It feels to me like I did a lot of functional analysis terms in the Leap Day 2016 series. Its essay for the letter ‘S’, Surjective Map, is one of them. We have many ways of dividing up the kinds of functions we have. One of them is in how they use their range. A function has a set called the domain, and a set called the range, and they might be the same set, yes. The function pairs things in the domain with things in the range. Everything in the domain has to pair with something in the range. But we allow having things in the range that aren’t paired to anything in the domain. So we have jargon that tells us, quickly, whether there are unmatched pieces in the range.
Sometimes I write an essay and know it’s something I’m going to refer back to a lot. Sometimes I know it’s just going to sink without trace. Often these deserve it; the subject is something particular and not well-connected to other topics. Sometimes, one sinks without a trace and for not much good reason. Smooth is one of those curiously sunk pieces. It’s about a concept important to analysis. And also a piece that shows my obsession with pointing out cultural factors in mathematics: we care about ‘smooth’ because we’ve found it a useful thing to highlight. And yet it’s gotten no comments, only an average number of likes, and I don’t seem to have linked back to it in any essays where it might be useful. I may have forgotten I wrote the thing. So here’s a referral that maybe will help me remember I have it on hand, ready for future use.
I owe Mr Wu, author of the Singapore Maths Tuition blog, thanks for another topic for this A-to-Z. Statistics is a big field of mathematics, and so I won’t try to give you a course’s worth in 1500 words. But I have to start with a question. I seem to have ended at two thousand words.
Is statistics mathematics?
The answer seems obvious at first. Look at a statistics textbook. It’s full of algebra. And graphs of great sloped mounds. There’s tables full of four-digit numbers in back. The first couple chapters are about probability. They’re full of questions about rolling dice and dealing cards and guessing whether the sibling who just entered is the younger.
But then, why does Rutgers University have a Department of Mathematics and also a Department of Statistics? And considered so distinct as to have an interdisciplinary mathematics-and-statistics track? It’s not an idiosyncrasy of Rutgers. Many schools have the same division between mathematics and statistics. Some join them into a Department of Mathematics and Statistics. But the name hints at something just different about the field. Not too different, though. Physics and Chemistry and important threads of Economics and History are full of mathematics. But you never see a Department of Mathematics and History.
Thinking of the field’s history, though, and its use, tell us more. Some of the earliest work we now recognize as statistics was Arab mathematicians deciphering messages. This cryptanalysis is the observation that (in English) a three-letter word is very likely to be ‘the’, mildly likely to be ‘one’, and not likely to be ‘pyx’. A more modern forerunner is the Republic of Venice supposedly calculating that war with Milan would not be worth the winning. Or the gatherings of mortality tables, recording how many people of what age can be expected to die any year, and what from. (Mortality tables are another of Edmond Halley’s claims to fame, though it won’t displace his comet work.) Florence Nightingale’s charts explaining how more soldiers die of disease than in fighting the Crimean War. William Sealy Gosset sharing sample-testing methods developed at the Guinness brewery.
You see a difference in kind to a mathematical question like finding a square with the same area as this trapezoid. It’s not that mathematics is not practical; it’s always been. And it’s not that statistics lacks abstraction and pure mathematics content. But statistics wears practicality in a way that number theory won’t.
Practical about what? History and etymology tip us off. The early uses of things we now see as statistics are about things of interest to the State. Decoding messages. Counting the population. Following — in the study of annuities — the flow of money between peoples. With the industrial revolution, statistics sneaks into the factory. To have an economy of scale you need a reliable product. How do you know whether the product is reliable, without testing every piece? How can you test every beer brewed without drinking it all?
One great leg of statistics — it’s tempting to call it the first leg, but the history is not so neat as to make that work — is descriptive. This gives us things like mean and median and mode and standard deviation and quartiles and quintiles. These try to let us represent more data than we can really understand in a few words. We lose information in doing so. But if we are careful to remember the difference between the descriptive statistics we have and the original population? (nb, a word of the State) We might not do ourselves much harm.
Another great leg is inferential statistics. This uses tools with names like z-score and the Student t distribution. And talk about things like p-values and confidence intervals. Terms like correlation and regression and . This is about looking for causes in complex scenarios. We want to believe there is a cause to, say, a person’s lung cancer. But there is no tracking down what that is; there are too many things that could start a cancer, and too many of them will go unobserved. But we can notice that people who smoke have lung cancer more often than those who don’t. We can’t say why a person recovered from the influenza in five days. But we can say people who were vaccinated got fewer influenzas, and ones that passed quicker, than those who did not. We can get the dire warning that “correlation is not causation”, uttered by people who don’t like what the correlation suggests may be a cause.
Also by people being honest, though. In the 1980s geologists wondered if the sun might have a not-yet-noticed companion star. Its orbit would explain an apparent periodicity in meteor bombardments of the Earth. But completely random bombardments would produce apparent periodicity sometimes. It’s much the same way trees in a forest will sometimes seem to line up. Or imagine finding there is a neighborhood in your city with a high number of arrests. Is this because it has the highest rate of street crime? Or is the rate of street crime the same as any other spot and there are simply more cops here? But then why are there more cops to be found here? Perhaps they’re attracted by the neighborhood’s reputation for high crime. It is difficult to see through randomness, to untangle complex causes, and to root out biases.
The tools of statistics, as we recognize them, largely came together in the 19th and early 20th century. Adolphe Quetelet, a Flemish scientist, set out much early work, including introducing the concept of the “average man”. He studied the crime statistics of Paris for five years and noticed how regular the numbers were. The implication, to Quetelet — who introduced the idea of the “average man”, representative of societal matters — was that crime is a societal problem. It’s something we can control by mindfully organizing society, without infringing anyone’s autonomy. Put like that, the study of statistics seems an obvious and indisputable good, a way for governments to better serve their public.
So here is the dispute. It’s something mathematicians understate when sharing the stories of important pioneers like Francis Galton or Karl Pearson. They were eugenicists. Part of what drove their interest in studying human populations was to find out which populations were the best. And how to help them overcome their more-populous lessers.
I don’t have the space, or depth of knowledge, to fully recount the 19th century’s racial politics, popular scientific understanding, and international relations. Please accept this as a loose cartoon of the situation. Do not forget the full story is more complex and more ambiguous than I write.
One of the 19th century’s greatest scientific discoveries was evolution. That populations change in time, in size and in characteristics, even budding off new species, is breathtaking. Another of the great discoveries was entropy. This incorporated into science the nostalgic romantic notion that things used to be better. I write that figuratively, but to express the way the notion is felt.
There are implications. If the Sun itself will someday wear out, how long can the Tories last? It was easy for the aristocracy to feel that everything was quite excellent as it was now and dread the inevitable change. This is true for the aristocracy of any country, although the United Kingdom had a special position here. The United Kingdom enjoyed a privileged position among the Great Powers and the Imperial Powers through the 19th century. Note we still call it the Victorian era, when Louis Napoleon or Giuseppe Garibaldi or Otto von Bismarck are more significant European figures. (Granting Victoria had the longer presence on the world stage; “the 19th century” had a longer presence still.) But it could rarely feel secure, always aware that France or Germany or Russia was ready to displace it.
And even internally: if Darwin was right and reproductive success all that matters in the long run, what does it say that so many poor people breed so much? How long could the world hold good things? Would the eternal famines and poverty of the “overpopulated” Irish or Indian colonial populations become all that was left? During the Crimean War, the British military found a shocking number of recruits from the cities were physically unfit for service. In the 1850s this was only an inconvenience; there were plenty of strong young farm workers to recruit. But the British population was already majority-urban, and becoming more so. What would happen by 1880? 1910?
One can follow the reasoning, even if we freeze at the racist conclusions. And we have the advantage of a century-plus hindsight. We can see how the eugenic attitude leads quickly to horrors. And also that it turns out “overpopulated” Ireland and India stopped having famines once they evicted their colonizers.
Does this origin of statistics matter? The utility of a hammer does not depend on the moral standing of its maker. The Central Limit Theorem has an even stronger pretense to objectivity. Why not build as best we can with the crooked timbers of mathematics?
