one-to-one relationship exists between two interesting things can often be a start to doing more interesting problems, especially if you can show that the relationship also preserves some interesting interactions; if you have two ways to work out a problem, you can do the easier one.

Mark Anderson’s Andertoons (October 27) riffs on the place value for numbers written in the familiar Arabic style. As befitting a really great innovation, place value becomes invisible when you’re familiar with it; it takes a little sympathy and imagination to remember the alienness of the idea that a “2” means different things based on how many digits are to the right (or, if it’s a decimal, to the left) of it.

Anthony Blades’s charming Bewley (October 27) has one of the kids insisting that instinct alone is enough to do maths problems. The work comes out disastrously bad, of course, or there’d not be a comic strip. However, my understanding is that people do have some instinctive understanding even of problems that would seem to have little survival application. One test I’ve seen demonstrating this asks people to give, without thinking, their answer to whether a multiplication problem might be right or wrong. It’s pretty quick for most people to say that “7 times 9 equals 12” has to be wrong; to say that “7 times 9 equals 59” is wrong takes longer, and that seems to reflect an idea that 59 is, if not the right answer, at least pretty close to it. There’s an instinctive plausibility at work there and it’s amazing to think people should have that. Zach Weinersmith’s Saturday Morning Breakfast Cereal for October 31 circles around this idea, with one person having little idea what 1,892,491,287 times 7,798,721,415 divided by 82,493,726,631 might be, but being pretty sure that “4” isn’t it.

Saturday Morning Breakfast Cereal (October 30) also contains a mention of “cross products”, which are an interesting thing people learning vectors trip over. A cross product is defined for a pair of three-dimensional vectors, and the interesting thing is it’s a new vector that’s perpendicular to the two vectors multiplied together. The length of the cross product vector depends on the lengths of the two vectors multiplied together and the angle they make; the closer the two vectors multiplied together are, the smaller the cross product is, to the point that the cross product of two parallel vectors has length zero. The closer the two vectors multiplied together are to perpendicular the longer the cross product vector is.

More mysterious: if you swap the first vector and the second vector being cross-multiplied together, you get a cross product that’s the same size but pointing the opposite direction, pointing (say) down instead of up. Cross products have some areas where they’re particularly useful, especially in describing the movement of charged particles in magnetic fields.

(There’s something that looks a lot like the cross product which exists for seven-dimensional vectors, but I’ve never even heard of anyone who had a use for it, so, you don’t need to do anything about it.)

Eric the Circle (November 2), this one by “dDave”, presents the idea that that the points on a line might themselves be miniature Erics the Circle. What a line is made of is again one of those problems that straddles the lines between mathematics and philosophy. It seems to be one of the problems of infinity that Zeno’s Paradoxes outlined so perfectly thousands of years ago. To shorten it to the point it becomes misleading, is a line made up of things that have some width? If they’re infinitesimals, things with no width, then, how can an aggregate of things with no width come to have some width? But if they’re made up of things which have some width, how can there be infinitely many of them fitting into a finite space?

We can form good logical arguments about the convergence of infinite series — lining up, essentially, circles of ever-dwindling but ever-positive sizes so that the pile has a finite length — but that seems to suggest that space has to be made up of intervals of different widths, which seems silly; why couldn’t all the miniature circles be the same? In short, space is either infinitely divisible into identical things, or it is not, and neither one is completely satisfying.

Guy Gilchrist’s Nancy (November 2) uses math homework appearing in the clouds, although that’s surely because it’s easier to draw a division problem than it is to depict an assignment for social studies or English.

Todd Clark’s Lola (November 4) uses an insult-the-minor-characters variant of what seems to be the standard way of explaining fractions to kids, that of dividing a whole thing into smaller pieces and counting the number of smaller pieces. As physical interpretations of mathematical concepts goes I suppose that’s hard to beat.

I had thought the folks at Comic Strip Master Command got most of their mathematics-themed comics cleaned out ahead of the end of the school year (United States time zones) by last week, and then over the course of the weekend they went and published about a hundred million of them, so let me try catching up on that before the long dry spell of summer sets in. (And yet none of them mentioned monkeys writing Shakespeare; go figure.) I’m kind of expecting an all-mathematics-strips series tomorrow morning.

Jason Chatfield’s Ginger Meggs (June 12) puns a bit on negative numbers as also meaning downbeat or pessimistic ones. Negative numbers tend to make people uneasy, when they’re first encountered. It took western mathematics several centuries to be quite fully comfortable with them and that even with the good example of debts serving as a mental model of what negative numbers might mean. Descartes, for example, apparently used four separate quadrants, giving points their positions to the right and up, to the left and up, to the left and down, or to the right and down, from the origin point, rather than deal with negative numbers; and the Fahrenheit temperature scale was pretty much designed around the constraint that Daniel Fahrenheit shouldn’t have to deal with negative numbers in measuring the temperature in his hometown of the Netherlands. I have seen references to Immanuel Kant writing about the theoretical foundation of negative numbers, but not a clear explanation of just what he did, alas. And skepticism of exotic number constructs would last; they’re not called imaginary numbers because people appreciated the imaginative leaps that working with the square roots of negative numbers inspired.

