Iva Sallay, creator of the Find The Factors recreational mathematics puzzle and a kind friend to my blog, posted Yes, YOU Can Host a Playful Math Education Blog Carnival. It explains in quite good form how to join in Denise Gaskins’s roaming blog event. It tries to gather educational or recreational or fun or just delightful mathematics links.
Hosting the blog carnival is a great experience I recommend for mathematics bloggers at least once. I seem to be up to hosting it about once a year, most recently in September 2020. Most important in putting one together is looking at your mathematics reading with different eyes. Sallay, though, goes into specifics about what to look for, and how to find that.
I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:
Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.
That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:
Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by , in which case the answer is 1?
And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.
I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.
This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.
But you also start to see why mathematicians tend to avoid the symbol. We understand, for example, to mean . Carry that out and then there’s no ambiguity about
I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.
(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)
And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.
And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.
I’m not yet looking to discuss every comic strip with any mathematics mention. But something gnawed at me in this installment of Greg Evans and Karen Evans’s Luann. It’s about the classes Gunther says he’s taking.
The main characters in Luann are in that vaguely-defined early-adult era. They’re almost all attending a local university. They’re at least sophomores, since they haven’t been doing stories about the trauma and liberation of going off to school. How far they’ve gotten has been completely undefined. So here’s what gets me.
Gunther taking vector calculus? That makes sense. Vector calculus is a standard course if you’re taking any mathematics-dependent major. It might be listed as Multivariable Calculus or Advanced Calculus or Calculus III. It’s where you learn partial derivatives, integrals along a path, integrals over a surface or volume. I don’t know Gunther’s major, but if it’s any kind of science, yeah, he’s taking vector calculus.
Algebraic topology, though. That I don’t get. Topology at all is usually an upper-level course. It’s for mathematics majors, maybe physics majors. Not every mathematics major takes topology. Algebraic topology is a deeper specialization of the subject. I’ve only seen courses listed as algebraic topology as graduate courses. It’s possible for an undergraduate to take a graduate-level course, yes. And it may be that Gunther is taking a regular topology course, and the instructor prefers to focus on algebraic topology.
But even a regular topology course relies on abstract algebra. Which, again, is something you’ll get as an undergraduate. If you’re a mathematics major you’ll get at least two years of algebra. And, if my experience is typical, still feel not too sure about the subject. Thing is that Intro to Abstract Algebra is something you’d plausibly take at the same time as Vector Calculus. Then you’d get Abstract Algebra and then, if you wished, Topology.
So you see the trouble. I don’t remember anything in algebra-to-topology that would demand knowing vector calculus. So it wouldn’t mean Gunther took courses without taking the prerequisites. But it’s odd to take an advanced mathematics course at the same time as a basic mathematics course. Unless Gunther’s taking an advanced vector calculus course, which might be. Although since he wants to emphasize that he’s taking difficult courses, it’s odd to not say “advanced”. Especially if he is tossing in “algebraic” before topology.
And, yes, I’m aware of the Doylist explanation for this. The Evanses wanted courses that sound impressive and hard. And that’s all the scene demands. The joke would not be more successful if they picked two classes from my actual Junior year schedule. None of the characters have a course of study that could be taken literally. They’ve been university students full-time since 2013 and aren’t in their senior year yet. It would be fun, is all, to find a way this makes sense.
John Golden, MathHombre, was host this month for the Playful Math Education Blog Carnival. And this month’s collection of puzzles, essays, and creative mathematics projects. Among them are some quilts and pattern-block tiles, which manifest all that talk about the structure of mathematical objects and their symmetries in easy-to-see form. There’s likely to be something of interest there.
Among the wonderful things I discovered there is Math Zine Fest 2021. It’s as the name suggests, a bunch of zines — short printable magazines on a niche topic — put together for the end of February. I had missed this organizing, but hope to get to see later installments. I don’t know what zine I might make, but I must have something I could do.
I regret not having the time or energy to write something original about π for today. I hope you’ll accept this offering of past Reading the Comics posts covering the day, and some of my other π-related writings:
And then there’s comic strips. I seem to complain every year that there’s fewer Pi Day comic strips than I expected, which invites the question of just what I expect. Here’s, as best I can tell, the actual record:
I have not yet read today’s comics, so don’t know what they’ll offer. We shall see! Also, I apologize but some of the comics may have been removed from GoComics or Comics Kingdom, and so the links may be dead. I’m not happy about that. But if I wanted the essays discussing these strips to stay permanently sensible I’d have posted the comics on my own web site.
This is the 141st Playful Math Education Blog Carnival. And I will be taking this lower-key than I have past times I was able to host the carnival. I do not have higher keys available this year.
I will start by borrowing a page from Iva Sallay, kind creator and host of FindTheFactors.com, and say some things about 141. I owe Iva Sallay many things, including this comfortable lead-in to the post, and my participation in the Playful Math Education Blog Carnival. She was also kind enough to send me many interesting blogs and pages and I am grateful.
141 is a centered pentagonal number. It’s like 1 or 6 or 16 that way. That is, if I give you six pennies and ask you to do something with it, a natural thing is one coin in the center and a pentagon around that. With 16 coins, you can add a nice regular pentagon around that, one that reaches three coins from vertex to vertex. 31, 51, 76, and 106 are the next couple centered pentagonal numbers. 181 and 226 are the next centered pentagonal numbers. The units number in these follow a pattern, too, in base ten. The last digits go 1-6-6-1, 1-6-6-1, 1-6-6-1, and so on.
141’s also a hendecagonal number. That is, arrange your coins to make a regular 11-sided polygon. 1 and then 11 are hendecagonal numbers. Then 30, 58, 95, and 141. 196 and 260 are the next couple. There are many of these sorts of polygonal numbers, for any regular polygon you like.
141 is also a Hilbert Prime, a class of number I hadn’t heard of before. It’s still named for the Hilbert of Hilbert’s problems. 141 is not a prime number, which you notice from adding up the digits. But a Hilbert Prime is a different kind of beast. These come from looking at counting numbers that are one more than a whole multiple of four. So, numbers like 1, 5, 9, 13, and so on. This sequence describes a lot of classes of numbers. A Hilbert Prime, at least as some number theorists use it, is a Hilbert Number that can’t be divided by any other Hilbert Number (other than 1). So these include 5, 9, 13, 17, and 21, and some of those are already not traditional primes. There are Hilbert Numbers that are the products of different sets of Hilbert Primes, such as 441 or 693. (441 is both 21 times 21 and also 9 times 49. 693 is 9 times 77 and also 21 times 33) So I don’t know what use Hilbert Primes are specifically. If someone knows, I’d love to hear.
Also, at the risk of causing trouble, The Aperiodical also hosts a monthly Carnival of Mathematics. It’s a similar gathering of interesting mathematics content. It doesn’t look necessarily for educational or playful pieces.
The Reflective Educator posted Precision In Language. This is about one of the hardest bits of teaching. That is to say things which are true and which can’t be mis-remembered as something false. Author David Wees points out an example of this hazard, as kids apply rules outside their context.
Simon Gregg’s essay The Gardener and the Carpenter follows a connected theme. The experience students have with a thing can be different depending on how the teacher presents it. The lead example of Gregg’s essay is about the different ways students played with a toy depending on how the teacher prompted them to explore it.
Now I can come to more bundles of things to teach. Colleen Young gathered Maths at school … and at home, bundles of exercises and practice sheets. One of the geometry puzzles, about the missing lengths in the perimeter of a hexagon, brings me a smile as this is a sort of work I’ve been doing for my day job.
Starting Points Maths has a page of Radian Measure — Intro. The goal here is building comfort in the use of radians as angle measure. Mathematicians tend to think in radians. The trigonometric functions for radian measure behave well. Derivatives and integrals are easy, for example. We do a lot of derivatives and integrals. The measures look stranger, is all, especially as they almost always involve fractions times π.
Lowry also has Helping Your Child Learn Time, using both analog and digital clocks. That lets me mention a recent discussion with my love, who teaches. My love’s students were not getting the argument that analog clocks can offer a better sense of how time is elapsing. I had what I think a compelling argument: an analog clock is like a health bar, a digital clock like the count of hit points. Logic tells me this will communicate well.
YummyMath’s Fall Equinox 2020 describes some of the geometry of the equinoxes. It also offers questions about how to calculate the time of daylight given one’s position on the Earth. This is one of the great historic and practical uses for trigonometry.
To some play! Miguel Barral wrote Much More Than a Diversion: The Mathematics of Solitaire. There are many kinds of solitaire, which is ultimately just a game that can be played alone. They’re all subject to study through game theory. And to questions like “what is the chance of winning”? That’s often a question best answered by computer simulation. Working out that challenge helped create Monte Carlo methods. These can find approximate solutions to problems too difficult to find perfect solutions for.
Conditional probability is fun. It’s full of questions easy to present and contradicting intuition to solve. Wayne Chadburn’s Big Question explores one of them. It’s based on a problem which went viral a couple years ago, called “Hannah’s Sweet”. I missed the problem when it was getting people mad. But Chadburn explores how to think through the problem.
Now to some deeper personal interests. I am an amusement park enthusiast: I’ve ridden at least 250 different roller coasters at least once each. This includes all the wooden Möbius-strip roller coasters out there. Also all three racing merry-go-rounds. The oldest roller coaster still standing. And I had hoped, this year, to get to the centennial years for the Jackrabbit roller coaster at Kennywood Amusement Park (Pittsburgh) and Jack Rabbit roller coaster at Seabreeze Park (Rochester, New York). Jackrabbit (with spelling variants) used to be a quite popular roller coaster name.
So plans went awry and it seems unlikely we’ll get to any amusement parks this year. No county fairs or carnivals. We can still go to virtual ones, though. Amusement parks and midway games inspire many mathematical questions. So let’s take some in.
Michigan State University’s Connected Mathematics Program set up set up a string of carnival-style games. The event’s planners figured on then turning the play money into prize raffles but you can also play games. Some are legitimate midway games, such as plinko, spinner wheels, or racing games, too.