It is in my lifetime that a popular racist book claimed science proved that Black people were intellectual inferiors to White people. This on the basis of supposedly significant differences in the populations’ IQ scores. It proposed that racism wasn’t a thing, or at least nothing to do anything about. It would be mere “realism”. Intelligence Quotients, incidentally, are another idea we can trace to Francis Galton. But an IQ test is not objective. The best we can say is it might be standardized. This says nothing about the biases built into the test, though, or of the people evaluating the results.
So what if some publisher 25 years ago got suckered into publishing a bad book? And racist chumps bought it because they liked its conclusion?
The past is never fully past. In the modern environment of surveillance capitalism we have abundant data on any person. We have abundant computing power. We can find many correlations. This gives people wild ideas for “artificial intelligence”. Something to make predictions. Who will lose a job soon? Who will get sick, and from what? Who will commit a crime? Who will fail their A-levels? At least, who is most likely to?
These seem like answerable questions. One can imagine an algorithm that would answer them fairly. And make for a better world, one which concentrates support around the people most likely to need it. If we were wise, we would ask our friends in the philosophy department about how to do this. Or we might just plunge ahead and trust that since an algorithm runs automatically it must be fair. Our friends in the philosophy department might have some advice there too.
Consider, for example, the body mass index. It was developed by our friend Adolphe Quetelet, as he tried to understand the kinds of bodies in the population. It is now used to judge whether someone is overweight. Weight is treated as though it were a greater threat to health than actual illnesses are. Your diagnosis for the same condition with the same symptoms will be different — and on average worse — if your number says 25.2 rather than 24.8.
We must do better. We can hope that learning how tools were used to injure people will teach us to use them better, to reduce or to avoid harm. We must fight our tendency to latch on to simple ideas as the things we can understand in the world. We must not mistake the greater understanding we have from the statistics for complete understanding. To do this we must have empathy, and we must have humility, and we must understand what we have done badly in the past. We must catch ourselves when we repeat the patterns that brought us to past evils. We must do more than only calculate.
This and the rest of the 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be gathered at this link. And I am looking for V, W, and X topics to write about. Thanks for your thoughts, and thank you for reading.
Bernard Riemann is one of those figures you can’t be a mathematics major without learning about. His name attaches to an enormous amount of analysis. One Riemann-named thing every mathematician learns very well is the Riemann Sum. It’s the first analysis model we use to explain why integration works. And we can put together a version of this for numerical integration. Its greatest use, though, is that we can use it to justify other ways to integrate that are easier to actually use. Great little utility.
Part of why I write these essays is to save future time. If I have an essay explaining some complex idea, then in the future, I can use a link and a short recap of the central idea. There’s some essays that have been perennials. I think I’ve linked to polynomials more than anything else on this site. And then some disappear, even though they seem to be about good useful subjects. Riemann sphere, from the Leap Day 2016 sequence, is one of those disappeared topics. This is one of the ways to convert between “shapes on the plane” and “shapes on the sphere”. There’s no way to perfectly move something from the plane to the sphere, or vice-versa. The Riemann Sphere is an approach which preserves the interior angles. If two lines on the plane intersect at a 25 degree angle, their representation on the sphere will intersect at a 25 degree angle. But everything else may get strange.
I have again Elke Stangl, author of elkemental Force, to thank for the subject this week. Again, Stangl’s is a blog of wide-ranging theme interests. And it’s got more poetry this week again, this time haikus about the Dirac delta function.
I also have Kerson Huang, of the Massachusetts Institute of Technology and of Nanyang Technological University, to thank for much insight into the week’s subject. Huang published this A Critical History of Renormalization, which gave me much to think about. It’s likely a paper that would help anyone hoping to know the history of the technique better.
There is a mathematical model, the Ising Model, for how magnets work. The model has the simplicity of a toy model given by a professor (Wilhelm Lenz) to his grad student (Ernst Ising). Suppose matter is a uniform, uniformly-spaced grid. At each point on the grid we have either a bit of magnetism pointed up (value +1) or down (value -1). It is a nearest-neighbor model. Each point interacts with its nearest neighbors and none of the other points. For a one-dimensional grid this is easy. It’s the stuff of thermodynamics homework for physics majors. They don’t understand it, because you need the hyperbolic trigonometric functions. But they could. For two dimensions … it’s hard. But doable. And interesting. It describes important things like phase changes. The way that you can take a perfectly good strong magnet and heat it up until it’s an iron goo, then cool it down to being a strong magnet again.
For such a simple model it works well. A lot of the solids we find interesting are crystals, or are almost crystals. These are molecules arranged in a grid. So that part of the model is fine. They do interact, foremost, with their nearest neighbors. But not exclusively. In principle, every molecule in a crystal interacts with every other molecule. Can we account for this? Can we make a better model?
Yes, many ways. Here’s one. It’s designed for a square grid, the kind you get by looking at the intersections on a normal piece of graph paper. Each point is in a row and a column. The rows are a distance ‘a’ apart. So are the columns.
Now draw a new grid, on top of the old. Do it by grouping together two-by-two blocks of the original. Draw new rows and columns through the centers of these new blocks. Put at the new intersections a bit of magnetism. Its value is the mean of whatever the four blocks around it are. So, could be 1, could be -1, could be 0, could be ½, could be -½. There’s more options. But look at what we have. It’s still an Ising-like model, with interactions between nearest-neighbors. There’s more choices for what value each point can have. And the grid spacing is now 2a instead of a. But it all looks pretty similar.
And now the great insight, that we can trace to Leo P Kadanoff in 1966. What if we relabel the distance between grid points? We called it 2a before. Call it a, now, again. What’s important that’s different from the Ising model we started with?
There’s the not-negligible point that there’s five different values a point can have, instead of two. But otherwise? In the operations we do, not much is different. How about in what it models? And there it’s interesting. Think of the original grid points. In the original scaling, they interacted only with units one original-row or one original-column away. Now? Their average interacts with the average of grid points that were as far as three original-rows or three original-columns away. It’s a small change. But it’s closer to reflecting the reality of every molecule interacting with every other molecule.
You know what happens when mathematicians get one good trick. We figure what happens if we do it again. Take the rescaled grid, the one that represents two-by-two blocks of the original. Rescale it again, making two-by-two blocks of these two-by-two blocks. Do the same rules about setting the center points as a new grid. And then re-scaling. What we have now are blocks that represent averages of four-by-four blocks of the original. And that, imperfectly, let a point interact with a point seven original-rows or original-columns away. (Or farther: seven original-rows down and three original-columns to the left, say. Have fun counting all the distances.) And again: we have eight-by-eight blocks and even more range. Again: sixteen-by-sixteen blocks and double the range again. Why not carry this on forever?
This is renormalization. It’s a specific sort, called the block-spin renormalization group. It comes from condensed matter physics, where we try to understand how molecules come together to form bulks of matter. Kenneth Wilson stretched this over to studying the Kondo Effect. This is a problem in how magnetic impurities affect electrical resistance. (It’s named for Jun Kondo.) It’s great work. It (in part) earned Wilson a Nobel Prize. But the idea is simple. We can understand complex interactions by making them simple ones. The interactions have a natural scale, cutting off at the nearest neighbor. But we redefine ‘nearest neighbor’, again and again, until it reaches infinitely far away.
This problem, and its solution, come from thermodynamics. Particularly, statistical mechanics. This is a bit ahistoric. Physicists first used renormalization in quantum mechanics. This is all right. As a general guideline, everything in statistical mechanics turns into something in quantum mechanics, and vice-versa. What quantum mechanics lacked, for a generation, was logical rigor for renormalization. This statistical mechanics approach provided that.
Renormalization in quantum mechanics we needed because of virtual particles. Quantum mechanics requires that particles can pop into existence, carrying momentum, and then pop back out again. This gives us electromagnetism, and the strong nuclear force (which holds particles together), and the weak nuclear force (which causes nuclear decay). Leave gravity over on the side. The more momentum in the virtual particle, the shorter a time it can exist. It’s actually the more energy, the shorter the particle lasts. In that guise you know it as the Uncertainty Principle. But it’s momentum that’s important here. This means short-range interactions transfer more momentum, and long-range ones transfer less. And here we had thought forces got stronger as the particles interacting got closer together.
