The Summer 2017 Mathematics A To Z: Topology

Today’s glossary entry comes from Elke Stangl, author of the Elkemental Force blog. I’ll do my best, although it would have made my essay a bit easier if I’d had the chance to do another topic first. We’ll get there.

Topology.

Start with a universe. Nice thing to have around. Call it ‘M’. I’ll get to why that name.

I’ve talked a fair bit about weird mathematical objects that need some bundle of traits to be interesting. So this will change the pace some. Here, I request only that the universe have a concept of “sets”. OK, that carries a little baggage along with it. We have to have intersections and unions. Those come about from having pairs of sets. The intersection of two sets is all the things that are in both sets simultaneously. The union of two sets is all the things that are in one set, or the other, or both simultaneously. But it’s hard to think of something that could have sets that couldn’t have intersections and unions.

So from your universe ‘M’ create a new collection of things. Call it ‘T’. I’ll get to why that name. But if you’ve formed a guess about why, then you know. So I suppose I don’t need to say why, now. ‘T’ is a collection of subsets of ‘M’. Now let’s suppose these four things are true.

First. ‘M’ is one of the sets in ‘T’.

Second. The empty set ∅ (which has nothing at all in it) is one of the sets in ‘T’.

Third. Whenever two sets are in ‘T’, their intersection is also in ‘T’.

Fourth. Whenever two (or more) sets are in ‘T’, their union is also in ‘T’.

Got all that? I imagine a lot of shrugging and head-nodding out there. So let’s take that. Your universe ‘M’ and your collection of sets ‘T’ are a topology. And that’s that.

Yeah, that’s never that. Let me put in some more text. Suppose we have a universe that consists of two symbols, say, ‘a’ and ‘b’. There’s four distinct topologies you can make of that. Take the universe plus the collection of sets {∅}, {a}, {b}, and {a, b}. That’s a topology. Try it out. That’s the first collection you would probably think of.

Here’s another collection. Take this two-thing universe and the collection of sets {∅}, {a}, and {a, b}. That’s another topology and you might want to double-check that. Or there’s this one: the universe and the collection of sets {∅}, {b}, and {a, b}. Last one: the universe and the collection of sets {∅} and {a, b} and nothing else. That one barely looks legitimate, but it is. Not a topology: the universe and the collection of sets {∅}, {a}, and {b}.

The number of toplogies grows surprisingly with the number of things in the universe. Like, if we had three symbols, ‘a’, ‘b’, and ‘c’, there would be 29 possible topologies. The universe of the three symbols and the collection of sets {∅}, {a}, {b, c}, and {a, b, c}, for example, would be a topology. But the universe and the collection of sets {∅}, {a}, {b}, {c}, and {a, b, c} would not. It’s a good thing to ponder if you need something to occupy your mind while awake in bed.

With four symbols, there’s 355 possibilities. Good luck working those all out before you fall asleep. Five symbols have 6,942 possibilities. You realize this doesn’t look like any expected sequence. After ‘4’ the count of topologies isn’t anything obvious like “two to the number of symbols” or “the number of symbols factorial” or something.

Are you getting ready to call me on being inconsistent? In the past I’ve talked about topology as studying what we can know about geometry without involving the idea of distance. How’s that got anything to do with this fiddling about with sets and intersections and stuff?

So now we come to that name ‘M’, and what it’s finally mnemonic for. I have to touch on something Elke Stangl hoped I’d write about, but a letter someone else bid on first. That would be a manifold. I come from an applied-mathematics background so I’m not sure I ever got a proper introduction to manifolds. They appeared one day in the background of some talk about physics problems. I think they were introduced as “it’s a space that works like normal space”, and that was it. We were supposed to pretend we had always known about them. (I’m translating. What we were actually told would be that it “works like R3”. That’s how mathematicians say “like normal space”.) That was all we needed.

Properly, a manifold is … eh. It’s something that works kind of like normal space. That is, it’s a set, something that can be a universe. And it has to be something we can define “open sets” on. The open sets for the manifold follow the rules I gave for a topology above. You can make a collection of these open sets. And the empty set has to be in that collection. So does the whole universe. The intersection of two open sets in that collection is itself in that collection. The union of open sets in that collection is in that collection. If all that’s true, then we have a manifold.

And now the piece that makes every pop mathematics article about topology talk about doughnuts and coffee cups. It’s possible that two topologies might be homeomorphic to each other. “Homeomorphic” is a term of art. But you understand it if you remember that “morph” means shape, and suspect that “homeo” is probably close to “homogenous”. Two things being homeomorphic means you can match their parts up. In the matching there’s nothing left over in the first thing or the second. And the relations between the parts of the first thing are the same as the relations between the parts of the second thing.

So. Imagine the snippet of the number line for the numbers larger than -π and smaller than π. Think of all the open sets you can use to cover that. It will have a set like “the numbers bigger than 0 and less than 1”. A set like “the numbers bigger than -π and smaller than 2.1”. A set like “the numbers bigger than 0.01 and smaller than 0.011”. And so on.

Now imagine the points that exist on a circle, if you’ve omitted one point. Let’s say it’s the unit circle, centered on the origin, and that what we’re leaving out is the point that’s exactly to the left of the origin. The open sets for this are the arcs that cover some part of this punctured circle. There’s the arc that corresponds to the angles from 0 to 1 radian measure. There’s the arc that corresponds to the angles from -π to 2.1 radians. There’s the arc that corresponds to the angles from 0.01 to 0.011 radians. You see where this is going. You see why I say we can match those sets on the number line to the arcs of this punctured circle. There’s some details to fill in here. But you probably believe me this could be done if I had to.