Hooda Math’s Carnival Fun offers a series of games, many of them Flash, a fair number HTML5, and mostly for kindergraden through 8th grade. There are a lot of mathematics games here, along with some physics and word games.
Specific rides, though, are always beautiful and worth looking at. Ann-Marie Pendrill’s Rotating swings—a theme with variations looks at rotating swing rides. These have many kinds of motion and many can be turned into educational problems. Pendrill looks at some of them. There are other articles recommended by this, which seem relevant, but this was the only article I found which I had permission to read in full. Your institution might have better access.
Lin McMullin’s The Scrambler, or A Family of Vectors at the Amusement Park looks at the motion of the most popular thrill ride out there. (There are more intense rides. But they’re also ones many people feel are too much for them. Few people in a population think the Scrambler is too much for them.) McMullin uses the language of vectors to examine what path the rider traces out during a ride, and what they say about velocity and acceleration. These are all some wonderful shapes.
And Amusement Parks
Many amusement parks host science and mathematics education days. In fact I’ve never gone to the opening day of my home park, Michigan’s Adventure, as that’s a short four-hour day filled with area kids. Many of the parks do have activity pages, though, suggesting the kinds of things to think about at a park. Some of the mathematics is things one can use; some is toying with curiosity.
Here’s The State Fair of Texas’s Grade 6 STEM games. I don’t know whether there’s a more recent edition. But also imagine that tasks like counting the traffic flow or thinking about what energies are shown at different times in a ride do not age.
Dorney Park, in northeastern Pennsylvania, was never my home park, but it was close. And I’ve had the chance to visit several times. People with Kutztown University, regional high schools, and Dorney Park prepared Coaster Quest – Geometry. These include a lot of observations and measurements all tied to specific rides at the park. (And a side fact, fun for me: Dorney Park’s carousel used to be at Lake Lansing Amusement Park, a few miles from me. Lake Lansing’s park closed in 1972, and the carousel spent several decades at Cedar Point in Ohio before moving to Pennsylvania. The old carousel building at Lake Lansing still stands, though, and I happened to be there a few weeks ago.)
A 2018 posting on Social Mathematics asks: Do height restrictions matter to safety on Roller Coasters? Of course they do, or else we’d have more roller coasters that allowed mice to ride. The question is how much the size restriction matters, and how sensitive that dependence is. So the leading question is a classic example of applying mathematics to the real world. This includes practical subtleties like if a person 39.5 inches tall could ride safely, is it fair to round that off to 40 inches? It also includes the struggle to work out how dangerous an amusement park is.
Speaking from my experience as a rider and lover of amusement parks: don’t try to plead someone’s “close enough”. You’re putting an unfair burden on the ride operator. Accept the rules as posted. Everybody who loves amusement parks has their disappointment stories; accept yours in good grace.
This leads me into planning amusement park fun. School Specialty’s blog particularly offers PLAY & PLAN: Amusement Park. This is a guide to building an amusement park activity packet for any primary school level. It includes, by the way, some mention of the historical and cultural aspects. That falls outside my focus on mathematics with a side of science here. But there is a wealth of culture in amusement parks, in their rides, their attractions, and their policies.
Let me resume the fun, by looking to imaginary amusement parks. TeachEngineering’s Amusement Park Ride: Ups and Downs in Design designs and builds model “roller coasters”. This from foam tubes, toothpicks, masking tape, and marbles. It’s easier to build a ride in Roller Coaster Tycoon but that will always lack some of the thrill of having a real thing that doesn’t quite do what you want. The builders of Son Of Beast had the same frustration.
The Brunswick (Ohio) City Schools published a nice Amusement Park Map Project. It also introduces students to coordinate systems. This by having them lay out and design their own amusement park. It includes introductions to basic shapes. I am surprised reading the requirements that merry-go-rounds aren’t included, as circles. I am delighted that the plan calls for eight to ten roller coasters and a petting zoo, though. That plan works for me.
I am hosting, later this month, the 141st installment of Denise Gaskins’s Playful Math Education Blog Carnival. If you’ve seen recently any mathematics piece — a blog, a YouTube video, a magazine article — that you found educational or enlightening or just fun, please, share it with me in comments so I can share it with the wider world.
I mentioned yesterday Iva Sallay’s hosting of the 140th Playful Math Education Blog Carnival. This is a collection of pieces of educational, recreational, or otherwise just delightful mathematics posts. I’d said I hoped I might have the energy to host one again this year and, you know? Denise Gaskins, who organizes this monthly event, took me up on the offer.
So, if you write, or read, or are just aware of a good mathematics or mathematics-related blog, please, leave me a comment! I’ll need all the help I can get finding things worth sharing. Anything that you’ve learned from, or that’s delighted you, is worth it. It’ll teach and delight other people too.
Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.
Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.
The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.
And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.
It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.
Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.
Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.
In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.
That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …
This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.
Fermat’s Two-Squares Theorem, discussed here, is not the famous one about . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.
What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.
I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?
Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.
These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.
I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.
Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.
I hope to publish the March 2020 Playful Math Education Blog Carnival tomorrow. If you’ve recently seen any web site that shares and explains some aspect of mathematics or mathematics education that interested or delighted you, please, share it with me, so I can share it with more people. If you do, please, let me know of your own projects, besides that, so I can mention that to this month’s audience. Thank you.
So the first bit of news: I’m hosting the Playful Math Education Blog Carnival later this month. This is a roaming blog link party, sharing blogs that delight or educate, or ideally both, about mathematics. As mentioned the other day Iva Sallay of Find the Factors hosted the 135th of these. My entry, the 136th, I plan to post sometime the last week of March.
And I’ll need help! If you’ve run across a web site, YouTube video, blog post, or essay that discusses something mathematical in a way that makes you grin, please let me know, and let me share it with the carnival audience.
This Saturday is March 14th, which we’ve been celebrating as Pi Day. I remain skeptical that it makes a big difference in people’s view of mathematics or in their education. But an afternoon spent talking about mathematics with everyone agreeing that, for today, we won’t complain about how hard it always was or how impossible we always found it, is pleasant. And that’s a good thing. I don’t know how much activity there’ll be for it, since the 14th is a weekend day this year. And the Covid-19 problem has got all the schools in my state closed through to April, so any calendar relevance is shattered.
But I have some things in the archive anyway. Last year I gathered Six Or Arguably Four Things For Pi Day, a collection of short essays about ways to calculate π well or poorly, and about some of the properties we’re pretty sure that π has, even if we can’t prove it. Also this fascinating physics problem that yields the digits of π.
And the middle of March often brings out Comic Strip Master Command. It looks like I’ve had at least five straight Pi Day editions of Reading the Comics, although most of them cover strips from more than just the 14th of March. From the past:
The weekday Doonesbury has been in reruns for a very long while. Recently it’s been reprinting strips from the 1990s and something that I remember producing Very Worried Editorials, back in the day.
Garry Trudeau’s Doonesbury for the 17th reprints a sequence that starts off with the dread menace and peril of Grade Inflation, the phenomenon in which it turns out students of the generational cohort after yours are allowed to get A’s. (And, to a lesser extent, the phenomenon in which instructors respond to the treatment of education as a market by giving the “customers” the grades they’re “buying”.) The strip does depict an attitude common towards mathematics, though, the idea that it must be a subject immune to Grade Inflation: “aren’t there absolute answers”? If we are careful to say what we mean by an “absolute answer” then, sure.
But grades? Oh, there is so much subjectivity as to what goes into a course. And into what level to teach that course at. How to grade, and how harshly to grade. It may be easier, compared to other subjects, to make mathematics grading more consistent year-to-year. One can make many problems that test the same skill and yet use different numbers, at least until you get into topics like abstract algebra where numbers stop being interesting. But the factors that would allow any course’s grade to inflate are hardly stopped by the department name.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is a strip about using a great wall of equations as emblem of deep, substantial thought. The equations depicted are several meaningful ones. The top row is from general relativity, the Einstein Field Equations. These relate the world-famous Ricci curvature tensor with several other tensors, describing how mass affects the shape of space. The P = NP line describes a problem of computational science with an unknown answer. It’s about whether two different categories of problems are, in fact, equivalent. The line about is a tensor-based scheme to describe the electromagnetic field. The next two lines look, to me, like they’re deep in Schrödinger’s Equation, describing quantum mechanics. It’s possible Weinersmith has a specific problem in mind; I haven’t spotted it.
Ruben Bolling’s Super-Fun-Pak Comix for the 18th is one of the Guy Walks Into A Bar line, each of which has a traditional joke setup undermined by a technical point. In this case, it’s the horse counting in base four, in which representation the number 2 + 2 is written as 10. Really, yes, “10 in base four” is the number four. I imagine properly the horse should say “four” aloud. But it is quite hard to read the symbols “10” as anything but ten. It’s not as though anyone looks at the hexadecimal number “4C” and pronounces it “76”, either.
Garry Trudeau’s Doonesbury for the 19th twisted the Grade Inflation peril to something that felt new in the 90s: an attack on mathematics as “Eurocentric”. The joke depends on the reputation of mathematics as finding objectively true things. Many mathematicians accept this idea. After all, once we’ve seen a proof that we can do the quadrature of a lune, it’s true regardless of what anyone thinks of quadratures and lunes, and whether that person is of a European culture or another one.
But there are several points to object to here. The first is, what’s a quadrature? … This is a geometric thing; it’s finding a square that’s the same area as some given shape, using only straightedge and compass constructions. The second is, what’s a lune? It’s a crescent moon-type shape (hence the name) that you can make by removing the overlap from two circles of specific different radiuses arranged in a specific way. It turns out you can find the quadrature for the lune shape, which makes it seem obvious that you should be able to find the quadrature for a half-circle, a way easier (to us) shape. And it turns out you can’t. The third question is, who cares about making squares using straightedge and compass? And the answer is, well, it’s considered a particularly elegant way of constructing shapes. To the Ancient Greeks. And to those of us who’ve grown in a mathematics culture that owes so much to the Ancient Greeks. Other cultures, ones placing more value on rulers and protractors, might not give a fig about quadratures and lunes.