In principle, there is no upper limit to how much momentum one of these virtual particles can have. And, worse, the original particle can interact with its virtual particle. This by exchanging another virtual particle. Which is even higher-energy and shorter-range. The virtual particle can also interact with the field that’s around the original particle. Pairs of virtual particles can exchange more virtual particles. And so on. What we get, when we add this all together, seems like it should be infinitely large. Every particle the center of an infinitely great bundle of energy.
Renormalization, the original renormalization, cuts that off. Sets an effective limit on the system. The limit is not “only particles this close will interact” exactly. It’s more “only virtual particles with less than this momentum will”. (Yes, there’s some overlap between these ideas.) This seems different to us mere dwellers in reality. But to a mathematical physicist, knowing that position and momentum are conjugate variables? Limiting one is the same work as limiting the other.
This, when developed, left physicists uneasy. It’s for good reasons. The cutoff is arbitrary. Its existence, sure, but we often deal with arbitrary cutoffs for things. When we calculate a weather satellite’s orbit we do not care that other star systems exist. We barely care that Jupiter exists. Still, where to put the cutoff? Quantum Electrodynamics, using this, could provide excellent predictions of physical properties. But shouldn’t we get different predictions with different cutoffs? How do we know we’re not picking a cutoff because it makes our test problem work right? That we’re not picking one that produces garbage for every other problem? Read the writing of a physicist of the time and — oh, why be coy? We all read Richard Feynman, his QED at least. We see him sulking about a technique he used to brilliant effect.
Wilson-style renormalization answered Feynman’s objections. (Though not to Feynman’s satisfaction, if I understand the history right.) The momentum cutoff serves as a scale. Or if you prefer, the scale of interactions we consider tells us the cutoff. Different scales give us different quantum mechanics. One scale, one cutoff, gives us the way molecules interact together, on the scale of condensed-matter physics. A different scale, with a different cutoff, describes the particles of Quantum Electrodynamics. Other scales describe something more recognizable as classical physics. Or the Yang-Mills gauge theory, as describes the Standard Model of subatomic particles, all those quarks and leptons.
Renormalization offers a capsule of much of mathematical physics, though. It started as an arbitrary trick to avoid calculation problems. In time, we found a rationale for the trick. But found it from looking at a problem that seemed unrelated. On learning the related trick well, though, we see they’re different aspects of the same problem. It’s a neat bit of work.
This and all the other 2020 A-to-Z essays should be at this link. Essays from every A-to-Z series should be gathered at this link. I am looking eagerly for topics for the letters S, T, and U, and am scouting ahead for V, W, and X topics also. Thanks for your thoughts, and thank you for reading.
You know what? I should probably get as much of November done ahead of schedule as possible. So to that end, I’ll also open up the next three letters of the alphabet. If you’d like me to try explaining a mathematics term that starts with V, W, or X, please leave a comment saying so. Also please let me know what your home blog, YouTube channel, Twitter feed, or whatnot is, so I can give that some attention too. I’m also really eager to find other X words; this is a difficult part of the alphabet. And, I’m open to considering re-doing past essay topics, if I have some new angle on them. Don’t be unreasonably afraid to ask.
Topics I’ve already covered, starting with the letter ‘V’, are:
- Vertex (graph theory) (2015)
- Vector (Leap Day 2016)
- Voronoi Diagram (End 2016)
- Volume Forms (2017)
- Volume (2018)
- Versine (2019)
Topics I’ve already covered, starting with the letter ‘W’, are:
- Well-Posed Problem (2015)
- Wlog (Leap Day 2016)
- Weierstrass Function (End 2016)
- Well-Ordering Principle (2017)
- Witch of Agnesi (2018)
- Wallis Products (2019)
Topics I’ve already covered, starting with the letter ‘X’, are:
- Xor (2015)
- X-Intercept (Leap Day 2016)
- Xi Function (End 2016)
- X (2017)
- Extreme Value Theorem (2018)
- Chi-Squared Test (2019)
Thanks for reading. The All 2020 A-to-Z essays should all be at this link. And all essays from all of my A-to-Z sequences should be at this link.
I feel like I talk group theory a lot in these A-to-Z sequences. Some of that’s deserved. Group theory underlies a lot of modern mathematics. Part of it is surely that it made the deepest impression on me, as a mathematics major, even though my work ended up not touching groups often. Quotient Groups are at that nice intersection of being important yet having a misleading name. You’re introduced to them after learning about groups, which have an operation that works like addition/subtraction; and then rings, which have addition/subtraction plus multiplication. Surely a quotient group is just a ring with division, right? No, it is not. But, lucky thing, there’s one quotient group you certainly know and feel familiar with. You’ll see.
In summer 2015 I picked all the topics for my A-to-Z; I didn’t work up the courage to ask for topics until the next time around. Some, I remember why I chose. I’m not sure why I picked Quintile, as a statistics term, rather than quartile. Both are legitimate terms, and circle around a similar idea. That is that we need to know how data is distributed: what range of numbers are common, what ones are rare. I wonder if I wasn’t saving ‘quartile’ for some later A-to-Z, for fear of running out of Q terms. Or if I felt that quartiles were familiar enough that quintiles would seem a touch strange. That is the sort of thing I’d likely do.
I continue my tradition of doing these monthly readership reviews just a little too far into the month to feel sensible. Well, I’m trying to publish more things on the weekdays and have three of those five committed, while the A-to-Z goes on.
In September I posted only 18 pieces. That’s all right. There was more to them: 15,922 words posted in total. This comes to an average of 936.6 words per posting, way up from August’s 634.3. It’s my most wordy month this year, so far. My year-to-date average post has been 694 words, around here.
Those 18, on average enormous, posts drew 2,422 page views. I like seeing that sort of number, since it’s above the twelve-month running average of 2,383.3 page views. There were 1,643 unique visitors, again above the twelve-month running average of 1,622.8. And I’m really amazed by that since the twelve-month running average includes that fluke last October where something like five thousand more people than usual came in and looked at my post about linear programming.
It was an engaged month, too. There were 80 things liked in September, above the average of 62.3. And 32 comments, beating the 17.4 average.
The per-posting figures were similarly above the twelve-month running averages. 134.6 views per posting, above the 125.3 running average. 91.3 unique visitors per posting, above the 85.0 running average. 4.4 likes per posting, compared to a 3.3 running average. 1.8 comments per posting, compared to a 1.0 running average. I’m going to be felling good about this month until that happens again.
I wanted to look at the most popular posts from August and September around here. August because, you know, there’s stuff posted the last week of the month that gets readers early in the new month. It doesn’t seem fair to rule them out as popular posts just because the kalends work against them. Turns out nothing from late August was among the most popular stuff. There was a tie for fifth place, though, as sometimes happens. So here’s the six most popular posts of September:
- Playful Math Education Blog Carnival 141
- My All 2020 Mathematics A to Z: John von Neumann
- So I’m hosting the 141th Playful Math Education Blog Carnival
- My All 2020 Mathematics A to Z: Gottfried Wilhelm Leibniz
- My All 2020 Mathematics A to Z: Big-O and Little-O Notation
- How August 2020 Saw People Finding Non-Comics Things Here
I always feel strange when the monthly readership post is one of the most popular things here. It implies I should do more of just writing up past glories.
October started with me having 1,535 posts here. They have collected 113,769 views, from 63,868 logged unique visitors.
Each Wednesday, I hope to publish an A-to-Z essay. You can see that, and all this year’s essays, at this link. This year’s and all past A-to-Z essays should be at this link. And I am open for topics starting S, T, or U, if you’d like to see me explain something.
You can be a regular reader here by clicking the “Follow Nebusresearch” button on the page. Or add my essays’ feed to whatever your RSS reader is. If you don’t have an RSS reader, get a free account at Dreamwidth or Livejournal. You can add any RSS feed to your friends page from https://www.dreamwidth.org/feeds/ or https://www.livejournal.com/syn as you like.