There’s two (or three) great branches of topology. One is called “algebraic topology”. It’s the one that makes for fun pop mathematics articles about imaginary rubber sheets. It’s called “algebraic” because this field makes it natural to study the holes in a sheet. And those holes tend to form groups and rings, basic pieces of Not That Algebra. The field (I’m told) can be interpreted as looking at functors on groups and rings. This makes for some neat tying-together of subjects this A To Z round.

The other branch is called “differential topology”, which is a great field to study because it sounds like what Mister Spock is thinking about. It inspires awestruck looks where saying you study, like, Bayesian probability gets blank stares. Differential topology is about differentiable functions on manifolds. This gets deep into mathematical physics.

As you study mathematical physics, you stop worrying about ever solving specific physics problems. Specific problems are petty stuff. What you like is solving whole classes of problems. A steady trick for this is to try to find some properties that are true about the problem regardless of what exactly it’s doing at the time. This amounts to finding a manifold that relates to the problem. Consider a central-force problem, for example, with planets orbiting a sun. A planet can’t move just anywhere. It can only be in places and moving in directions that give the system the same total energy that it had to start. And the same linear momentum. And the same angular momentum. We can match these constraints to manifolds. Whatever the planet does, it does it without ever leaving these manifolds. To know the shapes of these manifolds — how they are connected — and what kinds of functions are defined on them tells us something of how the planets move.

The maybe-third branch is “low-dimensional topology”. This is what differential topology is for two- or three- or four-dimensional spaces. You know, shapes we can imagine with ease in the real world. Maybe imagine with some effort, for four dimensions. This kind of branches out of differential topology because having so few dimensions to work in makes a lot of problems harder. We need specialized theoretical tools that only work for these cases. Is that enough to count as a separate branch? It depends what topologists you want to pick a fight with. (I don’t want a fight with any of them. I’m over here in numerical mathematics when I’m not merely blogging. I’m happy to provide space for anyone wishing to defend her branch of topology.)

But each grows out of this quite general, quite abstract idea, also known as “point-set topology”, that’s all about sets and collections of sets. There is much that we can learn from thinking about how to collect the things that are possible.

• gaurish 5:31 pm on Thursday, 14 September, 2017 Permalink | Reply

I am really happy that you didn’t start with “Topology is also known as rubber sheet geometry”.

Like

• Joseph Nebus 1:46 am on Friday, 15 September, 2017 Permalink | Reply

Although I never know precisely what I’m going to write before I put in the first paragraph, I did resolve that I was going to put off rubber sheets, as well as coffee cups, as long as I possibly could.

Liked by 1 person

• elkement (Elke Stangl) 7:33 am on Tuesday, 19 September, 2017 Permalink | Reply

Great post! I was interested in your take as there are different ways to introduce manifolds in theoretical physics – I worked through different General Relativity textbooks / courses in parallel: One lecturer insisted that you need to treat that stuff “with the rigor of a mathematician”, and he went to great lengths to point out why a manifold is different from “normal space”. Others use the typical physicist’s approach of avoiding all specialized terms like fiber bundles and pushbacks, calling everything a “vector field” and “space”, only alluding to comprehensible familiar structures that sort of work in the same way – and somehow still managed to get across the messages and theorems in the end. But the rigorous lecturer said that it was exactly confusing the actual space (or spacetime) and a manifold that had stalled and confused Einstein for many years – so I suppose one should really learn the mathematics thoroughly here …
On the other hand from what you say it seems to me that manifolds have sort of emerged as a tool in physics, and so Einstein had to create or inspire new mathematics as he went along … while today we can build on this and after we learned the rigorous stuff it is probably OK to fall back into the typical physicist’s mode. (Landau / Lifshitz are my favorite resource in the latter class – the treat GR very concisely in the volume on the Classical Theory of Fields, part of their 10-volume Course of Theoretical Physics – and they use hardly any specialized term related to topologies).

Like

• Joseph Nebus 8:10 pm on Friday, 22 September, 2017 Permalink | Reply

Thank you so. Well, I’ve shared just how I got introduced to manifolds myself. I come from a more mathematical physics background and it’s a little surprising how often things would be introduced casually, trusting that the precise details would be filled in later. Sometimes they even were. I don’t think that’s idiosyncratic to my school, although it was a heavily applied-mathematics department. (The joke was that we had two tracks, Applied Mathematics and More Applied Mathematics.)

I’m not very well-studied in the history of modern physics, at least not in how the mathematical models develop. But I think that you have a good read on it, that we started to get manifolds because they solved some very specific niche problems well. And then treated rigorously they promised more, and then people started looking for problems they could solve. I think that’s probably more common a history for mathematical structures than people realize. But, as you point out, that doesn’t mean everyone’s going to see the tool as worth learning how to use.

Liked by 1 person

Reading the Comics, April 15, 2017: Extended Week Edition

It turns out last Saturday only had the one comic strip that was even remotely on point for me. And it wasn’t very on point either, but since it’s one of the Creators.com strips I’ve got the strip to show. That’s enough for me.

Henry Scarpelli and Craig Boldman’s Archie for the 8th is just about how algebra hurts. Some days I agree.

Henry Scarpelli and Craig Boldman’s Archie for the 8th of April, 2017. Do you suppose Archie knew that Dilton was listening there, or was he just emoting his fatigue to himself?