This before we get into deeper questions. For example, if we grant that some mathematical thing is objectively true, independent of the culture which finds it, then what role does the proof play? It can’t make the thing more or less true. It doesn’t eve matter whether the proof is flawed, or whether it convinces anyone. It seems to imply a mathematician isn’t actually needed for their mathematics. This runs contrary to intuition.
Anyway, this gets off the point of the student here, who’s making a bad-faith appeal to multiculturalism to excuse laziness. It’s difficult to imagine a culture that doesn’t count, at least, even if they don’t do much work with numbers like 144. Granted that, it seems likely they would recognize that 12 has some special relationship with 144, even if they don’t think too much of square roots as a thing.
Goldenoj suggested my topic for today’s essay. It delighted me because I had no idea what it was. It wasn’t even listed on Mathworld, where I start all my research for these essays. It turned out to be something that I use all the time, but that I learned so long ago that it’s faded to invisibility. I didn’t even know that the concept had a name. So that makes it a great topic for an essay like this. I hope.
I once interviewed for a job I didn’t expect to get (or take). I would have taught for a university that provided courses for United States armed forces dependents. One bit of small talk that I thought went well had my potential department head mention a weird little quirk. United States-raised children were unusually good in multiplying stuff by 25. I had a ready hypothesis: the United States (and Canada) have a quarter-dollar coin. Many other countries just don’t, making do with 20-cent and 50-cent pieces instead. The potential department head said that was a good observation. United States-raised kids got practice turning four 25’s into a block of 100.
And this is the thing labelled as unitizing. A unit is, in this context, the thing we think of as “one thing”. This can be dollars, or feet of distance, or loaves of bread, or weeks of paid vacation. Whatever we need to measure. A unit often is made up of tinier pieces, cents or inches or slices or days. It can often be bundled up into bigger ones. Unitizing is about finding the bundle of things that makes the work one wants to do easy to understand.
This is a difficult topic for me to write about. I find it hard to notice myself doing it. But, for example, consider counting. Most people have a fair time counting up to five or six things at a glance. Eighteen things? There’s no telling that at a glance. What you can do, though, is notice that they group together, a block of six things here, another six here, another six there. Then the mass of things has turned into a manageable several collections of manageable counts of things. And, if we need to reverse the process, we can do that. Recognize that the 36 little triangular-wedge game tokens can be given out nine each to the four players. They can in turn arrange six of the tokens into an attractive complete wheel, and make do with the three remainder.
Slices of things turn up a good bit in thought about unitizing. One of particular delight that I found is this paper, by Susan J Lamon. It’s The Development of Unitizing: Its Role in Children’s Partitioning Strategies. Lamon investigated how children understand quantity, and the paper describes several experiments. A typical example is asking children how to evenly divide four pizzas among six people. And how their strategies change if all the pizzas are cut beforehand, versus whether they have to make the cuts themselves. Or how the question changes if things that are not pizza are considered. One child had different cutting strategies for four pizzas versus four cookies. The good reason: cookies are harder to slice than pizzas. You need to be more economical with your cuts so you don’t ruin the food.
And what kids found to be units depended on what was being divided. Four pizzas with different toppings would be divided differently from four identical pizzas. Four Chinese dinners were split by different strategies too. One child explained it just didn’t seem right to call what each person got four-sixths of each dinners. Lamon speculates this reflects cultural conventions about meals that are often eaten in common, and that feels right to me.
There’s obvious uses to this unitizing, in figuring how to divide pizzas and cases of 24 pop cans. There are subtler uses. Positional notation depends on unitizing. We group ten individual things into a new block, and denote it as something in a tens column. Or ten individual blocks-of-ten, which we denote as something in a hundreds column. And we go the other way as we need, when subtracting or dividing.
When I was learning base-ten (and other) arithmetic, they taught me to think of exchanging ten pennies for a dime, or ten dimes for a dollar, or back the other way. To someone hoarding pennies so as to afford things from the bookmobile the practice working out units worked well.
With that context you see why it’s hard to point out what’s happening. You aren’t reading a pop mathematics blog unless you’re quite at ease with calculation. That there is a particular skill done becomes invisible due to its ubiquity. It takes special circumstances to see it again.
Three of the strips I have for this installment feature kids around mathematics talk. That’s enough for a theme name.
Gary Delainey and Gerry Rasmussen’s Betty for the 23rd is a strip about luck. It’s easy to form the superstitious view that you have a finite amount of luck, or that you have good and bad lucks which offset each other. It feels like it. If you haven’t felt like it, then consider that time you got an unexpected $200, hours before your car’s alternator died.
If events are independent, though, that’s just not so. Whether you win $600 in the lottery this week has no effect on whether you win any next week. Similarly whether you’re struck by lightning should have no effect on whether you’re struck again.
Except that this assumes independence. Even defines independence. This is obvious when you consider that, having won $600, it’s easier to buy an extra twenty dollars in lottery tickets and that does increase your (tiny) chance of winning again. If you’re struck by lightning, perhaps it’s because you tend to be someplace that’s often struck by lightning. Probability is a subtler topic than everyone acknowledges, even when they remember that it is such a subtle topic.
Darrin Bell’s Candorville for the 23rd jokes about the uselessness of arithmetic in modern society. I’m a bit surprised at Lemont’s glee in not having to work out tips by hand. The character’s usually a bit of a science nerd. But liking science is different from enjoying doing arithmetic. And bad experiences learning mathematics can sour someone on the subject for life. (Which is true of every subject. Compare the number of people who come out of gym class enjoying physical fitness.)
If you need some Internet Old, read the comments at GoComics, which include people offering dire warnings about what you need in case your machine gives the wrong answer. Which is technically true, but for this application? Getting the wrong answer is not an immediately awful affair. Also a lot of cranky complaining about tipping having risen to 20% just because the United States continues its economic punishment of working peoples.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is some wordplay. Mathematicians often need to find minimums of things. Or maximums of things. Being able to do one lets you do the other, as you’d expect. If you didn’t expect, think about it a moment, and then you expect it. So min and max are often grouped together.
Paul Trap’s Thatababy for the 26th is circling around wordplay, turning some common shape names into pictures. This strip might be aimed at mathematics teachers’ doors. I’d certainly accept these as jokes that help someone learn their shapes.
I’m posting this for several sordid reasons. First is that I want to test whether WordPress has changed something in how pingbacks — a post linking to another post — get handled. Second is I want to get my post count for the month up from its pitifully low number. I’m at something like negative four posts for all April. Third is that oh, yes, it is about that time of the semester when a kind of student is trying to study just hard enough to get a 79.6 percent in their classwork. So they want to study up to an 86.2 on the final and not waste their efforts studying up to an 86.5.
So here’s a couple tables I set up years ago. They show, for some common breakdowns of how much the final exam is worth, and what your class average is before going into the finals, what you’d need to get a 60, 65, 70, 80, or 90.
Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd has a neat variation on story problems. Zoe’s given the assignment to make her own. I don’t remember getting this as homework, in elementary school, but it’s hard to see why I wouldn’t. It’s a great exercise: not just set up an arithmetic problem to solve, but a reason one would want to solve it.
Composing problems is a challenge. It’s a skill, and you might be surprised that when I was in grad school we didn’t get much training in it. We were just taken to be naturally aware of how to identify a skill one wanted to test, and to design a question that would mostly test that skill, and to write it out in a question that challenged students to identify what they were to do and how to do it, and why they might want to do it. But as a grad student I wasn’t being prepared to teach elementary school students, just undergraduates.
Mastroianni and Hart’s B.C. for the 23rd is a joke in the funny-definition category, this for “chaos theory”. Chaos theory formed as a mathematical field in the 60s and 70s, and it got popular alongside the fractal boom in the 80s. The field can be traced back to the 1890s, though, which is astounding. There was no way in the 1890s to do the millions of calculations needed to visualize any good chaos-theory problem. They had to develop results entirely by thinking.
Wiley’s definition is fine enough about certain systems being unpredictable. Wiley calls them “advanced”, although they don’t need to be that advanced. A compound pendulum — a solid rod that swings on the end of another swinging rod — can be chaotic. You can call that “advanced” if you want but then people are going to ask if you’ve had your mind blown by this post-singularity invention, the “screw”.
What makes for chaos is not randomness. Anyone knows the random is unpredictable in detail. That’s no insight. What’s exciting is when something’s unpredictable but deterministic. Here it’s useful to think of continental divides. These are the imaginary curves which mark the difference in where water runs. Pour a cup of water on one side of the line, and if it doesn’t evaporate, it eventually flows to the Pacific Ocean. Pour the cup of water on the other side, it eventually flows to the Atlantic Ocean. These divides are often wriggly things. Water may mostly flow downhill, but it has to go around a lot of hills.
So pour the water on that line. Where does it go? There’s no unpredictability in it. The water on one side of the line goes to one ocean, the water on the other side, to the other ocean. But where is the boundary? And that can be so wriggly, so crumpled up on itself, so twisted, that there’s no meaningfully saying. There’s just this zone where the Pacific Basin and the Atlantic Basin merge into one another. Any drop of water, however tiny, dropped in this zone lands on both sides. And that is chaos.
Neatly for my purposes there’s even a mountain at a great example of this boundary. Triple Divide Peak, in Montana, rests on the divides between the Atlantic and the Pacific basins, and also on the divide between the Atlantic and the Arctic oceans. (If one interprets the Hudson Bay as connecting to the Arctic rather than the Atlantic Ocean, anyway. If one takes Hudson Bay to be on the Atlantic Ocean, then Snow Dome, Alberta/British Columbia, is the triple point.) There’s a spot on this mountain (or the other one) where a spilled cup of water could go to any of three oceans.
John Graziano’s Ripley’s Believe It Or Not for the 23rd mentions one of those beloved bits of mathematics trivia, the birthday problem. That’s finding the probability that no two people in a group of some particular size will share a birthday. Or, equivalently, the probability that at least two people share some birthday. That’s not a specific day, mind you, just that some two people share a birthday. The version that usually draws attention is the relatively low number of people needed to get a 50% chance there’s some birthday pair. I haven’t seen the probability of 70 people having at least one birthday pair before. 99.9 percent seems plausible enough.