My essays are announcedon Twitter as @nebusj. My twitter is nearly abandoned, though. Only sometimes does Safari let it load. If you actually want to social-media talk with me look to the mathematics-themed Mathstodon and my account @email@example.com. It’s low-volume over there, but it’s pleasant. If you really need me, well, leave a comment. I try to get back to those soon enough. Thank you for reading.
I’m happy to have a subject from Elke Stangl, author of elkemental Force. That’s a fun and wide-ranging blog which, among other things, just published a poem about proofs. You might enjoy.
One delight, and sometimes deadline frustration, of these essays is discovering things I had not thought about. Researching quadratic forms invited the obvious question of what is a form? And that goes undefined on, for example, Mathworld. Also in the textbooks I’ve kept. Even ones you’d think would mention, like R W R Darling’s Differential Forms and Connections, or Frigyes Riesz and Béla Sz-Nagy’s Functional Analysis. Reluctantly I started thinking about what we talk about when discussing forms.
Quadratic forms offer some hints. These take a vector in some n-dimensional space, and return a scalar. Linear forms, and cubic forms, do the same. The pattern suggests a form is a mapping from a space like to or maybe to . That looks good, but then we have to ask: isn’t that just an operator? Also: then what about differential forms? Or volume forms? These are about how to fill space. There’s nothing scalar in that. But maybe these are both called forms because they fill similar roles. They might have as little to do with one another as red pandas and giant pandas do.
Enlightenment comes after much consideration or happening on Wikipedia’s page about homogenous polynomials. That offers “an algebraic form, or simply form, is a function defined by a homogeneous polynomial”. That satisfies. First, because it gets us back to polynomials. Second, because all the forms I could think of do have rules based in homogeneous polynomials. They might be peculiar polynomials. Volume forms, for example, have a polynomial in wedge products of differentials. But it counts.
A function’s homogenous if it scales a particular way. Evaluate it at some set of coordinates x, y, z, (more variables if you need). That’s some number (let’s say). Take all those coordinates and multiply them by the same constant; let me call that α. Evaluate the function at α x, α y α z, (α times more variables if you need). Then that value is αk times the original value of f. k is some constant. It depends on the function, but not on what x, y, z, (more) are.
For a quadratic form, this constant k equals 4. This is because in the quadratic form, all the terms in the polynomial are of the second degree. So, for example, is a quadratic form. So is ; the x times the y brings this to a second degree. Also a quadratic form is . So is .
This can have many variables. If we have a lot, we have a couple choices. One is to start using subscripts, and to write the form something like:
This is respectable enough. People who do a lot of differential geometry get used to a shortcut, the Einstein Summation Convention. In that, we take as implicit the summation instructions. So they’d write the more compact . Those of us who don’t do a lot of differential geometry think that looks funny. And we have more familiar ways to write things down. Like, we can put the collection of variables into an ordered n-tuple. Call it the vector . If we then think to put the numbers into a square matrix we have a great way of writing things. We have to manipulate the a little to make the matrix, but it’s nothing complicated. Once that’s done we can write the quadratic form as:
This uses matrix multiplication. The vector we assume is a column vector, a bunch of rows one column across. Then we have to take its transposition, one row a bunch of columns across, to make the matrix multiplication work out. If we don’t like that notation with its annoying superscripts? We can declare the bare ‘x’ to mean the vector, and use inner products:
This is easier to type at least. But what does it get us?
Looking at some quadratic forms may give us an idea. practically begs to be matched to an , and the name “the equation of a circle”. is less familiar, but to the crowd reading this, not much less familiar. Fill that out to and we have a hyperbola. If we have and let that then we have an ellipse, something a bit wider than it is tall. Similarly is a hyperbola still, just anamorphic.
If we expand into three variables we start to see spheres: just begs to equal . Or ellipsoids: , set equal to some (positive) , is something we might get from rolling out clay. Or hyperboloids: or , set equal to , give us nice shapes. (We can also get cylinders: equalling some positive number describes a tube.)
How about ? This also wants to be an ellipse. , to pick an easy number, is a rotated ellipse. The long axis is along the line described by . The short axis is along the line described by . How about — let me make this easy. ? The equation describes a hyperbola, but a rotated one, with the x- and y-axes as its asymptotes.
Do you want to take any guesses about three-dimensional shapes? Like, what might represent? If you’re thinking “ellipsoid, only it’s at an angle” you’re doing well. It runs really long in one direction, along the plane described by . It runs medium-size along the plane described by . It runs pretty short along the z-axis. We could run some more complicated shapes. Ellipses pointing in weird directions. Hyperboloids of different shapes. They’ll have things in common.
One is that they have obviously important axes. Axes of symmetry, particularly. There’ll be one for each dimension of space. An ellipse has a long axis and a short axis. An ellipsoid has a long, a middle, and a short. (It might be that two of these have the same length. If all three have the same length, you have a sphere, my friend.) A hyperbola, similarly, has two axes of symmetry. One of them is the midpoint between the two branches of the hyperbola. One of them slices through the two branches, through the points where the two legs come closest together. Hyperboloids, in three dimensions, have three axes of symmetry. One of them connects the points where the two branches of hyperboloid come closest together. The other two run perpendicular to that.
We can go on imagining more dimensions of space. We don’t need them. The important things are already there. There are, for these shapes, some preferred directions. The ones around which these quadratic-form shapes have symmetries. These directions are perpendicular to each other. These preferred directions are important. We call them “eigenvectors”, a partly-German name.
Eigenvectors are great for a bunch of purposes. One is that if the matrix A represents a problem you’re interested in? The eigenvectors are probably a great basis to solve problems in it. This is a change of basis vectors, which is the same work as doing a rotation. And it’s happy to report this change of coordinates doesn’t mess up the problem any. We can rewrite the problem to be easier.
And, roughly, any time we look at reflections in a Euclidean space, there’s a quadratic form lurking around. This leads us into interesting places. Looking at reflections encourages us to see abstract algebra, to see groups. That space can be rotated in infinitesimally small pieces gets us a kind of group named a Lie (pronounced ‘lee’) Algebra. Quadratic forms give us a way of classifying those.
Quadratic forms work in number theory also. There’s a neat theorem, the 15 Theorem. If a quadratic form, with integer coefficients, can produce all the integers from 1 through 15, then it can produce all positive numbers. For example, can, for sets of integers x, y, z, and w, add up to any positive number you like. (It’s not guaranteed this will happen. can’t produce 15.) We know of at least 54 combinations which generate all the positive integers, like and and such.
There’s more, of course. There always is. I spent time skimming Quadratic Forms and their Applications, Proceedings of the Conference on Quadratic Forms and their Applications. It was held at University College Dublin in July of 1999. It’s some impressive work. I can think of very little that I can describe. Even Winfried Scharlau’s On the History of the Algebraic Theory of Quadratic Forms, from page 229, is tough going. Ina Kersten’s Biography of Ernst Witt, one of the major influences on quadratic forms, is accessible. I’m not sure how much of the particular work communicates.
It’s easy at least to know what things this field is about, though. The things that we calculate. That they connect to novel and abstract places shows how close together arithmetic and dynamical systems and topology and group theory and number theory are, despite appearances.
Thanks for reading this. Today’s and all the other 2020 A-to-Z essays should be at this link. Both the All-2020 and past A-to-Z essays should be at this link. And I am looking for letter S, T, and U topics for the coming weeks. I’m grateful for your thoughts.
I am as surprised as anyone to be this near the end of the All 2020 A-to-Z. But, also, I am hoping to stockpile a couple of essays for the first weeks of November. I expect that to be an even more emotionally trying time and would like to have as little work, even fun work like this, as possible then.
So please, in comments, suggest mathematical terms starting with the letters S, T, or U, or that can be reasonably phrased as something with those letters. Also please list any blogs, YouTube channels, books, anything that you’ve written or would like to see publicized.
I’m probably going to put out an appeal for the letter V soon, also, since that’s also scheduled for an early-November publication.