Steve Breen and Mike Thompson’s Grand Avenue (June 12) served notice that, just like last summer, Grandma is going to make sure the kids experience mathematics as a series of chores they have to endure through an otherwise pleasant summer break.

Mike Twohy’s That’s Life (June 12) might be a marginal inclusion here, but it does refer to a lab mouse that’s gone from merely counting food pellets to cost-averaging them. The mathematics abilities of animals are pretty amazing things, certainly, and I’d also be impressed by an animal that was so skilled in abstract mathematics that it was aware “how much does a thing cost?” is a pretty tricky question when you look hard at it.

Jim Scancarelli’s Gasoline Alley (June 13) features a punch line that’s familiar to me — it’s what you get by putting a parrot and the subject of geometry together — although the setup seems clumsy to me. I think that’s because the kid has to bring up geometry out of nowhere in the first panel. Usually the setup as I see it is more along the lines of “what geometric figure is drawn by a parrot that then leaves the room”, which I suppose also brings geometry up out of nowhere to start off, really. I guess the setup feels clumsy to me because I’m trying to imagine the dialogue as following right after the previous day’s, so the flow of the conversation feels odd.

Eric the Circle (June 14), this one signed “andel”, riffs on the popular bit of mathematics trivia that in a randomly selected group of 22 people there’s about a fifty percent chance that some pair of them will share a birthday; that there’s a coincidental use for 22 in estimating π is, believe it or not, something I hadn’t noticed before.

Pab Sungenis’s New Adventures of Queen Victoria (June 14) plays with infinities, and whether the phrase “forever and a day” could actually mean anything, or at least anything more than “forever” does. This requires having a clear idea what you mean by “forever” and, for that matter, by “more”. Normally we compare infinitely large sets by working out whether it’s possible to form pairs which match one element of the first set to one element of the second, and seeing whether elements from either set have to be left out. That sort of work lets us realize that there are just as many prime numbers as there are counting numbers, and just as many counting numbers as there are rational numbers (positive and negative), but that there are more irrational numbers than there are rational numbers. And, yes, “forever and a day” would be the same length of time as “forever”, but I suppose the Innamorati (I tried to find his character’s name, but I can’t, so, Pab Sungenis can come in and correct me) wouldn’t do very well if he promised love for the “power set of forever”, which would be a bigger infinity than “forever”.

Mark Anderson’s Andertoons (June 15) is actually roughly the same joke as the Ginger Meggs from the 12th, students mourning their grades with what’s really a correct and appropriate use of mathematics-mentioning terminology.

Keith Knight’s The Knight Life (June 16) introduces a “personal statistician”, which is probably inspired by the measuring of just everything possible that modern sports has gotten around to doing. But the notion of keeping track of just what one is doing, and how effectively, is old and, at least in principle, sensible. It’s implicit in budgeting (time, money, or other resources) that you are going to study what you do, and what you want to do, and what’s required by what you want to do, and what you can do. And careful tracking of what one’s doing leads to what’s got to be a version of the paradox of Achilles and the tortoise, in which the time (and money) spent on recording the fact of one’s recordings starts to spin out of control. I’m looking forward to that. Don’t read the comments.

Max Garcia’s Sunny Street (June 16) shows what happens when anthropomorphized numerals don’t appear in Scott Hilburn’s The Argyle Sweater for too long a time.

Wuff and Morgenthaler’s WuMo (December 16) uses a spray of a bit of mathematics to stand in for “something just too complicated to understand”, and even uses a caricature of Albert Einstein to represent the person who’s just too smart to be understood. I’m a touch disappointed that, as best I can tell, the equations sprayed out don’t mean anything; I’ve enjoyed WuMo — a new comic to North American audiences — so far and kind of expected they would get an irrelevant detail like that plausibly right.

I’m also interested that sixty years after his death the portrait of Einstein still hasn’t been topped as an image for The Really, Really Smart Guy. Possibly nobody since him has managed to combine being both incredibly important — even if it weren’t for relativity, Einstein would be an important figure in science for his work in quantum mechanics, and if he didn’t have relativity or quantum mechanics, he’d still be important for statistical mechanics — and iconic-looking, which I guess really means he let his hair grow wild. I wonder if Stephen Hawking will be able to hold some of that similar pop cultural presence.