The birthday problem usually gets calculated something like this: Grant that one person has a birthday. That’s one day out of either 365 or 366, depending on whether we consider leap days. Consider a second person. There are 364 out of 365 chances that this person’s birthday is not the same as the first person’s. (Or 365 out of 366 chances. Doesn’t make a real difference.) Consider a third person. There are 363 out of 365 chances that this person’s birthday is going to be neither the first nor the second person’s. So the chance that all three have different birthdays is . Consider the fourth person. That person has 362 out of 365 chances to have a birthday none of the first three have claimed. So the chance that all four have different birthdays is . And so on. The chance that at least two people share a birthday is 1 minus the chance that no two people share a birthday.
As always happens there are some things being assumed here. Whether these probability calculations are right depends on those assumptions. The first assumption being made is independence: that no one person’s birthday affects when another person’s is likely to be. Obvious, you say? What if we have twins in the room? What if we’re talking about the birthday problem at a convention of twins and triplets? Or people who enjoyed the minor renown of being their city’s First Babies of the Year? (If you ever don’t like the result of a probability question, ask about the independence of events. Mathematicians like to assume independence, because it makes a lot of work easier. But assuming isn’t the same thing as having it.)
The second assumption is that birthdates are uniformly distributed. That is, that a person picked from a room is no more likely to be born the 13th of February than they are the 24th of September. And that is not quite so. September births are (in the United States) slightly more likely than other months, for example, which suggests certain activities going on around New Year’s. Across all months (again in the United States) birthdates of the 13th are slightly less likely than other days of the month. I imagine this has to be accounted for by people who are able to select a due date by inducing delivery. (Again if you need to attack a probability question you don’t like, ask about the uniformity of whatever random thing is in place. Mathematicians like to assume uniform randomness, because it akes a lot of work easier. But assuming it isn’t the same as proving it.)
Do these differences mess up the birthday problem results? Probably not that much. We are talking about slight variations from uniform distribution. But I’ll be watching Ripley’s to see if it says anything about births being more common in September, or less common on 13ths.
I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.
John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.
I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.
(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.
Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.
Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.
Nobody had a suggested topic starting with ‘W’ for me! So I’ll take that as a free choice, and get lightly autobiogrpahical.
Witch of Agnesi.
I know I encountered the Witch of Agnesi while in middle school. Eighth grade, if I’m not mistaken. It was a footnote in a textbook. I don’t remember much of the textbook. What I mostly remember of the course was how much I did not fit with the teacher. The only relief from boredom that year was the month we had a substitute and the occasional interesting footnote.
It was in a chapter about graphing equations. That is, finding curves whose points have coordinates that satisfy some equation. In a bit of relief from lines and parabolas the footnote offered this:
In a weird tantalizing moment the footnote didn’t offer a picture. Or say what an ‘a’ was doing in there. In retrospect I recognize ‘a’ as a parameter, and that different values of it give different but related shapes. No hint what the ‘8’ or the ‘4’ were doing there. Nor why ‘a’ gets raised to the third power in the numerator or the second in the denominator. I did my best with the tools I had at the time. Picked a nice easy boring ‘a’. Picked out values of ‘x’ and found the corresponding ‘y’ which made the equation true, and tried connecting the dots. The result didn’t look anything like a witch. Nor a witch’s hat.
It was one of a handful of biographical notes in the book. These were a little attempt to add some historical context to mathematics. It wasn’t much. But it was an attempt to show that mathematics came from people. Including, here, from Maria Gaëtana Agnesi. She was, I’m certain, the only woman mentioned in the textbook I’ve otherwise completely forgotten.
We have few names of ancient mathematicians. Those we have are often compilers like Euclid whose fame obliterated the people whose work they explained. Or they’re like Pythagoras, credited with discoveries by people who obliterated their own identities. In later times we have the mathematics done by, mostly, people whose social positions gave them time to write mathematics results. So we see centuries where every mathematician is doing it as their side hustle to being a priest or lawyer or physician or combination of these. Women don’t get the chance to stand out here.
Today of course we can name many women who did, and do, mathematics. We can name Emmy Noether, Ada Lovelace, and Marie-Sophie Germain. Challenged to do a bit more, we can offer Florence Nightingale and Sofia Kovalevskaya. Well, and also Grace Hopper and Margaret Hamilton if we decide computer scientists count. Katherine Johnson looks likely to make that cut. But in any case none of these people are known for work understandable in a pre-algebra textbook. This must be why Agnesi earned a place in this book. She’s among the earliest women we can specifically credit with doing noteworthy mathematics. (Also physics, but that’s off point for me.) Her curve might be a little advanced for that textbook’s intended audience. But it’s not far off, and pondering questions like “why ? Why not ?” is more pleasant, to a certain personality, than pondering what a directrix might be and why we might use one.
The equation might be a lousy way to visualize the curve described. The curve is one of that group of interesting shapes you get by constructions. That is, following some novel process. Constructions are fun. They’re almost a craft project.
For this we start with a circle. And two parallel tangent lines. Without loss of generality, suppose they’re horizontal, so, there’s lines at the top and the bottom of the curve.
Take one of the two tangent points. Again without loss of generality, let’s say the bottom one. Draw a line from that point over to the other line. Anywhere on the other line. There’s a point where the line you drew intersects the circle. There’s another point where it intersects the other parallel line. We’ll find a new point by combining pieces of these two points. The point is on the same horizontal as wherever your line intersects the circle. It’s on the same vertical as wherever your line intersects the other parallel line. This point is on the Witch of Agnesi curve.
Now draw another line. Again, starting from the lower tangent point and going up to the other parallel line. Again it intersects the circle somewhere. This gives another point on the Witch of Agnesi curve. Draw another line. Another intersection with the circle, another intersection with the opposite parallel line. Another point on the Witch of Agnesi curve. And so on. Keep doing this. When you’ve drawn all the lines that reach from the tangent point to the other line, you’ll have generated the full Witch of Agnesi curve. This takes more work than writing out , yes. But it’s more fun. It makes for neat animations. And I think it prepares us to expect the shape of the curve.
It’s a neat curve. Between it and the lower parallel line is an area four times that of the circle that generated it. The shape is one we would get from looking at the derivative of the arctangent. So there’s some reasons someone working in calculus might find it interesting. And people did. Pierre de Fermat studied it, and found this area. Isaac Newton and Luigi Guido Grandi studied the shape, using this circle-and-parallel-lines construction. Maria Agnesi’s name attached to it after she published a calculus textbook which examined this curve. She showed, according to people who present themselves as having read her book, the curve and how to find it. And she showed its equation and found the vertex and asymptote line and the inflection points. The inflection points, here, are where the curve chances from being cupped upward to cupping downward, or vice-versa.
It’s a neat function. It’s got some uses. It’s a natural smooth-hill shape, for example. So this makes a good generic landscape feature if you’re modeling the flow over a surface. I read that solitary waves can have this curve’s shape, too.
And the curve turns up as a probability distribution. Take a fixed point. Pick lines at random that pass through this point. See where those lines reach a separate, straight line. Some regions are more likely to be intersected than are others. Chart how often any particular line is the new intersection point. That chart will (given some assumptions I ask you to pretend you agree with) be a Witch of Agnesi curve. This might not surprise you. It seems inevitable from the circle-and-intersecting-line construction process. And that’s nice enough. As a distribution it looks like the usual Gaussian bell curve.
It’s different, though. And it’s different in strange ways. Like, for a probability distribution we can find an expected value. That’s … well, what it sounds like. But this is the strange probability distribution for which the law of large numbers does not work. Imagine an experiment that produces real numbers, with the frequency of each number given by this distribution. Run the experiment zillions of times. What’s the mean value of all the zillions of generated numbers? And it … doesn’t … have one. I mean, we know it ought to, it should be the center of that hill. But the calculations for that don’t work right. Taking a bigger sample makes the sample mean jump around more, not less, the way every other distribution should work. It’s a weird idea.
Imagine carving a block of wood in the shape of this curve, with a horizontal lower bound and the Witch of Agnesi curve as the upper bound. Where would it balance? … The normal mathematical tools don’t say, even though the shape has an obvious line of symmetry. And a finite area. You don’t get this kind of weirdness with parabolas.
(Yes, you’ll get a balancing point if you actually carve a real one. This is because you work with finitely-long blocks of wood. Imagine you had a block of wood infinite in length. Then you would see some strange behavior.)
It teaches us more strange things, though. Consider interpolations, that is, taking a couple data points and fitting a curve to them. We usually start out looking for polynomials when we interpolate data points. This is because everything is polynomials. Toss in more data points. We need a higher-order polynomial, but we can usually fit all the given points. But sometimes polynomials won’t work. A problem called Runge’s Phenomenon can happen, where the more data points you have the worse your polynomial interpolation is. The Witch of Agnesi curve is one of those. Carl Runge used points on this curve, and trying to fit polynomials to those points, to discover the problem. More data and higher-order polynomials make for worse interpolations. You get curves that look less and less like the original Witch. Runge is himself famous to mathematicians, known for “Runge-Kutta”. That’s a family of techniques to solve differential equations numerically. I don’t know whether Runge came to the weirdness of the Witch of Agnesi curve from considering how errors build in numerical integration. I can imagine it, though. The topics feel related to me.
I understand how none of this could fit that textbook’s slender footnote. I’m not sure any of the really good parts of the Witch of Agnesi could even fit thematically in that textbook. At least beyond the fact of its interesting name, which any good blog about the curve will explain. That there was no picture, and that the equation was beyond what the textbook had been describing, made it a challenge. Maybe not seeing what the shape was teased the mathematician out of this bored student.
And next is ‘X’. Will I take Mr Wu’s suggestion and use that to describe something “extreme”? Or will I take another topic or suggestion? We’ll see on Friday, barring unpleasant surprises. Thanks for reading.
I’m not sure there is a theme to the back half of last week’s mathematically-based comic strips. If there is, it’s about showing some origins of things. I’ll go with that title, then.