I am open to revisiting topics I looked at in the past, if I think I can do better, or can cover a different aspect of them. So for reference, the topics I’ve already covered starting with the letter ‘S’ were:
- Step (2015)
- Surjective Map (Leap Day 2016)
- Smooth (End 2016)
- Sárközy’s Theorem (2017)
- Sorites Paradox (2018)
- Sample Space (2019)
Topics I’ve already covered, starting with the letter ‘T’, are:
- Tensor (2015)
- Transcendental Number (Leap Day 2016)
- Tree (End 2016)
- Topology (2017)
- Tiling (2018)
- Taylor Series (2019)
Topics I’ve already covered, starting with the letter ‘U’, are:
- Unbounded (2015)
- Uncountable (Leap Day 2016)
- Unlink (End 2016)
- Ulam’s Spiral (2017)
- Unit Fraction (2018)
- Unitizing (2019)
Thanks for reading, and for all your thoughts.
And in last year’s A-to-Z I published one of those essays already becoming a favorite. I haven’t had much chance to link back to it. So let me fix that. My 2019 Mathematics A To Z: Platonic focuses on the Platonic Solids, and questions like why we might find them interesting. Also, what Platonic solids look like in spaces of other than three dimensions. Three-dimensional space has five Platonic solids. There are six Platonic Solids in four dimensions. How many would you expect in a five-dimensional space? Or a ten-dimensional one? The answer may surprise you!
As I did the 2015 A-to-Z I learned how to do them in a way that feels me. In writing about the meaning of Proper, I found an important part of my voice. That’s the part which began with a corny mathematician’s joke. It also shows something I have forgotten how to do: it explains the whole thing, even with a joke to warm things up, in maybe 500 words. Well, I was publishing three A-to-Z essays a week back then; something had to go.
Laura, the author of MathSux2, offered this week’s A-to-Z term. (I apologize for it being late but the Playful Math Education Blog Carnival 141 work took a lot out of me.) She writes the blog weekly, and hosts a YouTube channel of mathematics videos also. I’m glad to have the topic to discuss.
We learn to count permutations before we know what they are. There are good reasons to. Counting permutations gives us numbers that are big, and therefore interesting, fast. Counting is easy to motivate. Humans like counting. Counting is useful. Many probability questions are best answered by counting all the ways to arrange things, and how many of those arrangements are desirable somehow.
The count of permutations asks how many ways there are to put some things in order. If some of the things are identical, the number is smaller. Calculating the count may be a little tedious, but it’s not hard. We calculate, rather than “really” count, because — well, list all the possible ways to arrange the letters of the word ‘DEMONSTRATION’. I bet you turn that listing over to a computer too. But what is the computer counting?
If we’re trying to do this efficiently we have some system. Start with ‘DEMONSTRATION’. Then, say, swap the last two letters: ‘DEMONSTRATINO’. Then, mm, move the ‘N’ to the antepenultimate position: ‘DEMONSTRATNIO’. Then, oh, swap the last two letters again: ‘DEMONSTRATNOI’.
Then, oh, move the ‘N’ to the third-to-the-last position: ‘DEMONSTRANTIO’. What next? Oh, swap the last two letters again: ‘DEMONSTRANTOI’. Or, move what been the last letter to the antepenultimate position: ‘DEMONSTRANOTI’. And swap the last two letters once more: ‘DEMONSTRANOIT’.
Enough of that, you and my spellchecker say. I agree. What is it that all this is doing? What does that tell us about what a permutation is?
An obvious thing. Each new variation of the order came from swapping two letters of an earlier one. We needed a sequence of swaps to get to ‘DEMONSTRANOIT’. But each swap was of only two things. It’s a good thing to observe.
Another obvious thing. There’s no letters in ‘DEMONSTRANOIT’ or any of the other variations that weren’t in ‘DEMONSTRATION’. All that’s changed is the order.
This all has listed eight permutations, counting the original ‘DEMONSTRATION’ as one. There are, calculations tell me, 778,377,592 to go.
Would the number of permutations be different if we were shuffling around different things? If instead of the letters in the word ‘DEMONSTRATION’ it were, say, the numerals in the sequence ‘1234567897045’? Or the sequence of symbols ‘!@#$%^&*(&)$%’ instead? No, and that it would not is another clue about what permutations are.
Another thing, obvious in retrospect. Grant that we’ve been making new permutations by taking a sequence of letters (numerals, symbols) and swapping a pair. We got from ‘DEMONSTRATION’ to ‘DEMONSTRATINO’ by swapping the last two letters. What happens if we swap the last two letters again? We get ‘DEMONSTRATION’, a sequence of letters all right, although one already on our list of permutations.
One more thing, obvious once you’ve seen it. Imagine we had not started with ‘DEMONSTRATION’ but instead ‘DEMONSTRATNIO’. But that we followed the same sequences of swappings. Would we have come up with different permutations? … At least for the first couple permutations? Or would they be the same permutations, listed in a different order?
You’ve been kind, letting me call these things “permutations” before I say what a permutation is. It’s relied on a casual, intuitive idea of a permutation. It’s a shuffling around of some set of things. This is the casual idea that mathematicians rely on for a permutation. Sure we can make the idea precise. How hard will that be?
It’s not hard in form. The permutation is the rearranging of things into a new order. The hard part is the concept. It’s not “these symbols in this order” that’s the permutation. It’s the act of putting them in this new order that is. So it’s “swap the 12th and the 13th symbols”. Or, “move the 13th symbol to 11th place, the 11th symbol to 12th, and the 12th symbol to 13th place”.
We can describe each permutation as a function. All the permutation functions have the same domain and the same range. And the range is the domain. The function is a bijection. Every item in the domain matches exactly one item in the range, and vice-versa. There’s some sequence for the elements in the domain. And the rule for the function describes how that sequence changes.
So one permutation is “swap the 12th and the 13th elements”. Another permutation is “swap the 11th and the 12th elements”. Since the range of one function is the domain of another, we can compose the together. That is, we can “swap the 12th and the 13th elements, and then swap the 11th and the 12th elements”. This gets us another permutation. The effect of these two permutations, in this order, is “make the 13th element the 11th, make the 11th element the 12th, and make the 12th element the 13th”. The order we do these permutations in counts. “Swap the 11th and the 12th elements, and then swap the 12th and the 13th” gets us a different net effect. That one is “make the 12th element the 11th, make the 13th element the 12th, and make the 11th element the 13th”. Composition of functions does not commute.
That functions compose is normal enough. That their composition doesn’t commute is normal enough too. These functions are a bit odd in that we don’t care what the domain-and-range is. We only care that we can index the elements in it. That leads us to some new observations.
The big one is that the set of all these permutations is a group. I mean the way mathematicians mean group. That is, we have a set of items. These are the functions, the permutations. The instructions, like, “make the 12th element the 11th and the 13th element the 12th”, or “the 12th element the 13th”. We also need a group action, a thing that works like addition does for real numbers. That’s composition. That is, doing one permutation and then the other, to get a new permutation out of it. That new permutation is itself one of the permutations we’d had. We can’t compose permutations and get something that’s not a permutation. No amount of swapping around the letters of ‘DEMONSTRATION’ will get us ‘DEMONSTRATIONERS’.
When we talk about how permutations as a group work, we want to give individual permutations names. That ends up being letters. These are often Greek letters. I don’t know why we can’t use the ordinary Latin alphabet. I suppose someone who liked Greek letters wrote a really good textbook and everyone copies that. So instead of speaking about x and y, we’ll get α and β. Sometimes σ and τ. Or, quite often π, especially if we need a bunch of permutations. Then we get π1, π2, π3, and so on. πj. All the way to πN. For the young mathematics major it might be the first time seeing π used for something not at all circle-related. It’s a weird sensation. Still, αβ is the composition of permutation α with permutation β. This means, do permutation β first, and then permutation α on whatever that result is. This is the same way that f(g(x)) means “evaluate g(x) first, and then figure out what f( that ) is”.