Ruben Bolling’s Super-Fun-Pak Comix for the 8th is an installation of They Came From The Third Dimension. “Dimension” is one of those oft-used words that’s come loose of any technical definition. We use it in mathematics all the time, at least once we get into Introduction to Linear Algebra. That’s the course that talks about how blocks of space can be stretched and squashed and twisted into each other. You’d expect this to be a warmup act to geometry, and I guess it’s relevant. But where it really pays off is in studying differential equations and how systems of stuff changes over time. When you get introduced to dimensions in linear algebra they describe degrees of freedom, or how much information you need about a problem to pin down exactly one solution.

It does give mathematicians cause to talk about “dimensions of space”, though, and these are intuitively at least like the two- and three-dimensional spaces that, you know, stuff moves in. That there could be more dimensions of space, ordinarily inaccessible, is an old enough idea we don’t really notice it. Perhaps it’s hidden somewhere too.

Amanda El-Dweek’s Amanda the Great of the 9th started a story with the adult Becky needing to take a mathematics qualification exam. It seems to be prerequisite to enrolling in some new classes. It’s a typical set of mathematics anxiety jokes in the service of a story comic. One might tsk Becky for going through university without ever having a proper mathematics class, but then, I got through university without ever taking a philosophy class that really challenged me. Not that I didn’t take the classes seriously, but that I took stuff like Intro to Logic that I was already conversant in. We all cut corners. It’s a shame not to use chances like that, but there’s always so much to do.

Mark Anderson’s Andertoons for the 10th relieves the worry that Mark Anderson’s Andertoons might not have got in an appearance this week. It’s your common kid at the chalkboard sort of problem, this one a kid with no idea where to put the decimal. As always happens I’m sympathetic. The rules about where to move decimals in this kind of multiplication come out really weird if the last digit, or worse, digits in the product are zeroes.

Mel Henze’s Gentle Creatures is in reruns. The strip from the 10th is part of a story I’m so sure I’ve featured here before that I’m not even going to look up when it aired. But it uses your standard story problem to stand in for science-fiction gadget mathematics calculation.

Dave Blazek’s Loose Parts for the 12th is the natural extension of sleep numbers. Yes, I’m relieved to see Dave Blazek’s Loose Parts around here again too. Feels weird when it’s not.

Bill Watterson’s Calvin and Hobbes rerun for the 13th is a resisting-the-story-problem joke. But Calvin resists so very well.

John Deering’s Strange Brew for the 13th is a “math club” joke featuring horses. Oh, it’s a big silly one, but who doesn’t like those too?

Dan Thompson’s Brevity for the 14th is one of the small set of punning jokes you can make using mathematician names. Good for the wall of a mathematics teacher’s classroom.

Shaenon K Garrity and Jefferey C Wells’s Skin Horse for the 14th is set inside a virtual reality game. (This is why there’s talk about duplicating objects.) Within the game, the characters are playing that game where you start with a set number (in this case 20) tokens and take turn removing a couple of them. The “rigged” part of it is that the house can, by perfect play, force a win every time. It’s a bit of game theory that creeps into recreational mathematics books and that I imagine is imprinted in the minds of people who grow up to design games.

Reading the Comics, April 5, 2016: April 5, 2016 Edition

I’ve mentioned I like to have five or six comic strips for a Reading The Comics entry. On the 5th, it happens, I got a set of five all at once. Perhaps some are marginal for mathematics content but since when does that stop me? Especially when there’s the fun of a single-day Reading The Comics post to consider. So here goes:

Mark Anderson’s Andertoons is a student-resisting-the-problem joke. And it’s about long division. I can’t blame the student for resisting. Long division’s hard to learn. It’s probably the first bit of arithmetic in which you just have to make an educated guess for an answer and face possibly being wrong. And this is a problem that’ll have a remainder in it. I think I remember early on in long division finding a remainder left over feeling like an accusation. Surely if I’d done it right, the divisor would go into the original number a whole number of times, right? No, but you have to warm up to being comfortable with that.

Ted Key’s Hazel rerun the 5th of April, 2016. Were the Pac-Man ghosts ever called space demons? It seems like they might’ve been described that way in some boring official manual that nobody ever read. I always heard them as “ghosts” anyway.

Ted Key’s Hazel feels less charmingly out-of-date when you remember these are reruns. Ted Key — who created Peabody’s Improbable History as well as the sitcom based on this comic panel — retired in 1993. So Hazel’s attempt to create a less abstract version of the mathematics problem for Harold is probably relatively time-appropriate. And recasting a problem as something less abstract is often a good way to find a solution. It’s all right to do side work as a way to get the work you want to do.

John McNamee’s Pie Comic is a joke about the uselessness of mathematics. Tch. I wonder if the problem here isn’t the abstractness of a word like “hypotenuse”. I grant the word doesn’t evoke anything besides “hypotenuse”. But one irony is that hypotenuses are extremely useful things. We can use them to calculate how far away things are, without the trouble of going out to the spot. We can imagine post-apocalyptic warlords wanting to know how far things are, so as to better aim the trebuchets.

Percy Crosby’s Skippy is a rerun from 1928, of course. It’s also only marginally on point here. The mention of arithmetic is irrelevant to the joke. But it’s a fine joke and I wanted people to read it. Longtime readers know I’m a Skippy fan. (Saturday’s strip follows up on this. It’s worth reading too.)