Bill Holbrook’s On The Fastrack for the 21st is another in the curious thread of strips about Fi talking about mathematics. She’s presented as doing a good job inspiring kids to appreciate mathematics as a fun, exciting, interesting thing to think about. It’s good work. And I hope this does not sound like I am envious of a more successful, if fictional, mathematics popularizer. But I don’t see much in the strip of her doing this side job well. That is, of making the case that mathematics is worth the time spent on it. That’s a lot to ask given the confines of a syndicated daily newspaper comic strip, yes. What we can expect is some hint of what the actual good argument would look like. But this particular day’s strip rings false to me, for example. I don’t see how “here’s some pizza — but first, here’s a pop quiz” makes mathematics look as something other than a chore.
Pizza area offers many ways into mathematical ideas. How the area depends on the size of the pizza, for example. How the area depends on the shape, even independently of the size. How to slice a pizza fairly, especially if it’s not to be between four or six or eight people. What is the strangest shape you could make that would give people equal areas? Just the way slices intersect at angles inspires neat little geometry problems. How you might arrange toppings opens up symmetries and tilings, which are surprisingly big areas of mathematics. Setting problems on a pizza gives them a tangibility that could help capture young minds, surely. But I can’t make myself believe that this is a conversation to have when the pizza is entering the room.
Mike Peters’s Mother Goose and Grimm for the 22nd is a lottery joke. So if we suppose this was written about the last time the Powerball jackpot reached a half-billion dollars we can work out how far ahead of publication Mike Peters is working. One solid argument against ever buying a lottery ticket is, as Grimm notes, that you have zero chance of winning. (I’m open to an argument based on expectation value. And even more, I don’t object to people spending a reasonable bit of disposable income “foolishly”.) Mother Goose argues that her chances are vastly worse if she doesn’t buy a ticket. This is true. Are her chances “astronomically” worse? … That depends. A one in three hundred million chance (to use, roughly, the Powerball odds) is so small that it won’t happen to you. Is that any different than a zero in three hundred million chance [*]? Or than a six in three hundred million chance? In any case it won’t happen to you.
[*] Do you actually have zero chance of winning if you don’t have a ticket? I say no, you don’t. Someone might give you a winning ticket. Maybe you find one as a bookmark in a library book. Maybe you find it on the street and figure, what the heck, I’ll check. Unlikely? Sure. But impossible? Hardly.
Johnny Hart’s Back to BC for the 22nd has the form of the world’s oldest story problem. It could also be a joke about the discovery of the concept of zero and the struggle to understand it as a number. Given that clams are used as currency in the BC setting it also shows how finance has driven mathematical development. So the strip actually packs a fair bit of stuff into two panels. … And I’ll admit I’m not quite sure the joke parses, but if you read it quickly it looks like a good enough joke.
Johnny Hart’s Back to BC for the 24th is a more obvious joke. And it’s built on the learning abilities of animals, and the number sense of animals. A large animal stomping a foot evokes, to me at least, Clever Hans. This is a horse presented in the early 20th century as being able to actually do arithmetic. The horse would be given a question and would stop his hoof enough times to get to the right answer. However good the horse’s number sense might be, he had quite good behavioral sense. It turned out — after brilliant and pioneering work in animal cognition — that Hans was observing his trainer’s body language. When Wilhelm von Osten was satisfied that there’d been the right number of stomps, the horse stopped. This is sometimes presented as Hans `merely’ taking subconscious cues from his trainer. But consider how carefully the horse must be observing an animal with a very different body, and how it must have understood cues of satisfaction. I can’t call that `mere’. And the work of tracking down a signal that von Osten himself did not know he was sending (and, apparently, never accepted that he did) is also amazing. It serves as a reminder how hard biologists and zoologists have to work.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th gives a bit of Dad History about perspective. And, particularly, why artists didn’t seem to use it much before the 16th century. It gets more blatantly tied to mathematics by pointing out how it took ten thousand years of civilization to get Cartesian coordinates. We can argue about how many years civilization has been around. But it does seem strange that we went along for certainly the majority of that time without Cartesian coordinates. They seem so obvious it’s almost hard to not think of them. Many good ideas have such a legacy.
It’s easy to say why older pictures didn’t use perspective, though. For the most part, artists didn’t think perspective gave them something they wanted to show. Ancient peoples knew of perspective. It’s not as if ancient peoples were any dumber than we are, or any less able to look at square tiles held at different angles and at different distances. But we can convey information about the importance of things, or the flow of action of things, using position and relative size. That can be more important than showing that yes, an artist is aware that a square building far away looks small.
I’m less sure what I know about the history of coordinate systems, though, and particularly why it took until René Descartes to describe them. We have a legend of Descartes laying in bed, watching a fly on the tiled ceiling, and realizing he could describe where the fly was by what row and column of tile it was on. (In the past I have written this as though it happened. In writing this essay I went looking for a primary source and found nobody seems to have one. I shall try not to pass it on again without being very clear that it is just a legend.) But there have been tiled floors and walls and ceilings for a very long time. There have been flies even longer. Why didn’t anyone notice this?
One answer may be that they did. We just haven’t heard about it, because it was found by someone who didn’t catch the interest of a mathematical community. There’s likely a lot of such lost mathematics out there. But still, why not? Wouldn’t anyone with a mathematical inclination see that this is plainly a great discovery? And maybe not. What made Cartesian coordinates great was the realization that arithmetic and geometry, previously seen as separate liberal arts, were duals. A problem in one had an expression as a problem in the other. If you don’t make that connection, then Cartesian coordinates don’t solve any problems you have. They’re just a new way to index things you didn’t need indexed. So that would slow down using them any.
Greetings one and all! Come, gather round! Wonder and spectate and — above all else — tell your friends of the Playful Mathematics Blog Carnival! Within is a buffet of delights and treats, fortifications for the mind and fire for the imagination.
121 is a special number. When I was a mere tot, growing in the wilds of suburban central New Jersey, it stood there. It held a spot of privilege in the multiplication tables on the inside front cover of composition books. On the forward diagonal, yet insulated from the borders. It anchors the safe interior. A square number, eleventh of that set in the positive numbers.
121 is more than just a square. It is the lone square known to be the sum of the first several powers of a prime number: it is , a fantastic combination. If there is another square that is such a sum of primes, it is unknown to any human — and must be at least 35 digits long.
We look now for a moment at some astounding animals. From the renowned Dr Nic: Introducing Cat Maths cards, activities, games and lessons — a fine collection of feline companions, such toys as will enterain them. A dozen attributes each; twenty-seven value cards. These cats, and these cards, and these activity puzzles, promise games and delights, to teach counting, subtraction, statistics, and inference!
Next and no less incredible is the wooly Mathstodon. Christian Lawson-Perfect hosts this site, an instance of the open-source Twitter-like service Mastodon. Its focus: a place for people interested in mathematicians to write of what they know. To date over 1,300 users have joined, and have shared nearly 25,000 messages. You need not join to read many of these posts — your host here has yet to — but may sample its wares as you like.
The Second Tent
121 is one of only two perfect squares known to be four less than the cube of a whole number. The great Fermat conjectured that 4 and 121 are the only such numbers; no one has found a counter-example. Nor a proof.
Friends, do you know the secret to popularity? There is an astonishing truth behind it. Elias Worth of the MathSection blog explains the Friendship Paradox. This mind-warping phenomenon tells us your friends have more friends than you do. It will change forever how you look at your followers and following accounts.
And now to thoughts of learning. Stepping forward now is Monica Utsey, @Liveonpurpose47 of Chocolate Covered Boy Joy. Her declaration: “I incorporated Montessori Math materials with my right brain learner because he needed literal representations of the work we were doing. It worked and we still use it.” See now for yourself the representations, counting and comparing and all the joys of several aspects of arithmetic.
Take now a moment for your own fun. Blog Carnival patron and organizer Denise Gaskins wishes us to know: “The fun of mathematical coloring isn’t limited to one day. Enjoy these coloring resources all year ’round!” Happy National Coloring Book Day offers the title, and we may keep the spirit of National Coloring Book Day all the year round.
121 is a star number, the fifth of that select set. 121 identical items can be tiled to form a centered hexagon. You may have seen it in the German game of Chinese Checkers, as the board of that has 121 holes.
We come back again to teaching. “Many homeschoolers struggle with teaching their children math. Here are some tips to make it easier”, offers Denise Gaskins. Step forth and benefit from this FAQ: Struggling with Arithmetic, a collection of tips and thoughts and resources to help make arithmetic the more manageable.
Step now over to the arcade, and to the challenge of Pac-Man. This humble circle-inspired polygon must visit the entirety of a maze, and avoid ghosts as he does. Matthew Scroggs of Chalk Dust Magazine here seeks and shows us Optimal Pac-Man. Graph theory tells us there are thirteen billion different paths to take. Which of them is shortest? Which is fastest? Can it be known, and can it help you through the game?
121 is furthermore the sixth of the centered octagonal numbers. 121 of a thing may be set into six concentric octagons of one, then two, then three, then four, then five, and then six of them on a side.
Step now back to the amazing Mathstodon. Gaze in wonder at the account @dudeney_puzzles. Since the September of 2017 it has brought out challenges from Henry Ernest Dudeney’s Amusements in Mathematics. Puzzles given, yes, with answers that follow along. The impatient may find Dudeney’s 1917 book on Project Gutenberg among other places.
The Fifth Tent
Sum the digits of 121; you will find that you have four. Take its prime factors, 11 and 11, and sum their digits; you will find that this is four again. This makes 121 a Smith number. These marvels of the ages were named by Albert Wilansky, in honor of his brother-in-law, a man known to history as Harold Smith, and whose telephone number of 4,937,775 was one such.
And now to an astounding challenge. Imagine an assassin readies your death. Can you protect yourself? At all? Tai-Danae Bradley invites you to consider: Is the Square a Secure Polygon? This question takes you on a tour of geometries familiar and exotic. Learn how mathematicians consider how to walk between places on a torus — and the lessons this has for a square room. The fate of the universe itself may depend on the methods described herein — the techniques used to study it relate to those that study whether a physical system can return to its original state. And then J2kun turned this into code, Visualizing an Assassin Puzzle, for those who dare to program it.