That’s all fine for naming them. But we would also like a good way to describe what a permutation does. There are several good forms. They all rely on indexing the elements, using the counting numbers: 1, 2, 3, 4, and so on. The notation I’ll share is called cycle notation. It’s easy to type. You write it within nice ordinary parentheses: (11 12) means “put the 11th element in slot 12, and the 12th element in slot 11”. (11, 12, 13) means “put the 11th element in slot 12, the 12th element in slot 13, and the 13th element in slot 11”. You can even chain these together: (10, 11)(12, 13) means “put the 10th element in slot 11 and the 11th element in slot 10; also, put the 12th element in slot 13, and the 13th element in slot 12”.
In that notation, writing (9), for example, means “put the 9th element in slot 9”. Or if you prefer, “leave element 9 alone”. Or we don’t mention it at all. The convention is that if something isn’t mentioned, leave it where it is.
This by the way is where we get the identity element. The permutation (1)(2)(3)(4)(etc) doesn’t actually swap anything. It counts as a permutation. Doing this is the equivalent of adding zero to a number.
This cycle notation makes it not hard to figure out the composition of permutations. What does (1 2)(1 3) do? Well, the (1 3) swaps the first and the third items. The (1 2), next, swaps what’s become the first and the second items. The effect is the same as the permutation (2 3 1). You can get pretty good at this sort of manipulation, in time.
You may also consider: if (1 2)(1 3) is the same as (2 3 1), then isn’t (2 3 1) the same as (1 2)(1 3)? Sure. But, like, can we write a longer permutation, like, (1 3 5 2 4), as the product of some smaller permutations? And we can. If it’s convenient, we can write it as a string of swaps, exchanging pairs of elements. This was the first “obvious” thing I had listed. A long enough chain of pairwise swaps will, in time, swap everything.
We call the group made of all these permutations the Symmetric Group of the set. Since it doesn’t matter what the underlying set is, just the number of elements in it, we can abbreviate this with the number of elements. S2. S4. SN. Symmetric Groups are among the first groups you meet in abstract algebra that aren’t, like, integers modulo 12 or symmetries of a triangle. It’s novel enough to be interesting and to not be completely sure you’re doing it right.
You never leave the Symmetric Group, though, not if you stay in algebra. It has powerful consequences. It ties, for example, into the roots of polynomials. The structure of S5 tells us there must exist fifth-degree polynomials we can’t solve by ordinary arithmetic and root operations. That is, there’s no version of the quadratic equation for high-order polynomials, and never can be.
There are more groups to build from permutations. The next one that you meet in Intro to Abstract Algebra is the Alternating Group. It’s made of only the even permutations. Those are the permutations made from an even number of swaps. (There are also odd permutations, which are what you imagine. They can’t make a group, though. No identity element.) They’re great for recapturing dread and uncertainty once you think you’ve got a handle on the Symmetric Group.
They lead to other groups too, and even rings. The Levi-Civita symbol describes whether a set of indices gives an even or an odd permutation (or neither). It makes life easier when we work on determinants and tensors and Jacobians. These tie in to the geometry of space, and how that affects physics. It also gets a supporting role in cross products. There are many cryptography schemes that have permutations at their core.
So this is a bit of what permutations are, and what they can get us.
Today’s and all the other 2020 A-to-Z essays should be at this link. Both the All-2020 and past A-to-Z essays should be at this link. Thanks for reading.
Edward Dunne, executive editor of Mathematical Reviews, published a short piece about Otto Neugebauer in the October 2020 Notices of the American Mathematical Society. Anyone interested in mathematics or science history knows Neugebauer’s name. He’s renowned for teaching the modern world how much the ancient world knew of mathematics and astronomy. Much of what we know about Babylonian computing we owe to Neugebauer’s work. Also our understanding of the Alexandrian Christian and Jewish calendars.
What I did not know, and am grateful to Dunne for explaining, was of Neugebauer’s principled stance, and actions, against racism and against toxic nationalism. And that is Dunne’s piece. Dunne regularly publishes at a blog on the American Mathematical Society 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.
I first want to thank Denise Gaskins for organizing the Playful Math Education Blog Carnival. It must be always a challenging and exhausting task and to carry it on for years is a great effort. The plan for the next several hosts of the Carnival is here, and if you would like to host a month, it’s a good place to volunteer.
For myself, you’re already looking at my mathematics blog. My big, ambitious project for this year is The All 2020 Mathematics A-to-Z. Each Wednesday I try to publish a long-form piece explaining some piece of mathematics. This week, I should reach the letter P. If you’d like to suggest a topic for the letters Q or R please leave a comment here. My other major project, Reading the Comics and writing about their mathematical content, is on hiatus. I’ll likely get back to it once the A-to-Z is finished.
One of my newer regular readers is Laura, teacher and tutor and author of the MathSux2: Putting math into normal people language blog. There’s new essays every week.
A friend knowing me well shared the Stand-Up Maths video Why is there no equation for the perimeter of an ellipse? The friend knew me well. I once assigned the problem, without working it out, to a vector-calculus class. The integral to do this formula is easy to write. It’s one of the many, many integrals that can’t be done. Attempting to do it leads to fascinating formulas, as seen in the video. And also to elliptic curves, a major research topic in mathematics.
Christian Lawson-Perfect, writing at The Aperiodical, looked at The enormous difficulty of telling the truth about escalators with statistics. Lawson-Perfect saw a sign claiming the subway station’s escalators worked 95% of the time. What did that mean? Defining what it means to have “escalators working” is a challenge. And it’s hard to define “95% of the time” in a way that harmonizes with our intuitions.
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.
I do not have a Desmos account. It’s been long enough since I had a real class that I haven’t yet joined the site. This may need to change. Christopher Sewards posted a set of activities in Permutations and Combinations which may be useful. There’s three so far and they may be joined by more. This I learned through Dan Meyer’s weekly roundup of links.
Meyer’s also made me aware of TheCalt, a mathematics tournament to be held the 17th of October. They’re taking signups even now. Here’s a page with three sample problems for guidance.
Sarah Carter similarly attempts a Monday Must-Reads collection at the MathEqualsLove blog. Given the disruptions of this year this was the first in the series in months. This collects a good number of links, many of them about being interesting while doing online classes.
Helene Osana writes Mathematical thinking begins in the early years with dialogue and real-world exploration. This is an essay about priming the mathematical thinking for the youngest children, those up to about five years old. One can encourage kids with small, casual activities that don’t look like education.
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.
Also crossing my desk this month was a couple-year-old article Melinda D Anderson published in The Atlantic. How Does Race Affect a Student’s Math Education? Mathematics affects a pose of being a culturally-independent, value-neutral study. The conclusions it draws might be. But what we choose to study, and how we choose to study it, is not. And how we teach it is socially biased and determined. So here are thoughts about that.
The last several links describe things we know thanks to modern psychology and neuroscience studies. Nicklas Balboa and Richard D Glaser published in Psychology Today Three Habits That Reduce Conversational Success. There are conversations which are, effectively, teaching attempts. To be aware of how those attempts go wrong, and how to fix them, is surely worth while.
Ben Orlin, of the popular Math With Bad Drawings blog, wrote Democracy isn’t math. But it isn’t NOT math. He contributed recently to David Litt’s Democracy In One Book Or Less. The broad goal of democracy, the setting of social rules by common consensus, might not be mathematical. When we look to the practical matters of implementing this, though, then we get a lot of mathematics. I have not read Litt’s book, or any recently-published book, so can’t say anything about its contents. I bet it includes Arrow’s Impossibility Theorem, though.
Anyone attempting to teach this year is having a heck of a time. Sarah Carter offered Goals for the 2020-2021 School Year – PANDEMIC STYLE as an attempt to organize planning. And shared her goals, which may help other people too.
Emelina Minero offered 8 Strategies to Improve Participation in Your Virtual Classroom. Class participation was always the most challenging part of my teaching, when I did any of that, and this was face-to-face. Online is a different experience, with different challenges. That there is usually the main channel of voice chat and the side channel of text offers new ways to get people to share, though.