Bill Griffith’s Zippy the Pinhead for the 5th of April, 2016. Not what Neil DeGrasse Tyson is like when he’s not been getting enough sleep.

Bill Griffith’s Zippy the Pinhead has picked up some quantum mechanics talk. At least he’s throwing around the sorts of things we see in pop science and, er, pop mathematical talk about the mathematics of cutting-edge physics. I’m not aware of any current models of everything which suppose there to be fourteen, or seventeen, dimensions of space. But high-dimension spaces are common points of speculation. Most of those dimensions appear to be arranged in ways we don’t see in the everyday world, but which leave behind mathematical traces. The crack about God not playing dice with the universe is famously attributed to Albert Einstein. Einstein was not comfortable with the non-deterministic nature of quantum mechanics, that there is this essential randomness to this model of the world.

Packing For Higher Dimensions

You may have heard of the sphere-packing problem. If you haven’t, let me brief you. It’s a problem about how to pack a bunch of spheres. Particularly, it’s about how to place spheres, all the same size, so there’s as little wasted space as possible.

It’s not an easy problem. Johannes Kepler, whom you remember as the astronomer with the gold nose because you’ve mixed him up with Tycho Brahe, studied it. He conjectured, in 1611, that the best packing you could do was the “close packing”. You know this pattern because it’s what a stack of oranges ends up being. We believe he was right. A computer-assisted proof was published in 2005.

But if we’re comfortable with mathematics we know a sphere, or a ball, doesn’t have to be something as boring as the balls we have in the real world. We could consider a circle to be a two-dimensional sphere. We could make something four-dimensional that looks a lot like a sphere. Or five-dimensional. Or 800-dimensional, if we have some reason to do this. (We do!) And optimization problems can be strange things. How many dimensions of space something has can affect how easy or hard a problem is. But just having more dimensions doesn’t mean the problem is harder. Sometimes having a vaster space means the problem becomes easier.

There’s recently been a breakthrough in the eight dimension. A paper by Maryna S Viazovska, with the Berlin Mathematical School and the Humboldt University of Berlin, seems to have worked out the densest possible packing for eight-dimensional spheres. And better, it ties into this beautiful pattern known as the E8 lattice. The MathsByAGirl blog recently posted an essay about that, and I’d like to recommend folks over there.

And, because I’m like this, I’d like to point folks over to one of my old essays. I’d got to wondering what the least efficient sphere packings were. The answers might surprise you.

Reading the Comics, January 8, 2015: Rerun-Heavy Edition

I couldn’t think of what connective theme there might be to the mathematically-themed comic strips of the last couple days. It finally struck me: there’s a lot of reruns in this. That’ll do. Most of them are reruns from before I started writing about comics so much in these parts.

Bill Watterson’s Calvin and Hobbes for the 5th of January (a rerun, of course, from the 7th of January, 1986) is a kid-resisting-the-test joke. The particular form is trying to claim a religious exemption from mathematics tests. I sometimes see attempts to claim that mathematics is a kind of religion since, after all, you have to believe it’s true. I’ll grant that you do have to assume some things without proof. Those are the rules of logical inference, and the axioms of the field, particularly. But I can’t make myself buy a definition of “religion” that’s just “something you believe”.

But there are religious overtones to a lot of mathematics. The field promises knowable universal truths, things that are true regardless of who and in what context might know them. And the study of mathematical infinity seems to inspire thoughts of God. Amir D Aczel’s The Mystery Of The Aleph: Mathematics, The Kabbala, and the Search for Infinity is a good read on the topic. Addition is still not a kind of religion, though.

Bud Grace’s The Piranha Club for the 6th of January, 2015.

Bud Grace’s The Piranha Club for the 6th of January uses the ability to do arithmetic as proof of intelligence. It’s a kind of intelligence, sure. There’s fun to be had in working out a square root in your head, or on paper. But there’s really no need for it now that we’ve got calculator technology, except for what it teaches you about how to compute.

Ruben Bolling’s Super-Fun-Pak Comix for the 6th of June is an installment of A Voice From Another Dimension. It’s just what the title suggests, and of course it would have to be a three-panel comic. The idea that creatures could live in more, or fewer, dimensions of space is a captivating one. It’s challenging to figure how it could work, though. Spaces of one or two dimensions don’t seem like they would allow biochemistry to work. And, as I understand it, chemistry itself seems unlikely to work right in four or more dimensions of space too. But it’s still fun to think about.

David L Hoyt and Jeff Knurek’s Jumble for the 7th of January is a counting-number joke. It does encourage asking whether numbers are created or discovered, which is a tough question. Counting numbers like “four” are so familiar and so apparently universal that they don’t seem to be constructs. (Even if they are, animals have an understanding of at least small counting numbers like these.) But if “four” is somehow not a human construct, then what about “4,000, 000,000, 000,000, 000,000, 000,000, 000,000”, a number so large it’s hard to think of something we have that many of that we can visualize. And even if that is, “one fourth” seems a bit different from that, and “four i” — the number which, squared, gives us negative 16 — seems qualitatively different. But if they’re constructs, then why do they correspond well to things we can see in the real world?

David L Hoyt and Jeff Knurek’s Jumble for the 7th of January, 2016. The link will likely expire around mid-February.

Greg Curfman’s Meg Classics for the 7th of January originally ran the 19th of September, 1997. It’s about a kid distractingly interested in multiplication. You get these sometimes. My natural instinct is to put the bigger number first and the smaller number second in a multiplication. “2 times 27” makes me feel nervous in a way “27 times 2” never will.