Have you overcome this challenge? Then step into the world of linear algebra, and this delight from the Mathstodon account of Christian Lawson-Perfect. The puzzle is built on the wonders of eigenvectors, those marvels of matrix multiplication. They emerge from multiplication longer or shorter but unchanged in direction. Lawson-Perfect uses whole numbers, represented by Scrabble tiles, and finds a great matrix with a neat eigenvalue. Can you prove that this is true?
The Sixth Tent
Another wonder of the digits of 121. Take them apart, then put them together again. Contorted into the form 112 they represent the same number. 121 is, in the base ten commonly used in the land, a Friedman Number, second of that line. These marvels, in the Arabic, the Roman, or even the Mayan numerals schemes, are named for Erich Friedman, a figure of mystery from the Stetson University.
Denise Gaskins coordinates the Playful Mathematics Education Blog Carnival. Upcoming scheduled carnivals, including the chance to volunteer to host it yourself, or to recommend your site for mention, are listed here. And October’s 122nd Playful Mathematics Education Blog Carnival is scheduled to be hosted by Arithmophobia No More, and may this new host have the best of days!
This one I saw through John Allen Paulos’s twitter feed. He points out that it’s like the Collatz conjecture but is, in fact, proven. If you try this yourself don’t make the mistake of giving up too soon. You might figure, like start with 12. Sum the squares of its digits and you get 5, which is neither 1 nor anything in that 4-16-37-58-89-145-42-20 cycle. Not so! Square 5 and you get 25. Square those digits and add them and you get 29. Square those digits and add them and you get 40. And what comes next?
This is about a proof of Fermat’s Theorem of Sums of Two Squares. According to it, a prime number — let’s reach deep into the alphabet and call it p — can be written as the sum of two squares if and only if p is one more than a whole multiple of four. It’s a proof by using fixed point methods. This is a fun kind of proof, at least to my sense of fun. It’s an approach that’s got a clear physical interpretation. Imagine picking up a (thin) patch of bread dough, stretching it out some and maybe rotating it, and then dropping it back on the board. There’s at least one bit of dough that’s landed in the same spot it was before. Once you see this you will never be able to just roll out dough the same way. So here the proof involves setting up an operation on integers which has a fixed point, and that the fixed point makes the property true.
John D Cook, who runs a half-dozen or so mathematics-fact-of-the-day Twitter feeds, looks into calculating the volume of an egg. It involves calculus, as finding the volume of many interesting shapes does. I am surprised to learn the volume can be written out as a formula that depends on the shape of the egg. I would have bet that it couldn’t be expressed in “closed form”. This is a slightly flexible term. It’s meant to mean the thing can be written using only normal, familiar functions. However, we pretend that the inverse hyperbolic tangent is a “normal, familiar” function.
For example, there’s the surface area of an egg. This can be worked out too, again using calculus. It can’t be written even with the inverse hyperbolic cotangent, so good luck. You have to get into numerical integration if you want an answer humans can understand.
Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.
Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.
Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.
Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)
The end of the (US) semester snuck up on me but, in my defense, I’m not teaching this semester. If you know someone who needs me to teach, please leave me a note. But as a service for people who are just trying to figure out exactly how much studying they need to do for their finals, knock it off. You’re not playing a video game. It’s not like you can figure out how much effort it takes to get an 83.5 on the final and then put the rest of your energy into your major’s classes.
For those not interested in grade-grubbing, here’s some old-time radio. Vic and Sade was a longrunning 15-minute morning radio program written with exquisite care by Paul Rhymer. It’s not going to be to everyone’s taste. But if it is yours, it’s going to be really yours: a tiny cast of people talking not quite past one another while respecting the classic Greek unities. Part of the Overnightscape Underground is the Vic and Sadecast, which curates episodes of the show, particularly trying to explain the context of things gone by since 1940. This episode, from October 1941, is aptly titled “It’s Algebra, Uncle Fletcher”. Neither Vic nor Sade are in the episode, but their son Rush and Uncle Fletcher are. And they try to work through high school algebra problems. I’m tickled to hear Uncle Fletcher explaining mathematics homework. I hope you are too.
This week was another average-grade week of mathematically-themed comic strips. I wonder if I should track them and see what spurious correlations between events and strips turn up. That seems like too much work and there’s better things I could do with my time, so it’s probably just a few weeks before I start doing that.
Ruben Bolling’s Super-Fun-Pax Comics for the 19th is an installment of A Voice From Another Dimension. It’s in that long line of mathematics jokes that are riffs on Flatland, and how we might try to imagine spaces other than ours. They’re taxing things. We can understand some of the rules of them perfectly well. Does that mean we can visualize them? Understand them? I’m not sure, and I don’t know a way to prove whether someone does or does not. This wasn’t one of the strips I was thinking of when I tossed “shapes” into the edition title, but you know what? It’s close enough to matching.
Olivia Walch’s Imogen Quest for the 20th — and I haven’t looked, but it feels to me like I’m always featuring Imogen Quest lately — riffs on the Monty Hall Problem. The problem is based on a game never actually played on Monty Hall’s Let’s Make A Deal, but very like ones they do. There’s many kinds of games there, but most of them amount to the contestant making a choice, and then being asked to second-guess the choice. In this case, pick a door and then second-guess whether to switch to another door. The Monty Hall Problem is a great one for Internet commenters to argue about while the rest of us do something productive. The trouble — well, one trouble — is that whether switching improves your chance to win the car is that whether it does depends on the rules of the game. It’s not stated, for example, whether the host must open a door showing a goat behind it. It’s not stated that the host certainly knows which doors have goats and so chooses one of those. It’s not certain the contestant even wants a car when, hey, goats. What assumptions you make about these issues affects the outcome.
If you take the assumptions that I would, given the problem — the host knows which door the car’s behind, and always offers the choice to switch, and the contestant would rather have a car, and such — then Walch’s analysis is spot on.
Todd Clark’s Lola for the 20th does the old “when do I need to know algebra” question and I admit getting grumpy like this when people ask. Do French teachers have to put up with this stuff?
Brian Fies’s Mom’s Cancer rerun for the 23rd is from one of the delicate moments in her story. Fies’s mother just learned the average survival rate for her cancer treatment is about five percent and, after months of things getting haltingly better, is shaken. But as with most real-world probability questions context matters. The five-percent chance is, as described, the chance someone who’d just been diagnosed in the state she’d been diagnosed in would survive. The information that she’s already survived months of radiation and chemical treatment and physical therapy means they’re now looking at a different question. What is the chance she will survive, given that she has survived this far with this care?
Mark Anderson’s Andertoons for the 24th is the Mark Anderson’s Andertoons for the week. It’s a protesting-student kind of joke. For the student’s question, I’m not sure how many sides a polygon has before we can stop memorizing them. I’d say probably eight. Maybe ten. Of the shapes whose names people actually care about, mm. Circle, triangle, a bunch of quadrilaterals, pentagons, hexagons, octagons, maybe decagon and dodecagon. No, I’ve never met anyone who cared about nonagons. I think we could drop heptagons without anyone noticing either. Among quadrilaterals, ugh, let’s see. Square, rectangle, rhombus, parallelogram, trapezoid (or trapezium), and I guess diamond although I’m not sure what that gets you that rhombus doesn’t already. Toss in circles, ellipses, and ovals, and I think that’s all the shapes whose names you use.
And now the other half of last week’s comic strips. It was unusually rich in comics that come from Comics Kingdom or Creators.com, which have limited windows of access and therefore make me feel confident I should include the strips so my comments make any sense.
Rick Kirkman and Jerry Scott’s Baby Blues for the 9th mentions mathematics homework as a resolutely rage-inducing topic. It’s mathematics homework, obviously, or else it wouldn’t be mentioned around here. And even more specifically it’s Common Core mathematics homework. So it always is with attempts to teach subjects better. Especially mathematics, given how little confidence people have in their own mastery. I can’t blame parents for supposing any change to be just malice.
Chris Browne’s Hagar the Horrible for the 10th is about the numerous. There’s different kinds of limits. There’s the greatest number of things we can count in an instant. There’s a limit to how long a string of digits or symbols we can remember. There’s the biggest number of things we can visualize. And “visualize” is a slippery concept. I think I have a pretty good idea what we mean when we say “a thousand” of something. I could calculate how long it took me to do something a thousand times, or to write a thousand of something. I know that it was at about a thousand words that, last A To Z sequence, I got to feeling I should wrap up any particular essay. But did I see any particular difference between word 999 and word 1,000? No; what I really knew was “about enough paragraphs” and maybe “fills just over two screens in my text editor”. So do I know what a thousand is? Anyway, we all have our limits, acknowledge them or not.
Henry Scarpelli and Craig Boldman’s Archie rerun for the 17th is about Moose’s struggle with mathematics. Just writing “more or less” doesn’t fix an erroneous answer, true. But error margins, and estimates of where an answer should be, can be good mathematics. (Part of the Common Core that many parents struggle with is making the estimate of an answer the first step, and a refined answer later. Based on what I see crossing social media, this really offends former engineering majors who miss the value in having an expected approximate answer.) It’s part of how we define limits, and derivatives, and integrals, and all of calculus. But it’s in a more precise way than Moose tries to do.
Comic Strip Master Command hasn’t had many comics exactly on mathematical points the past week. I’ll make do. There are some that are close enough for me, since I like the comics already. And enough of them circle around people being nervous about doing mathematics that I have a title for this edition.
Tony Cochrane’s Agnes for the 24th talks about math anxiety. It’s not a comic strip that will do anything to resolve anyone’s mathematics anxiety. But it’s funny about its business. Agnes usually is; it’s one of the less-appreciated deeply-bizarre comics out there.
Charles Schulz’s Peanuts for the 24th reruns the comic from the 2nd of November, 1970. It has Sally discovering that multiplication is much easier than she imagined. As it is, she’s not in good shape. But if you accept ‘tooty-two’ as another name for ‘four’ and ‘threety-three’ as another name for ‘nine’, why not? And she might do all right in group theory. In that you can select a bunch of things, called ‘elements’, and describe their multiplication to fit anything you like, provided there’s consistency. There could be a four-forty-four if that seems to answer some question.
