My 2018 Mathematics A To Z: Manifold


Two commenters suggested the topic for today’s A to Z post. I suspect I’d have been interested in it if only one had. (Although Dina Yagoditch’s suggestion of the Menger Sponge is hard to resist.) But a double domination? The topic got suggested by Mr Wu, author of MathTuition88, and by John Golden, author of Math Hombre. My thanks to all for interesting things to think about.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Manifold.

So you know how in the first car you ever owned the alternator was always going bad? If you’re lucky, you reach a point where you start owning cars good enough that the alternator is not the thing always going bad. Once you’re there, congratulations. Now the thing that’s always going bad in your car will be the manifold. That one’s for my dad.

Manifolds are a way to do normal geometry on weird shapes. What’s normal geometry? It’s … you know, the way shapes work on your table, or in a room. The Euclidean geometry that we’re so used to that it’s hard to imagine it not working. Why worry about weird shapes? They’re interesting, for one. And they don’t have to be that weird to count as weird. A sphere, like the surface of the Earth, can be weird. And these weird shapes can be useful. Mathematical physics, for example, can represent the evolution of some complicated thing as a path drawn on a weird shape. Bringing what we know about geometry from years of study, and moving around rooms, to a problem that abstract makes our lives easier.

We use language that sounds like that of map-makers when discussing manifolds. We have maps. We gather together charts. The collection of charts describing a surface can be an atlas. All these words have common meanings. Mercifully, these common meanings don’t lead us too far from the mathematical meanings. We can even use the problem of mapping the surface of the Earth to understand manifolds.

If you love maps, the geography kind, you learn quickly that there’s no making a perfect two-dimensional map of the Earth’s surface. Some of these imperfections are obvious. You can distort shapes trying to make a flat map of the globe. You can distort sizes. But you can’t represent every point on the globe with a point on the paper. Not without doing something that really breaks continuity. Like, say, turning the North Pole into the whole line at the top of the map. Like in the Equirectangular projection. Or skipping some of the points, like in the Mercator projection. Or adding some cuts into a surface that doesn’t have them, like in the Goode homolosine projection. You may recognize this as the one used in classrooms back when the world had first begun.

But what if we don’t need the whole globedone in a single map? Turns out we can do that easy. We can make charts that cover a part of the surface. No one chart has to cover the whole of the Earth’s surface. It only has to cover some part of it. It covers the globe with a piece that looks like a common ordinary Euclidean space, where ordinary geometry holds. It’s the collection of charts that covers the whole surface. This collection of charts is an atlas. You have a manifold if it’s possible to make a coherent atlas. For this every point on the manifold has to be on at least one chart. It’s okay if a point is on several charts. It’s okay if some point is on all the charts. Like, suppose your original surface is a circle. You can represent this with an atlas of two charts. Each chart maps the circle, except for one point, onto a line segment. The two charts don’t both skip the same point. All but two points on this circle are on all the maps of this chart. That’s cool. What’s not okay is if some point can’t be coherently put onto some chart.

This sad fate can happen. Suppose instead of a circle you want to chart a figure-eight loop. That won’t work. The point where the figure crosses itself doesn’t look, locally, like a Euclidean space. It looks like an ‘x’. There’s no getting around that. There’s no atlas that can cover the whole of that surface. So that surface isn’t a manifold.

But many things are manifolds nevertheless. Toruses, the doughnut shapes, are. Möbius strips and Klein bottles are. Ellipsoids and hyperbolic surfaces are, or at least can be. Mathematical physics finds surfaces that describe all the ways the planets could move and still conserve the energy and momentum and angular momentum of the solar system. That cheesecloth surface stretched through 54 dimensions, is a manifold. There are many possible atlases, with many more charts. But each of those means we can, at least locally, for particular problems, understand them the same way we understand cutouts of triangles and pentagons and circles on construction paper.

So to get back to cars: no one has ever said “my car runs okay, but I regret how I replaced the brake covers the moment I suspected they were wearing out”. Every car problem is easier when it’s done as soon as your budget and schedule allow.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. What will I choose for ‘N’, later this week? I really should have decided that by now.

Convex hulls, the TSP, and long drives


The ‘God Plays Dice’ blog here makes me aware of a really fascinating little project: in 1998, apparently, Barry Stiefel decided to use a week off work to see the United States. All of them. And by dint of driving with a ruthlessness I can hardly believe, he managed it, albeit with some states just barely peeked at from the road, with Kentucky surprisingly hard to get at. Stiefel also managed, in a 2003 road expedition, to visit 21 states in a single carefully chosen day. I’m impressed.

This all seems connected to the Travelling Salesman Problem, or generally the problem of how to find the beset way to do something when there are so many ways to do it there’s no hope of testing all the possible combinations, but when there aren’t an infinite continuum of ways so that you can’t use calculus techniques to find a solution. Problems like this tend to be interesting ones: you can usually see exactly why the problem might be interesting, and you can work out a couple easy little toy cases, and if you fiddle around with toys that are a little more realistic you see just how hard they are.