Hector D Cantu and Carlos Castellanos’s Baldo for the 8th of January is a rerun from 2011. It’s an old arithmetic joke. I wouldn’t be surprised if George Burns and Gracie Allen did it. (Well, a little surprised. Gracie Allen didn’t tend to play quite that kind of dumb. But everybody tells some jokes that are a little out of character.)

The Set Tour, Part 8: Balls, Only Made Harder

I haven’t forgotten or given up on the Set Tour, don’t worry or celebrate. I just expected there to be more mathematically-themed comic strips the last couple days. Really, three days in a row without anything at ComicsKingdom or GoComics to talk about? That’s unsettling stuff. Ah well.

Sn

We are also starting to get into often-used domains that are a bit stranger. We are going to start seeing domains that strain the imagination more. But this isn’t strange quite yet. We’re looking at the surface of a sphere.

The surface of a sphere we call S2. The “S” suggests a sphere. The “2” means that we have a two-dimensional surface, which matches what we see with the surface of the Earth, or a beach ball, or a soap bubble. All these are sphere enough for our needs. If we want to say where we are on the surface of the Earth, it’s most convenient to do this with two numbers. These are a latitude and a longitude. The latitude is the angle made between the point we’re interested in and the equator. The longitude is the angle made between the point we’re interested in and a reference prime longitude.

There are some variations. We can replace the latitude, for example, with the colatitude. That’s the angle between our point and the north pole. Or we might replace the latitude with the cosine of the colatitude. That has some nice analytic properties that you have to be well into grad school to care about. It doesn’t matter. The details may vary but it’s all the same. We put in a number for the east-west distance and another for the north-south distance.

It may seem pompous to use the same system to say where a point is on the surface of a beach ball. But can you think of a better one? Pointing to the ball and saying “there”, I suppose. But that requires we go around with the beach ball pointing out spots. Giving two numbers saves us having to go around pointing.

(Some weenie may wish to point out that if we were clever we could describe a point exactly using only a single number. This is true. Nobody does that unless they’re weenies trying to make a point. This essay is long enough without describing what mathematicians really mean by “dimension”. “How many numbers normal people use to identify a point in it” is good enough.)

S2 is a common domain. If we talk about something that varies with your position on the surface of the earth, we’re probably using S2 as the domain. If we talk about the temperature as it varies with position, or the height above sea level, or the population density, we have functions with a domain of S2 and a range in R. If we talk about the wind speed and direction we have a function with domain of S2 and a range in R3, because the wind might be moving in any direction.

Of course, I wrote down Sn rather than just S2. As with Rn and with Rm x n, there is really a family of similar domains. They are common enough to share a basic symbol, and the superscript is enough to differentiate them.

What we mean by Sn is “the collection of points in Rn+1 that are all the same distance from the origin”. Let me unpack that a little. The “origin” is some point in space that we pick to measure stuff from. On the number line we just call that “zero”. On your normal two-dimensional plot that’s where the x- and y-axes intersect. On your normal three-dimensional plot that’s where the x- and y- and z-axes intersect.

And by “the same distance” we mean some set, fixed distance. Usually we call that the radius. If we don’t specify some distance then we mean “1”. In fact, this is so regularly the radius I’m not sure how we would specify a different one. Maybe we would write Snr for a radius of “r”. Anyway, Sn, the surface of the sphere with radius 1, is commonly called the “unit sphere”. “Unit” gets used a fair bit for shapes. You’ll see references to a “unit cube” or “unit disc” or so on. A unit cube has sides length 1. A unit disc has radius 1. If you see “unit” in a mathematical setting it usually means “this thing measures out at 1”. (The other thing it may mean is “a unit of measure, but we’re not saying which one”. For example, “a unit of distance” doesn’t commit us to saying whether the distance is one inch, one meter, one million light-years, or one angstrom. We use that when we don’t care how big the unit is, and only wonder how many of them we have.)

S1 is an exotic name for a familiar thing. It’s all the points in two-dimensional space that are a distance 1 from the origin. Real people call this a “circle”. So do mathematicians unless they’re comparing it to other spheres or hyperspheres.

This is a one-dimensional figure. We can identify a single point on it easily with just one number, the angle made with respect to some reference direction. The reference direction is almost always that of the positive x-axis. That’s the line that starts at the center of the circle and points off to the right.

S3 is the first hypersphere we encounter. It’s a surface that’s three-dimensional, and it takes a four-dimensional space to see it. You might be able to picture this in your head. When I try I imagine something that looks like the regular old surface of the sphere, only it has fancier shading and maybe some extra lines to suggest depth. That’s all right. We can describe the thing even if we can’t imagine it perfectly. S4, well, that’s something taking five dimensions of space to fit in. I don’t blame you if you don’t bother trying to imagine what that looks like exactly.

The need for S4 itself tends to be rare. If we want to prove something about a function on a hypersphere we usually make do with Sn. This doesn’t tell us how many dimensions we’re working with. But we can imagine that as a regular old sphere only with a most fancy job of drawing lines on it.

If we want to talk about Sn aloud, or if we just want some variation in our prose, we might call it an n-sphere instead. So the 2-sphere is the surface of the regular old sphere that’s good enough for everybody but mathematicians. The 1-sphere is the circle. The 3-sphere and so on are harder to imagine. Wikipedia asserts that 3-spheres and higher-dimension hyperspheres are sometimes called “glomes”. I have not heard this word before, and I would expect it to start a fight if I tried to play it in Scrabble. However, I do not do mathematics that often requires discussion of hyperspheres. I leave this space open to people who do and who can say whether “glome” is a thing.