The National Centre for Excellence in the Teaching of Mathematics offered Two Pleas to Maths Teachers at the Start of the School Year. This is about how to keep the unusual circumstances of the whole year from encouraging bad habits. This particularly since no one is on track, or near it.
S Leigh Nataro, of the MathTeacher24 blog, writes Learning Math is Social: We Are in This Together. Many teachers have gotten administrative guidance that … doesn’t … guide well. The easy joke is to say it never did. But the practical bits of most educational strategies we learn from long experience. There’s no comparable experience here. What are ways to reduce the size of the crisis? Nataro has thoughts.
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.
The Google Images picture gallery How Many? offers a soothing and self-directed counting puzzle. Each picture is a collection of things. How to count them, and even what you choose to count, is yours to judge.
Miss Konstantine of MathsHKO posted Area (Equal — Pythagorean Triples). Miss Konstantine had started with Pythagorean triplets, sets of numbers that can be the legs of a right triangle. And then explored other families of shapes that can have equal areas, including looking to circles and rings.
Sarah Carter makes another appearance here with New Puzzle: Only ‘Takes’ and ‘Adds’. This is in part about the challenge of finding new puzzles to make each week. And then an arithmetic challenge. Carter mentions how one presentation is quite nice for how it teaches so many rules of the puzzle.
Cassandra Lowry with the Australian Mathematical Sciences Institute offers Finding the Maths in Books. This is about how to read a book to find mathematical puzzles within. This is for children up to about second grade. The problems are about topics like counting and mapping and ordering.
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.
At Bedtime Math, Laura Overdeck wrote How Do Doggie Treats Taste? And spun this into some basic arithmetic problems built around the fun of giving dogs treats.
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.
Paul Godding’s 7 Puzzle Blog gives a string of recreational mathematics puzzles. Some include factoring, some include making expressions equal to particular numbers. They’re all things you can do when Slylock Fox printed the Six Differences puzzle too small for your eyes.
FractalKitty has a cute cartoon, No 5-second rule … about how the set of irrational numbers interacts with rationals in basic arithmetic.
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.
Resource Area For Teaching’s Carnival Math offers for preschool through grade six a semi-practical carnival game. There’s different goals for different education levels.
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.
I found interesting the talk about Math Midway, a touring exhibition meant to make mathematics ideas tactile. I’m not sure it’s still a going concern, though. Its schedule lists it as being at the Singapore Science Centre from February 2016 to present. But it’s not mentioned on the Singapore Science Centre’s page. (They do have a huge Tesla coil, though. Also they at least used to have an Albert Einstein animatronic, forever ascending and descending a rope. I enjoyed visiting it, although I would recommend going to the Tiger Balm Gardens as higher prioerity.) Still, exploring this did lead me to The National Museum of Mathematics, located in New York City. It has a fair number of exhibits and its events online.
But enough of the carnival as a generic theme. How about specific, actual rides and games? Theme Park Insider, one of the web’s top amusement-park-industry news, published Master the Midway: The Theme Park Insider Guide to Winning Carnival Games several years ago. The take from midway games is an expression of the Law of Large Numbers. The number of prizes won and their value will fluctuate day to day, but the averages will be predictable. And what players can do to better their chances is subject to reason.
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.)
And I have yet to make it to Six Flags America, but their Math & Science In Action page offers a similar roster of activities tied to that park. Six Flags America is their park in Maryland; the one in Illinois is Six Flags Great America.
Math Word Problems Solved offers a booklet of Amusement Park Word Problems Starring Pre-Algebra. These tie in to no particular amusement park. They do draw from real parks, though. For example it lists the highest point on the tallest steel roller coaster as 456 feet; it doesn’t name the ride, but that’s Kingda Ka, at Great Adventure. The highest point on the tallest wooden roller coaster is given as 218 feet, which was true at its 2009 publication: Son of Beast at Kings Island. Sad to say Son Of Beast closed in 2009, and was torn down in 2012. The current record heights in wooden coasters are T Express at Everland in South Korea, and Wildfire at Kolmården in Sweden. (Too much height is not really that good for wooden roller coasters.)
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.
And to step away from the fun a moment. Many aspects of the struggle to bring equality to Americans are reflected in amusement parks, or were fought by proxy in them. This is some serious matter, and is challenging to teach. Few amusement parks would mention segregation or racist attractions or policies except elliptically. (That midway game where you throw a ball at a clown’s face? The person taking the hit was not always a clown.) Claire Prentice’s The Lost Tribe of Coney Island: Headhunters, Luna Park, and the Man Who Pulled Off the Spectacle of the Century is a book I recommend. It reflects one slice of this history.
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 Howard County Public Schools Office published a Mathatastic Amusement Park worksheet. It uses the problem of finding things on a park map to teach about (Cartesian) coordinates in a well-motivated way.
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.
Cheryl Q Nelson and Nicole L Williams, writing for Mathematics Teacher, published the article Sprinklers and Amusement Parks: What Do They Have To Do With Geometry? Both (water) sprinklers and amusement park vendors are about covering spaces without waste. Someone might wonder at their hypothetical park where the bumper cars are one of the three most popular rides. I recommend a visit, when possible, to Conneaut Lake Park, in northwestern Pennsylvania. Their bumper cars are wild. Their roller coaster’s pretty great too.
And finally a bit of practical yet light news. Dickinson University was happy to share how The Traveling Salesman Problem Finds A Novel Application in Summer Student-Faculty Research Project. The Traveling Salesman Problem is the challenge to find the most efficient way to any set of points. It’s a problem both important and difficult. As you try to get to more points the problem (typically) gets far more difficult. I hadn’t seen it applied to amusement park itineraries before, but that’s a legitimate use. I am disappointed the press release did not share their work on most efficient routes around Hersheypark and Disney World. They did publish a comparison of ways to attack the problem.
And this closes the carnival, for today. If you’d like to follow this blog, please click the “Follow NebusReseearch” button the page. Or you can add the articles feed to your favorite RSS reader. My Twitter account @Nebusj is all but moribund. For whatever reason Safari often doesn’t want to let me see it. I am also present and active on Mathstodon. This is the mathematics-themed instance of Mastodon, as @Nebusj@mathstodon.xyz. I would be glad to have more people to chat with there. Thank you as ever for reading.
I’m going to take one more day, I think, preparing the Playful Math Education Blog Carnival. It’s hard work. But while you wait let me please share an older piece. In 2017 I wrote about Open Sets. These are important things, born of topology and offering us many useful tools. One of the best is that it lets us define “neighborhoods” and, along the way, “limits” and from that, “continuity”.
It was also a chance for me to finally think about one of those obvious nagging questions. There are open sets and there are closed sets. But it’s not the case that a set is either open or closed. A set can be not-open without being closed, and not-closed without being open. A set can even be both open and closed simultaneously. How can that turn out? And I learned that while “open” and “closed” are an obvious matched pair of words, they’re about describing very different traits of sets.
Occasionally an A-to-Z gives me the chance to naturally revisit an earlier piece. Orthonormal, from the Leap Day 2016 series, was one of those. It builds heavily on orthogonal, discussed the year before. When you know what the terms mean, of course it would. But getting to what the terms mean is part of the point of these essays.
Also, I hope to publish the 141th installment of the Playful Math Education Blog Carnival this weekend. If you’ve found a mathematics page, video, game, anything that delights or teaches or both, please mention in the comments. I’m eager to share it with more people.
Mr Wu, author of the Singapore Maths Tuition blog, asked me to explain a technical term today. I thought that would be a fun, quick essay. I don’t learn very fast, do I?
A note on style. I make reference here to “Big-O” and “Little-O”, capitalizing and hyphenating them. This is to give them visual presence as a name. In casual discussion they’re just read, or said, as the two words or word-and-a-letter. Often the Big- or Little- gets dropped and we just talk about O. An O, without further context, in my experience means Big-O.
The part of me that wants smooth consistency in prose urges me to write “Little-o”, as the thing described is represented with a lowercase ‘o’. But Little-o sounds like a midway game or an Eyerly Aircraft Company amusement park ride. And I never achieve consistency in my prose anyway. Maybe for the book publication. Until I’m convinced another is better, though, “Little-O” it is.