Hilary Price’s Rhymes with Orange for the 26th is a calculator joke, made explicitly magical. I’m amused but also wonder if those are small wizards or large mushrooms. And it brings up again the question: why do mathematics teachers care about seeing how you got the answer? Who cares, as long as the answer is right? And my answer there is that yeah, sometimes all we care about is the answer. But more often we care about why someone knows the answer is this instead of that. The argument about what makes this answer right — or other answers wrong — should make it possible to tell why. And it often will help inform other problems. Being able to use the work done for one problem to solve others, or better, a whole family of problems, is fantastic. It’s the sort of thing mathematicians naturally try to do.
The week was looking ready to be one where I have my five paragraphs about how something shows off a word problem and that’s it. And then Comic Strip Master Command turned up the flow of comics for Saturday. So, here’s my five paragraphs about something being word problems and we’ll pick up the other half of them soon.
Bill Whitehead’s Free Range for the 10th is an Albert Einstein joke. That’s usually been enough. That it mentions curved space, the exotic geometries that make general relativity so interesting, gives it a little more grounding as a mathematical comic. It’s a bit curious, surely, that curved space strikes people as so absurd. Nobody serious argues whether we live on a curved space, though, not when we see globes and think about shapes that cover a big part of the surface of the Earth. But there is something different about thinking of three-dimensional space as curved; it’s hard to imagine curved around what.
Brian Basset’s Red and Rover started some word problems on the 11th, this time with trains travelling in separate directions. The word problem seemed peculiar, since the trains wouldn’t be 246 miles apart at any whole number of hours. But they will be at a reasonable fraction more than a whole number of hours, so I guess Red has gotten to division with fractions.
Red and Rover are back at it the 12th with basically the same problem. This time it’s with airplanes. Also this time it’s a much worse problem. While you can do the problem still, the numbers are uglier. It’ll be just enough over two hours and ten minutes that I wonder if the numbers got rewritten away from some nicer set. For example, if the planes had been flying at 360 and 540 miles per hour, and the question was when they would be 2,100 miles apart, then you’d have a nice two-and-a-third hours.
The rest of last week had more mathematically-themed comic strips than Sunday alone did. As sometimes happens, I noticed an objectively unimportant detail in one of the comics and got to thinking about it. Whether I could solve the equation as posted, or whether at least part of it made sense as a mathematics problem. Well, you’ll see.
Patrick McDonnell’s Mutts for the 25th of September I include because it’s cute and I like when I can feature some comic in these roundups. Maybe there’s some discussion that could be had about what “equals” means in ordinary English versus what it means in mathematics. But I admit that’s a stretch.
Olivia Walch’s Imogen Quest for the 25th uses, and describes, the mathematics of a famous probability problem. This is the surprising result of how few people you need to have a 50 percent chance that some pair of people have a birthday in common. It then goes over to some other probability problems. The examples are silly. But the reasoning is sound. And the approach is useful. To find the chance of something happens it’s often easiest to work out the chance it doesn’t. Which is as good as knowing the chance it does, since a thing can either happen or not happen. At least in probability problems, which define “thing” and “happen” so there’s not ambiguity about whether it happened or not.
Piers Baker’s Ollie and Quentin rerun for the 26th I’m pretty sure I’ve written about before, although back before I included pictures of the Comics Kingdom strips. (The strip moved from Comics Kingdom over to GoComics, which I haven’t caught removing old comics from their pages.) Anyway, it plays on a core piece of probability. It sets out the world as things, “events”, that can have one of multiple outcomes, and which must have one of those outcomes. Coin tossing is taken to mean, by default, an event that has exactly two possible outcomes, each equally likely. And that is near enough true for real-world coin tossing. But there is a little gap between “near enough” and “true”.
Rick Stromoski’s Soup To Nutz for the 27th is your standard sort of Dumb Royboy joke, in this case about him not knowing what percentages are. You could do the same joke about fractions, including with the same breakdown of what part of the mathematics geek population ruins it for the remainder.
Nate Fakes’s Break of Day for the 28th is not quite the anthropomorphic-numerals joke for the week. Anthropomorphic mathematics problems, anyway. The intriguing thing to me is that the difficult, calculus, problem looks almost legitimate to me. On the right-hand-side of the first two lines, for example, the calculation goes from
This is a little sloppy. The first line ought to end in a ‘dt’, and the second ought to have a constant of integration. If you don’t know what these calculus things are let me explain: they’re calculus things. You need to include them to express the work correctly. But if you’re just doing a quick check of something, the mathematical equivalent of a very rough preliminary sketch, it’s common enough to leave that out.
It doesn’t quite parse or mean anything precisely as it is. But it looks like the sort of thing that some context would make meaningful. That there’s repeated appearances of , or , particularly makes me wonder if Frakes used a problem he (or a friend) was doing for some reason.
Comic Strip Master Command sent a nice little flood of comics this week, probably to make sure that I transitioned from the A To Z project to normal activity without feeling too lost. I’m going to cut the strips not quite in half because I’m always delighted when I can make a post that’s just a single day’s mathematically-themed comics. Last Sunday, the 24th of September, was such a busy day. I’m cheating a little on what counts as noteworthy enough to talk about here. But people like comic strips, and good on them for liking them.
Norm Feuti’s Gil for the 24th sees Gil discover and try to apply some higher mathematics. There’s probably a good discussion about what we mean by division to explain why Gil’s experiment didn’t pan out. I would pin it down to eliding the difference between “dividing in half” and “dividing by a half”, which is a hard one. Terms that seem almost alike but mean such different things are probably the hardest part of mathematics.
Russell Myers’s Broom Hilda looks like my padding. But the last panel of the middle row gets my eye. The squirrels talk about how on the equinox night and day “can never be of identical length, due to the angular size of the sun and atmospheric refraction”. This is true enough for the equinox. While any spot on the Earth might see twelve hours facing the sun and twelve hours facing away, the fact the sun isn’t a point, and that the atmosphere carries light around to the “dark” side of the planet, means daylight lasts a little longer than night.
Ah, but. This gets my mathematical modelling interest going. Because it is true that, at least away from the equator, there’s times of year that day is way shorter than night. And there’s times of year that day is way longer than night. Shouldn’t there be some time in the middle when day is exactly equal to night?
The easy argument for is built on the Intermediate Value Theorem. Let me define a function, with domain each of the days of the year. The range is real numbers. It’s defined to be the length of day minus the length of night. Let me say it’s in minutes, but it doesn’t change things if you argue that it’s seconds, or milliseconds, or hours, if you keep parts of hours in also. So, like, 12.015 hours or something. At the height of winter, this function is definitely negative; night is longer than day. At the height of summer, this function is definitely positive; night is shorter than day. So therefore there must be some time, between the height of winter and the height of summer, when the function is zero. And therefore there must be some day, even if it isn’t the equinox, when night and day are the same length
Mike Baldwin’s Cornered features an old-fashioned adding machine being used to drown an audience in calculations. Which makes for a curious pairing with …
Bill Amend’s FoxTrot, and its representation of “math hipsters”. I hate to encourage Jason or Marcus in being deliberately difficult. But there are arguments to make for avoiding digital calculators in favor of old-fashioned — let’s call them analog — calculators. One is that people understand tactile operations better, or at least sooner, than they do digital ones. The slide rule changes multiplication and division into combining or removing lengths of things, and we probably have an instinctive understanding of lengths. So this should train people into anticipating what a result is likely to be. This encourages sanity checks, verifying that an answer could plausibly be right. And since a calculation takes effort, it encourages people to think out how to arrange the calculation to require less work. This should make it less vulnerable to accidents.
I suspect that many of these benefits are what you get in the ideal case, though. Slide rules, and abacuses, are no less vulnerable to accidents than anything else is. And if you are skilled enough with the abacus you have no trouble multiplying 18 by 7, you probably would not find multiplying 17 by 8 any harder, and wouldn’t notice if you mistook one for the other.
Jef Mallett’s Frazz asserts that numbers are cool but the real insight is comparisons. And we can argue that comparisons are more basic than numbers. We can talk about one thing being bigger than another even if we don’t have a precise idea of numbers, or how to measure them. See every mathematics blog introducing the idea of different sizes of infinity.
Bill Whitehead’s Free Range features Albert Einstein, universal symbol for really deep thinking about mathematics and physics and stuff. And even a blackboard full of equations for the title panel. I’m not sure whether the joke is a simple absent-minded-professor joke, or whether it’s a relabelled joke about Werner Heisenberg. Absent-minded-professor jokes are not mathematical enough for me, so let me point once again to American Cornball. They’re the first subject in Christopher Miller’s encyclopedia of comic topics. So I’ll carry on as if the Werner Heisenberg joke were the one meant.
Heisenberg is famous, outside World War II history, for the Uncertainty Principle. This is one of the core parts of quantum mechanics, under which there’s a limit to how precisely one can know both the position and momentum of a thing. To identify, with absolutely zero error, where something is requires losing all information about what its momentum might be, and vice-versa. You see the application of this to a traffic cop’s question about knowing how fast someone was going. This makes some neat mathematics because all the information about something is bundled up in a quantity called the Psi function. To make a measurement is to modify the Psi function by having an “operator” work on it. An operator is what we call a function that has domains and ranges of other functions. To measure both position and momentum is equivalent to working on Psi with one operator and then another. But these operators don’t commute. You get different results in measuring momentum and then position than you do measuring position and then momentum. And so we can’t know both of these with infinite precision.
There are pairs of operators that do commute. They’re not necessarily ones we care about, though. Like, the total energy commutes with the square of the angular momentum. So, you know, if you need to measure with infinite precision the energy and the angular momentum of something you can do it. If you had measuring tools that were perfect. You don’t, but you could imagine having them, and in that case, good. Underlying physics wouldn’t spoil your work.
Probably the panel was an absent-minded professor joke.
Today Gaurish, of For the love of Mathematics, gives me the last subject for my Summer 2017 A To Z sequence. And also my greatest challenge: the Zeta function. The subject comes to all pop mathematics blogs. It comes to all mathematics blogs. It’s not difficult to say something about a particular zeta function. But to say something at all original? Let’s watch.