Something that all these Sn sets have in common are that they are the surfaces of spheres. They are just the boundary, and omit the interior. If we want a function that’s defined on the interior of the Earth we need to find a different domain.

• BunKaryudo 4:39 pm on Monday, 16 November, 2015 Permalink | Reply

Don’t let those weenies try explaining how to identify spots on the bubble with just one number. I was proud that my non-mathematical brain managed to more or less follow how to do it with two numbers. I don’t want a bunch of weenies spoiling it all and throwing me into confusion again.

Like

• Joseph Nebus 4:16 am on Tuesday, 17 November, 2015 Permalink | Reply

Two numbers is easy. If you’ve done latitude and longitude you’ve gotten the idea. Everything else is an implementation detail.

If you feel like a bit of a puzzle, you can work out how to go from the latitude and longitude to a single number that does represent a point on the surface of the sphere. Or vice-versa, from one big number to a latitude and longitude. (There are a lot of ways to do this. But there’s at least one really easy way.)

Liked by 1 person

• BunKaryudo 4:42 am on Tuesday, 17 November, 2015 Permalink | Reply

Yes, I like the two numbers way. I can understand that one. The one number version sounds harder. If there’s a really easy way to do it, that’s the one I’d use. I’m not confident I could follow anything more complicated.

Like

• Joseph Nebus 4:05 am on Friday, 20 November, 2015 Permalink | Reply

It does seem like two numbers is the natural way to represent points. Of course, that natural-ness reflects a cultural heritage. We’ve gotten very comfortable representing stuff with pairs of numbers, thanks to things like latitude-and-longitude, or cities with rectangular-grid layouts such as midtown Manhattan.

Liked by 1 person

• BunKaryudo 11:47 am on Friday, 20 November, 2015 Permalink | Reply

That’s interesting. I never thought of it being part of our cultural heritage before. Of course, the thing about ideas that come from our common cultural heritage is that when you’re actually part of the culture they can be rather difficult to spot. They just seem like common sense.

Like

• Joseph Nebus 6:57 am on Saturday, 21 November, 2015 Permalink | Reply

They do, yes. They seem like common sense, or even more insidiously they don’t even seem to be ideas at all. I feel that different most starkly when I look at things like those South American nations that would use webs of tied strings to represent numbers. Even seeing how they’re supposed to be read, I feel wholly lost. And that’s just numerals, almost the first thing you can do with mathematics.

Like

Reading the Comics, June 25, 2015: Not Making A Habit Of This Edition

I admit I did this recently, and am doing it again. But I don’t mean to make it a habit. I ran across a few comic strips that I can’t, even with a stretch, call mathematically-themed, but I liked them too much to ignore them either. So they’re at the end of this post. I really don’t intend to make this a regular thing in Reading the Comics posts.

Justin Boyd’s engagingly silly Invisible Bread (June 22) names the tuning “two steps below A”. He dubs this “negative C#”. This is probably an even funnier joke if you know music theory. The repetition of the notes in a musical scale could be used as an example of cyclic or modular arithmetic. Really, that the note above G is A of the next higher octave, and the note below A is G of the next lower octave, probably explains the idea already.

If we felt like, we could match the notes of a scale to the counting numbers. Match A to 0, B to 1, C to 2 and so on. Work out sharps and flats as you like. Then we could think of transposing a note from one key to another as adding or subtracting numbers. (Warning: do not try to pass your music theory class using this information! Transposition of keys is a much more subtle process than I am describing.) If the number gets above some maximum, it wraps back around to 0; if the number would go below zero, it wraps back around to that maximum. Relabeling the things in a group might make them easier or harder to understand. But it doesn’t change the way the things relate to one another. And that’s why we might call something F or negative C#, as we like and as we hope to be understood.

Hilary Price’s Rhymes With Orange for the 23rd of June, 2015.

Hilary Price’s Rhymes With Orange (June 23) reminds us how important it is to pick the correct piece of chalk. The mathematical symbols on the board don’t mean anything. A couple of the odder bits of notation might be meant as shorthand. Often in the rush of working out a problem some of the details will get written as borderline nonsense. The mathematician is probably more interested in getting the insight down. She’ll leave the details for later reflection.

Jason Poland’s Robbie and Bobby (June 23) uses “calculating obscure digits of pi” as computer fun. Calculating digits of pi is hard, at least in decimals, which is all anyone cares about. If you wish to know the 5,673,299,925th decimal digit of pi, you need to work out all 5,673,299,924 digits that go before it. There are formulas to work out a binary (or hexadecimal) digit of pi without working out all the digits that go before. This saves quite some time if you need to explore the nether-realms of pi’s digits.

The comic strip also uses Stephen Hawking as the icon for most-incredibly-smart-person. It’s the role that Albert Einstein used to have, and still shares. I am curious whether Hawking is going to permanently displace Einstein as the go-to reference for incredible brilliance. His pop culture celebrity might be a transient thing. I suspect it’s going to last, though. Hawking’s life has a tortured-genius edge to it that gives it Romantic appeal, likely to stay popular.