I’ve never driven in more than six United States states in a day, near as I can tell. I have driven across Pennsylvania, even though that means crossing the Appalachians, and as anyone from the East Coast knows deep down, the Appalachians are a fundamentally uncrossable geographic barrier and anyone claiming to have crossed it is surely mad.

God plays dice

I’ve been reading the book In Pursuit of the Traveling Salesman by William Cook lately. In Chapter 10, on “The Human Touch”, Cook mentions a paper, by James MacGregor and Thomas Ormerod, “Human performance on the traveling salesman problem”. One thing that this paper mentions is that humans reach solutions to the TSP by looking at global properties such as the convex hull — there’s a theorem that a tour must visit the vertices of the convex hull in order.

This reminds me of one of my favorite travelogues, Barry Stiefel’s fifty states in a week’s vacation, in which Stiefel visited all fifty states on a week’s vacation (doing the lower 48 by driving, and flying to Alaska and Hawaii) Stiefel writes, in regard to getting to Kentucky: Now things were going to get complicated. Until then, my route had been primarily one of following the inside perimeter…

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Reading The Comics, November 14, 2014: Rectangular States Edition


I have no idea why Comic Strip Master Command decided this week should see everybody do some mathematics-themed comic strips, but, so they did, and here’s my collection of the, I estimate, six hundred comic strips that touched on something recently. Good luck reading it all.

Samsons Dark Side of the Horse (November 10) is another entry on the theme of not answering the word problem.

Scott Adams’s Dilbert Classics (November 10) started a sequence in which Dilbert gets told the big boss was a geometry major, so, what can he say about rectangles? Further rumors indicate he’s more a geography fan, shifting Dilbert’s topic to the “many” rectangular states of the United States. Of course, there’s only two literally rectangular states, but — and Mark Stein’s How The States Got Their Shapes contains a lot of good explanations of this — many of the states are approximately rectangular. After all, when many of the state boundaries were laid out, the federal government had only vague if any idea what the landscapes looked like in detail, and there weren’t many existing indigenous boundaries the white governments cared about. So setting a proposed territory’s bounds to be within particular lines of latitude and longitude, with some modification for rivers or shorelines or mountain ranges known to exist, is easy, and can be done with rather little of the ambiguity or contradictory nonsense that plagued the eastern states (where, say, a colony’s boundary might be defined as where a river intersects a line of latitude that in fact it never touches). And while perfect rectangularity may be achieved only by Colorado and Wyoming, quite a few states — the Dakotas, Washington, Oregon, Missisippi, Alabama, Iowa — are rectangular enough.

Mikael Wulff and Anders Morgenthaler’s WuMo (November 10) shows that their interest in pi isn’t just a casual thing. They think about what those neglected and non-famous numbers get up to.

Sherman does poorly with mathematic problems that look more like short stories. And don't have pictures.
Jim Toomey’s Sherman’s Lagoon for the 11th of November, 2014. He’s got a point about pictures helping with this kind of problem.

Jim Toomey’s Sherman’s Lagoon starts a “struggling with mathematics homework” story on the 11th, with Sherman himself stumped by a problem that “looks more like a short story” than a math problem. By the 14th Megan points out that it’s a problem that really doesn’t make sense when applied to sharks. Such is the natural hazard in writing a perfectly good word problem without considering the audience.

Homework: Megan points out that a problem about walking in the rain with umbrellas doesn't make much sense for sharks or other underwater creatures.
Jim Toomey’s Sherman’s Lagoon for the 14th of November, 2014.

Mike Peters’s Mother Goose and Grimm (November 12) takes one of its (frequent) breaks from the title characters for a panel-strip-style gag about Roman numerals.

A Roman medic calls for an IV; the nurse asks if he means this numeral 4.
Mike Peters’s Mother Goose and Grimm for the 12th of November, 2014.

Darrin Bell’s Candorville (November 12) starts talking about Zeno’s paradox — not the first time this month that a comic strip’s gotten to the apparent problem of covering any distance when distance is infinitely divisible. On November 13th it’s extended to covering stretches of time, which has exactly the same problem. Now it’s worth reminding people, because a stunning number of them don’t seem to understand this, that Zeno was not suggesting that there’s no such thing as motion (or that he couldn’t imagine an infinite convergent sequence; it’s easy to think of a geometric construction that would satisfy any ancient geometer); he was pointing out that there’s things that don’t make perfect sense about it. Either distance (and time) are infinitely divisible into indistinguishable units, or they are not; and either way has implications that seem contrary to the way motion works. Perhaps they can be rationalized; perhaps they can’t; but when you can find a question that’s easy to pose and hard to answer, you’re probably looking at something really worth thinking hard about.

Bill Amend’s FoxTrot Classics (November 12, a rerun) puns on the various meanings of “irrational”. A fun little fact you might want to try proving sometime, though I wouldn’t fault you if you only tried it out for a couple specific numbers and decided the general case too much to do: any whole number — like 2, 3, 4, or so on — has a square root that’s either another whole number, or else has a square root that’s irrational. There’s not a case where, say, the square root is exactly 45.144 or something like that, though it might be close.