The spring semester of my sophomore year I had Intro to Complex Analysis. Monday Wednesday 7:30; a rare evening class, one of the few times I’d eat dinner and then go to a lecture hall. There I discovered something strange and wonderful. Complex Analysis is a far easier topic than Real Analysis. Both are courses about why calculus works. But why calculus for complex-valued numbers works is a much easier problem than why calculus for real-valued numbers works. It’s dazzling. Part of this is that Complex Analysis, yes, builds on Real Analysis. So Complex can take for granted some things that Real has to prove. I didn’t mind. Given the way I crashed through Intro to Real Analysis I was glad for a subject that was, relatively, a breeze.
As we worked through Complex Variables and Applications so many things, so very many things, got to be easy. The basic unit of complex analysis, at least as we young majors learned it, was in contour integrals. These are integrals whose value depends on the values of a function on a closed loop. The loop is in the complex plane. The complex plane is, well, your ordinary plane. But we say the x-coordinate and the y-coordinate are parts of the same complex-valued number. The x-coordinate is the real-valued part. The y-coordinate is the imaginary-valued part. And we call that summation ‘z’. In complex-valued functions ‘z’ serves the role that ‘x’ does in normal mathematics.
So a closed loop is exactly what you think. Take a rubber band and twist it up and drop it on the table. That’s a closed loop. Suppose you want to integrate a function, ‘f(z)’. If you can always take its derivative on this loop and on the interior of that loop, then its contour integral is … zero. No matter what the function is. As long as it’s “analytic”, as the terminology has it. Yeah, we were all stunned into silence too. (Granted, mathematics classes are usually quiet, since it’s hard to get a good discussion going. Plus many of us were in post-dinner digestive lulls.)
Integrating regular old functions of real-valued numbers is this tedious process. There’s sooooo many rules and possibilities and special cases to consider. There’s sooooo many tricks that get you the integrals of some functions. And then here, with complex-valued integrals for analytic functions, you know the answer before you even look at the function.
As you might imagine, since this is only page 113 of a 341-page book there’s more to it. Most functions that anyone cares about aren’t analytic. At least they’re not analytic everywhere inside regions that might be interesting. There’s usually some points where an interesting function ‘f(z)’ is undefined. We call these “singularities”. Yes, like starships are always running into. Only we rarely get propelled into other universes or other times or turned into ghosts or stuff like that.
So much of the rest of the course turns into ways to avoid singularities. Sometimes you can spackle them over. This is when the function happens not to be defined somewhere, but you can see what it ought to be. Sometimes you have to do something more. This turns into a search for “removable” singularities. And this does something so brilliant it looks illicit. You modify your closed loop, so that it comes up very close, as close as possible, to the singularity, but studiously avoids it. Follow this game of I’m-not-touching-you right and you can turn your integral into two parts. One is the part that’s equal to zero. The other is the part that’s a constant times whatever the function is at the singularity you’re removing. And that ought to be easy to find the value for. (Being able to find a function’s value doesn’t mean you can find its derivative.)
Those tricks were hard to master. Not because they were hard. Because they were easy, in a context where we expected hard. But after that we got into how to move singularities. That is, how to do a change of variables that moved the singularities to where they’re more convenient for some reason. How could this be more convenient? Because of chapter five, “Series”. In regular old calculus we learn how to approximate well-behaved functions with polynomials. In complex-variable calculus, we learn the same thing all over again. They’re polynomials of complex-valued variables, but it’s the same sort of thing. And not just polynomials, but things that look like polynomials except they’re powers of instead. These open up new ways to approximate functions, and to remove singularities from functions.
And then we get into transformations. These are about turning a problem that’s hard into one that’s easy. Or at least different. They’re a change of variable, yes. But they also change what exactly the function is. This reshuffles the problem. Makes for a change in singularities. Could make ones that are easier to work with.
One of the useful, and so common, transforms is called the Laplace-Stieltjes Transform. (“Laplace” is said like you might guess. “Stieltjes” is said, or at least we were taught to say it, like “Stilton cheese” without the “ton”.) And it tends to create functions that look like a series, the sum of a bunch of terms. Infinitely many terms. Each of those terms looks like a number times another number raised to some constant times ‘z’. As the course came to its conclusion, we were all prepared to think about these infinite series. Where singularities might be. Which of them might be removable.
These functions, these results of the Laplace-Stieltjes Transform, we collectively call ‘zeta functions’. There are infinitely many of them. Some of them are relatively tame. Some of them are exotic. One of them is world-famous. Professor Walsh — I don’t mean to name-drop, but I discovered the syllabus for the course tucked in the back of my textbook and I’m delighted to rediscover it — talked about it.
That world-famous one is, of course, the Riemann Zeta function. Yes, that same Riemann who keeps turning up, over and over again. It looks simple enough. Almost tame. Take the counting numbers, 1, 2, 3, and so on. Take your ‘z’. Raise each of the counting numbers to that ‘z’. Take the reciprocals of all those numbers. Add them up. What do you get?
A mass of fascinating results, for one. Functions you wouldn’t expect are concealed in there. There’s strips where the real part is zero. There’s strips where the imaginary part is zero. There’s points where both the real and imaginary parts are zero. We know infinitely many of them. If ‘z’ is -2, for example, the sum is zero. Also if ‘z’ is -4. -6. -8. And so on. These are easy to show, and so are dubbed ‘trivial’ zeroes. To say some are ‘trivial’ is to say that there are others that are not trivial. Where are they?
Professor Walsh explained. We know of many of them. The nontrivial zeroes we know of all share something in common. They have a real part that’s equal to 1/2. There’s a zero that’s at about the number . Also at . There’s one at about . Also about . (There’s a symmetry, you maybe guessed.) Every nontrivial zero we’ve found has a real component that’s got the same real-valued part. But we don’t know that they all do. Nobody does. It is the Riemann Hypothesis, the great unsolved problem of mathematics. Much more important than that Fermat’s Last Theorem, which back then was still merely a conjecture.
What a prospect! What a promise! What a way to set us up for the final exam in a couple of weeks.
I had an inspiration, a kind of scheme of showing that a nontrivial zero couldn’t be within a given circular contour. Make the size of this circle grow. Move its center farther away from the z-coordinate to match. Show there’s still no nontrivial zeroes inside. And therefore, logically, since I would have shown nontrivial zeroes couldn’t be anywhere but on this special line, and we know nontrivial zeroes exist … I leapt enthusiastically into this project. A little less enthusiastically the next day. Less so the day after. And on. After maybe a week I went a day without working on it. But came back, now and then, prodding at my brilliant would-be proof.
The Riemann Zeta function was not on the final exam, which I’ve discovered was also tucked into the back of my textbook. It asked more things like finding all the singular points and classifying what kinds of singularities they were for functions like instead. If the syllabus is accurate, we got as far as page 218. And I’m surprised to see the professor put his e-mail address on the syllabus. It was merely “bwalsh@math”, but understand, the Internet was a smaller place back then.
I finished the course with an A-, but without answering any of the great unsolved problems of mathematics.
The back half of last week’s mathematically themed comic strips aren’t all that deep. They make up for it by being numerous. This is how calculus works, so, good job, Comic Strip Master Command. Here’s what I have for you.
Mark Anderson’s Andertoons for the 20th marks its long-awaited return to these Reading The Comics posts. It’s of the traditional form of the student misunderstanding the teacher’s explanations. Arithmetic edition.
Marty Links’s Emmy Lou for the 20th was a rerun from the 22nd of September, 1976. It’s just a name-drop. It’s not like it matters for the joke which textbook was lost. I just include it because, what the heck, might as well.
Jef Mallett’s Frazz for the 21st uses the form of a story problem. It’s a trick question anyway; there’s really no way the Doppler effect is going to make an ice cream truck’s song unrecognizable, not even at highway speeds. Too distant to hear, that’s a possibility. Also I don’t know how strictly regional this is but the ice cream trucks around here have gone in for interrupting the music every couple seconds with some comical sound effect, like a “boing” or something. I don’t know what this hopes to achieve besides altering the timeline of when the ice cream seller goes mad.
Mark Litzler’s Joe Vanilla for the 21st I already snuck in here last week, in talking about ‘x’. The variable does seem like a good starting point. And, yeah, hypothesis block is kind of a thing. There’s nothing quite like staring at a problem that should be interesting and having no idea where to start. This happens even beyond grade school and the story problems you do then. What to do about it? There’s never one thing. Study it a good while, read about related problems a while. Maybe work on something that seems less obscure a while. It’s very much like writer’s block.
Ryan North’s Dinosaur Comics rerun for the 22nd straddles the borders between mathematics, economics, and psychology. It’s a problem about making forecasts about other people’s behavior. It’s a mystery of game theory. I don’t know a proper analysis for this game. I expect it depends on how many rounds you get to play: if you have a sense of what people typically do, you can make a good guess of what they will do. If everyone gets a single shot to play, all kinds of crazy things might happen.
Jef Mallet’s Frazz gets in again on the 22nd with some mathematics gibberish-talk, including some tossing around of the commutative property. Among other mistakes Caulfield was making here, going from “less is more to therefore more is less” isn’t commutation. Commutation is about binary operations, where you match a pair of things to a single thing. The operation commutes if it never matters what the order of the pair of things is. It doesn’t commute if it ever matters, even a single time, what the order is. Commutativity gets introduced in arithmetic where there are some good examples of the thing. Addition and multiplication commute. Subtraction and division don’t. From there it gets forgotten until maybe eventually it turns up in matrix multiplication, which doesn’t commute. And then it gets forgotten once more until maybe group theory. There, whether operations commute or not is as important a divide as the one between vertebrates and invertebrates. But I understand kids not getting why they should care about commuting. Early on it seems like a longwinded way to say what’s obvious about addition.
Bud Blake’s Tiger rerun for the 23rd starts with a real-world example of your classic story problem. I like the joke in it, and I also like Hugo’s look of betrayal and anger in the second panel. A spot of expressive art will do so good for a joke.