Paul Trap’s Thatababy (June 23) presents confusing brand-new letters and numbers. Letters are obviously human inventions though. They’ve been added to and removed from alphabets for thousands of years. It’s only a few centuries since “i” and “j” became (in English) understood as separate letters. They had been seen as different ways of writing the same letter, or the vowel and consonant forms of the same letter. If enough people found a proposed letter useful it would work its way into the alphabet. Occasionally the ampersand & has come near being a letter. (The ampersand has a fascinating history. Honestly.) And conversely, if we collectively found cause to toss one aside we could remove it from the alphabet. English hasn’t lost any letters since yogh (the Old English letter that looks like a 3 written half a line off) was dropped in favor of “gh”, about five centuries ago, but there’s no reason that it couldn’t shed another.

Numbers are less obviously human inventions. But the numbers we use are, or at least work like they are. Arabic numerals are barely eight centuries old in Western European use. Their introduction was controversial. People feared shopkeepers and moneylenders could easily cheat people unfamiliar with these crazy new symbols. Decimals, instead of fractions, were similarly suspect. Negative numbers took centuries to understand and to accept as numbers. Irrational numbers too. Imaginary numbers also. Indeed, look at the connotations of those names: negative numbers. Irrational numbers. Imaginary numbers. We can add complex numbers to that roster. Each name at least sounds suspicious of the innovation.

There are more kinds of numbers. In the 19th century William Rowan Hamilton developed quaternions. These are 4-tuples of numbers that work kind of like complex numbers. They’re strange creatures, admittedly, not very popular these days. Their greatest strength is in representing rotations in three-dimensional space well. There are also octonions, 8-tuples of numbers. They’re more exotic than quaternions and have fewer good uses. We might find more, in time.

Rina Piccolo’s entry in Six Chix for the 24th of June, 2015.

Rina Piccolo’s entry in Six Chix this week (June 24) draws a house with extra dimensions. An extra dimension is a great way to add volume, or hypervolume, to a place. A cube that’s 20 feet on a side has a volume of 203 or 8,000 cubic feet, after all. A four-dimensional hypercube 20 feet on each side has a hypervolume of 160,000 hybercubic feet. This seems like it should be enough for people who don’t collect books.

Morrie Turner’s Wee Pals (June 24, rerun) is just a bit of wordplay. It’s built on the idea kids might not understand the difference between the words “ratio” and “racial”.

Tom Toles’s Randolph Itch, 2 am (June 25, rerun) inspires me to wonder if anybody’s ever sold novelty 4-D glasses. Probably they have, sometime.

Now for the comics that I just can’t really make mathematics but that I like anyway:

Phil Dunlap’s Ink Pen (June 23, rerun) is aimed at the folks still lingering in grad school. Please be advised that most doctoral theses do not, in fact, end in supervillainy.

Darby Conley’s Get Fuzzy (June 25, rerun) tickles me. But Albert Einstein did after all say many things in his life, and not everything was as punchy as that line about God and dice.

Like

• Joseph Nebus 6:38 pm on Saturday, 4 July, 2015 Permalink | Reply

I suspect it’s more that we can build models of anything we’re interested in that will have numbers. And since we have many tools for manipulating numbers, we can understand things about so very many models. If our models are good, this gives us insight into the thing we’re interested in. That’s a little different from saying that numbers are in everything, although it’s pretty close for many applications.

Like

• ivasallay 4:30 pm on Monday, 29 June, 2015 Permalink | Reply

The comic about ratio and racial made me laugh a lot. Thank you so much for sharing that one!

Like

• Joseph Nebus 6:42 pm on Saturday, 4 July, 2015 Permalink | Reply

Glad you enjoyed. Wee Pals was an intriguing strip, I think because it does try so earnestly to speak for racial and sexual equality in a medium that isn’t really well-equipped for the discussion.

Like

Hypersphere.

If you asked someone to say what mathematicians do, there are, I think, three answers you’d get. One would be “they write out lots of decimal places”. That’s fair enough; that’s what numerical mathematics is about. One would be “they write out complicated problems in calculus”. That’s also fair enough; say “analysis” instead of “calculus” and you’re not far off. The other answer I’d expect is “they draw really complicated shapes”. And that’s geometry. All fair enough; this is stuff real mathematicians do.

Geometry has always been with us. You may hear jokes about never using algebra or calculus or such in real life. You never hear that about geometry, though. The study of shapes and how they fill space is so obviously useful that you sound like a fool saying you never use it. That would be like claiming you never use floors.

There are different kinds of geometry, though. The geometry we learn in school first is usually plane geometry, that is, how shapes on a two-dimensional surface like a sheet of paper or a computer screen work. Here we see squares and triangles and trapezoids and theorems with names like “side-angle-side congruence”. The geometry we learn as infants, and perhaps again in high school, is solid geometry, how shapes in three-dimensional spaces work. Here we see spheres and cubes and cones and something called “ellipsoids”. And there’s spherical geometry, the way shapes on the surface of a sphere work. This gives us great circle routes and loxodromes and tales of land surveyors trying to work out what Vermont’s northern border should be.

• sheldonk2014 7:11 pm on Wednesday, 10 June, 2015 Permalink | Reply

The study of shapes and how they fill a space,I wished someone had said that to me instead of hitting me over the head with the geometry book and throwing me out if class I just mite of learned something
Sheldon

Like

• Joseph Nebus 7:37 pm on Thursday, 11 June, 2015 Permalink | Reply

Terrible experience; I’m sorry you hadn’t had a better one. The subject can be hard going since a lot of what’s really interesting requires deep thought. But geometry can make appeals to intuition that, say, number theory can only wish it could.