Susan tries to figure out either what year she was in Grade Ten or what her age was back then. She admits she was never any good at math, although the real trouble might be she hasn't got a clear idea what she wants to calculate.
Sandra Bell-Lundy’s Between Friends for the 13th of November, 2014.

Sandra Bell-Lundy’sBetween Friends (November 13) shows one of those cases where mental arithmetic really is useful, as Susan tries to work out — actually, staring at it, I’m not precisely sure what she is trying to work out. Her and her coffee partner’s ages in Grade Ten, probably, or else just when Grade Ten was. That’s most likely her real problem: if you don’t know what you’re looking for it’s very difficult to find it. Don’t start calculating before you know what you’re trying to work out.

If I wanted to work out what year was 35 years ago I’d probably just use a hack: 35 years before 2014 is one year before “35 years before 2015”, which is a much easier problem to do. 35 years before 2015 is also 20 years before 2000, which is 1980, so subtract one and you get 1979. (Alternatively, I might remember it was 35 years ago that the Buggles’ “Video Killed The Radio Star” first appeared, which I admit is not a method that would work for everyone, or for all years.) If I wanted to work out my (and my partner’s) age in Grade Ten … well, I’d use a slightly different hack: I remember very well that I was ten years old in Grade Five (seriously, the fact that twice my grade was my age overwhelmed my thinking on my tenth birthday, which is probably why I had to stay in mathematics), so, add five to that and I’d be 15 in Grade Ten.

Bill Whitehead’s Free Range (November 13) brings up one of the most-quoted equations in the world in order to show off how kids will insult each other, which is fair enough.

Rick Detorie’s One Big Happy (November 13), this one a rerun from a couple years ago because that’s how his strip works on Gocomics, goes to one of its regular bits of the kid Ruthie teaching anyone she can get in range, and while there’s a bit more to arithmetic than just adding two numbers to get a bigger number, she is showing off an understanding of a useful sanity check: if you add together two (positive) numbers, you have to get a result that’s bigger than either of the ones you started with. As for the 14th, and counting higher, well, there’s not much she could do about that.

Steve McGarry’s Badlands (November 14) talks about the kind of problem people wish to have: how to win a lottery where nobody else picks the same numbers, so that the prize goes undivided? The answer, of course, is to have a set of numbers that nobody else picked, but is there any way to guarantee that? And this gets into the curious psychology of random numbers: there is absolutely no reason that 1-2-3-4-5-6, or for that matter 7-8-9-10-11-12, would not come up just as often as, say, 11-37-39-51-52-55, but the latter set looks more random. But we see some strings of numbers as obviously a pattern, while others we don’t see, and we tend to confuse “we don’t know the pattern” with “there is no pattern”. I have heard the lore that actually a disproportionate number of people pick such obvious patterns like 1-2-3-4-5-6, or numbers that form neat pictures on a lottery card, no doubt cackling at how much more clever they are than the average person, and guaranteeing that if such a string ever does come out there’ll a large number of very surprised lottery winners. All silliness, really; the thing to do, obviously, is buy two tickets with the exact same set of numbers, so that if you do win, you get twice the share of anyone else, unless they’ve figured out the same trick.

My August 2013 Statistics


As promised I’m keeping and publicizing my statistics, as WordPress makes them out, the better I hope to understand what I do well and what the rest is. I’ve had a modest uptick in views from July — 341 to 367 — as well as in unique visitors — 156 to 175 — although this means my views-per-visitor count has dropped from 2.19 to 2.10. That’s still my third-highest views-per-visitor count since WordPress started revealing that data to us.

The most popular articles of the past month were:

  1. Reading the Comics, September 11, 2012 (which I suspect reflects the date turning up high in Google searches, though there are many comics mentioned in it so perhaps it just casts a wide net);
  2. Just Answer 1/e Whenever Anyone Asks This Kind Of Question (part of the thread on the chance of the 1902-built Leap-the-Dips roller coaster having any of its original boards remaining)
  3. Just How Far Is The End Of The World? (the start of a string based on seeing the Sleeping Bear Dunes)
  4. Why I Say 1/e About This Roller Coaster (back to the Leap-the-Dips question)
  5. Augustin-Louis Cauchy’s birthday (a little biographical post which drew some comments because Cauchy was at the center of a really interesting time in mathematics)
  6. Professor Ludwig von Drake Explains Numerical Mathematics (which isn’t one of the top-five pieces this month but is a comic strip, so enjoy)

The countries sending me the most readers were the United States (202), Canada (30), and Denmark (19). Sending me just one each were Argentina, Bangladesh, Estonia, Finland, the Netherlands, Portugal, Singapore, Sri Lanka, Taiwan, and Viet Nam. Argentina, Estonia, and the Netherlands did the same last month; clearly I’m holding steady. And my readership in Slovenia doubled from last month’s lone reader.

I also learn that the search terms bringing people to me have been, most often, “trapezoid”, “descartes and the fly”, “joseph nebus”, “nebus wordpress”, and “any number the price is right”. I’ve had more entries than I realized mention The Price Is Right and ought to try making them easier for search engines to locate. My trapezoid work meanwhile — on their area and the number of different kinds of trapezoids — remains popular.