Big-O and Little-O Notation.
When I first went to college I had a campus post office box. I knew my box number. I also knew the length of the sluggish line for the combination lock code. The lock was a dial, lettered A through J. Being a young STEM-class idiot I thought, boy, would it actually be quicker to pick the lock than wait for the line? A three-letter combination, of ten options? That’s 1,000 possibilities. If I could try five a minute that’s, at worst, three hours 20 minutes. Combination might be anywhere in that set; I might get lucky. I could expect to spend 80 minutes picking my lock.
I decided to wait in line instead, and good that I did. I was unaware combination might not be a letter, like ‘A’. It could be the midway point between adjacent letters, like ‘AB’. That meant there were eight times as many combinations as I estimated, and I could expect to spend over ten hours. Even the slow line was faster than that. It transpired that my combination had two of these midway letters.
But that’s a little demonstration of algorithmic complexity. Also in cracking passwords by trial-and-error. Doubling the set of possible combination codes octuples the time it takes to break into the set. Making the combination longer would also work; each extra letter would multiply the cracking time by twenty. So you understand why your password should include “special characters” like punctuation, but most of all should be long.
We’re often interested in how long to expect a task to take. Sometimes we’re interested in the typical time it takes. Often we’re interested in the longest it could ever take. If we have a deterministic algorithm, we can say. We can count how many steps it takes. Sometimes this is easy. If we want to add two two-digit numbers together we know: it will be, at most, three single-digit additions plus, maybe, writing down a carry. (To add 98 and 37 is adding 8 + 7 to get 15, to add 9 + 3 to get 12, and to take the carry from the 15, so, 1 + 12 to get 13, so we have 135.) We can get a good quarrel going about what “a single step” is. We can argue whether that carry into the hundreds column is really one more addition. But we can agree that there is some smallest bit of arithmetic work, and work from that.
For any algorithm we have something that describes how big a thing we’re working on. It’s often ‘n’. If we need more than one variable to describe how big it is, ‘m’ gets called up next. If we’re estimating how long it takes to work on a number, ‘n’ is the number of digits in the number. If we’re thinking about a square matrix, ‘n’ is the number of rows and columns. If it’s a not-square matrix, then ‘n’ is the number of rows and ‘m’ the number of columns. Or vice-versa; it’s your matrix. If we’re looking for an item in a list, ‘n’ is the number of items in the list. If we’re looking to evaluate a polynomial, ‘n’ is the order of the polynomial.
In normal circumstances we don’t work out how many steps some operation does take. It’s more useful to know that multiplying these two long numbers would take about 900 steps than that it would need only 816. And so this gives us an asymptotic estimate. We get an estimate of how much longer cracking the combination lock will take if there’s more letters to pick from. This allowing that some poor soul will get the combination A-B-C.
There are a couple ways to describe how long this will take. The more common is the Big-O. This is just the letter, like you find between N and P. Since that’s easy, many have taken to using a fancy, vaguely cursive O, one that looks like . I agree it looks nice. Particularly, though, we write , where f is some function. In practice, we’ll see functions like or or . Usually something simple like that. It can be tricky. There’s a scheme for multiplying large numbers together that’s . What you will not see is something like , or or such. This comes to what we mean by the Big-O.
It’ll be convenient for me to have a name for the actual number of steps the algorithm takes. Let me call the function describing that g(n). Then g(n) is if once n gets big enough, g(n) is always less than C times f(n). Here c is some constant number. Could be 1. Could be 1,000,000. Could be 0.00001. Doesn’t matter; it’s some positive number.
There’s some neat tricks to play here. For example, the function ‘‘ is . It’s also and and . The function ‘ is also and those later terms, but it is not . And you can see why is right out.
There is also a Little-O notation. It, too, is an upper bound on the function. But it is a stricter bound, setting tighter restrictions on what g(n) is like. You ask how it is the stricter bound gets the minuscule letter. That is a fine question. I think it’s a quirk of history. Both symbols come to us through number theory. Big-O was developed first, published in 1894 by Paul Bachmann. Little-O was published in 1909 by Edmund Landau. Yes, the one with the short Hilbert-like list of number theory problems. In 1914 G H Hardy and John Edensor Littlewood would work on another measure and they used Ω to express it. (If you see the letter used for Big-O and Little-O as the Greek omicron, then you see why a related concept got called omega.)
What makes the Little-O measure different is its sternness. g(n) is if, for every positive number C, whenever n is large enough g(n) is less than or equal to C times f(n). I know that sounds almost the same. Here’s why it’s not.
If g(n) is , then you can go ahead and pick a C and find that, eventually, . If g(n) is , then I, trying to sabotage you, can go ahead and pick a C, trying my best to spoil your bounds. But I will fail. Even if I pick, like a C of one millionth of a billionth of a trillionth, eventually f(n) will be so big that . I can’t find a C small enough that f(n) doesn’t eventually outgrow it, and outgrow g(n).
This implies some odd-looking stuff. Like, that the function n is not . But the function n is at least , and and those other fun variations. Being Little-O compels you to be Big-O. Big-O is not compelled to be Little-O, although it can happen.
These definitions, for Big-O and Little-O, I’ve laid out from algorithmic complexity. It’s implicitly about functions defined on the counting numbers. But there’s no reason I have to limit the ideas to that. I could define similar ideas for a function g(x), with domain the real numbers, and come up with an idea of being on the order of f(x).
We make some adjustments to this. The important one is that, with algorithmic complexity, we assumed g(n) had to be a positive number. What would it even mean for something to take minus four steps to complete? But a regular old function might be zero or negative or change between negative and positive. So we look at the absolute value of g(x). Is there some value of C so that, when x is big enough, the absolute value of g(x) stays less than C times f(x)? If it does, then g(x) is . Is it the case that for every positive number C it’s true that g(x) is less than C times f(x), once x is big enough? Then g(x) is .
Fine, but why bother defining this?
A compelling answer is that it gives us a way to describe how different a function is from an approximation to that function. We are always looking for approximations to functions because most functions are hard. We have a small set of functions we like to work with. Polynomials are great numerically. Exponentials and trig functions are great analytically. That’s about all the functions that are easy to work with. Big-O notation particularly lets us estimate how bad an error we make using the approximation.
For example, the Runge-Kutta method numerically approximates solutions to ordinary differential equations. It does this by taking the information we have about the function at some point x to approximate its value at a point x + h. ‘h’ is some number. The difference between the actual answer and the Runge-Kutta approximation is . We use this knowledge to make sure our error is tolerable. Also, we don’t usually care what the function is at x + h. It’s just what we can calculate. What we want is the function at some point a fair bit away from x, call it x + L. So we use our approximate knowledge of conditions at x + h to approximate the function at x + 2h. And use x + 2h to tell us about x + 3h, and from that x + 4h and so on, until we get to x + L. We’d like to have as few of these uninteresting intermediate points as we can, so look for as big an h as is safe.
That context may be the more common one. We see it, particularly, in Taylor Series and other polynomial approximations. For example, the sine of a number is approximately:
This has consequences. It tells us, for example, that if x is about 0.1, this approximation is probably pretty good. So it is: the sine of 0.1 (radians) is about 0.0998334166468282 and that’s exactly what five terms here gives us. But it also warns that if x is about 10, this approximation may be gibberish. And so it is: the sine of 10.0 is about -0.5440 and the polynomial is about 1448.27.
The connotation in using Big-O notation here is that we look for small h’s, and for to be a tiny number. It seems odd to use the same notation with a large independent variable and with a small one. The concept carries over, though, and helps us talk efficiently about this different problem.
I hope this week to post the Playful Math Education Blog Carnival for September. Any educational or recreational or fun mathematics sites you know about would be greatly helpful to me and them. Thanks for your help.
Lastly, I am open for mathematics topics starting with P, Q, and R to write about next month. I’ve basically chosen my ‘P’ subject, though I’d be happy to hear alternatives for ‘Q’ and ‘R’ yet.
Thank you for reading.