Like

• elkement (Elke Stangl) 4:55 pm on Thursday, 29 June, 2017 Permalink | Reply

Ha – of course you covered it! I came back to this post as it was listed in your nice table today! I admit I wrote about hyperspheres recently in detail – im relation to statistical mechanics, so about spheres with awfully many dimensions – but I totally forgot about this posting of yours ;-)

Please allow for a shameless plug:
https://elkement.blog/2017/06/17/spheres-in-a-space-with-trillions-of-dimensions/
(Your explanation of Taylor expansions came in handy – otherwise it would have been even longer ;-))

Like

• Joseph Nebus 3:45 am on Monday, 3 July, 2017 Permalink | Reply

Thanks so kindly and I’m glad I could be of use! And please, do feel free to plug away. I’m hoping to get something like caught up on reading WordPress blogs in the next couple days as I recover from a vacation that was fantastic but consumed all my attention for a couple weeks. It’s turning out to be a great summer, just a little frantic of one while I’m in the middle of it.

Liked by 1 person

Reading the Comics, May 14, 2015: At The Cash Register Edition

This might not be the most exciting week of mathematically-themed comic strips. But it gives me the chance to be more autobiographical than usual. And it’s got more reruns than average, too.

Also, I’m trying out a new WordPress Theme. I’m a little suspicious of it myself, but will see what I think of it a week from now. Don’t worry, I remember the name of the old one in case I want to go back. Also, WordPress Master Command: stop hiding the option to live-preview themes instead of switching to them right away.

Norm Feuti’s Retail For the 11th of May, 2015.

Norm Feuti’s Retail (May 11) led off a week of “Epic Customer Fails” with an arithmetic problem. My own work in retail was so long ago and for so short a time I don’t remember this happening. But I can believe in a customer being confused this way. I think there is a tendency to teach arithmetic problems as a matter of “pick out the numbers, pick out the operation, compute that”. This puts an emphasis placed on computing quickly. That seems to invite too-quick calculation of not-quite the right things. That percentages are a faintly exotic construct to many people doesn’t help either.

My own retail customers-with-percentages story is duller. A customer asked about a book, I believe an SAT preparation book, which had a 20 percent (or whatever) off sticker. He specifically wanted to know whether 20 percent was taken off the price before the sales tax (6 percent) was calculated, or whether the registers added the sales tax and then took 20 percent off that total. I tried to reassure him that it didn’t matter, the resulting price would be the same. He tried to reassure me that it did matter because the sales tax should be calculated on the price paid, not reduced afterward. I believed, then and now, that he was right legally, but for the practical point of how much he had to pay it made no difference.

He judged me warily, but I worked out what the price paid would be, and he let me ring the book up. And the price came out about a dollar too high. The bar code had a higher price for the book than the plain-english corner said. He snorted “Ha!” and may have told me so. I explained the problem, showing the bar code version of the price (it’s in the upper-right corner of the bar code on books) and the price I’d used to calculate. He repeated that this was why he had asked, while I removed the wrong price and entered the thing manually so I could put in the lower price. And took the 20 percent off, and added sales tax, which came out to what I had said the price was.

I don’t believe I ever saw him again, but I would like the world to know that I was right. And the SAT prep book-maker needed to not screw up their bar codes.

• sheldonk2014 4:21 pm on Friday, 15 May, 2015 Permalink | Reply

Are you saying there is a mathematical theory to how and why we die or even when it where,I guess if you step back and I think for a second I can see this understanding
Sheldonn

Like

• Joseph Nebus 6:07 pm on Friday, 15 May, 2015 Permalink | Reply

Well, what do you mean by a theory? Since governments started raising money by selling annuities they’ve studied how long people are likely to live, and how many are likely to die — and from what causes — in any given area in a given year. That’s in some sense a theory of how and when people do die. And demographers study how people move about, and what for; that would look at where and why.

That’s whole populations, though. And it’s the somewhat strange idea that we could say pretty reliably that (say) 300 people in this area will die over the coming year, but have no idea of which ones, or just when in the year they would. We can use that for individual predictions, though: if you’re a person of this age range, and have this set of medical conditions, and so on, then we can find a group of people you’re generally like and suppose you’ll live about as long as typical for that group.

On an individual body experience, studying the mathematics of how bodies work — how nutrients and oxygen are spread out, how the body’s cells reproduce and decay and even die, how they thrive and how they break down — is a lot of good work to be done too. In graduate school I even did a course on medical mathematics, although that studied more very specific topics like how to infer how nerve pulses flow around the heart from the signals provided by an electrocardiogram. That is a theory about how bodies work, and why, although that might not be quite what you’re thinking of.

Like

• ivasallay 12:36 pm on Saturday, 16 May, 2015 Permalink | Reply

I always enjoy Retail, and you made it even better with your story about selling the SAT prep book!

Like

• Joseph Nebus 5:08 pm on Sunday, 17 May, 2015 Permalink | Reply

Aw, thanks. I’m glad you liked.

Like

• sheldonk2014 4:28 am on Monday, 18 May, 2015 Permalink | Reply

I am a two dimensiona person living in a three dimensional world

Like

• Joseph Nebus 5:31 pm on Wednesday, 20 May, 2015 Permalink | Reply

Ah, but if you play your cards right, you could still fill a volume.

Like

• ioanaiuliana 7:30 pm on Monday, 25 May, 2015 Permalink | Reply

When I saw the 1st comic I thought I have to share my retail story :)) : P
One of my favorite clothes shops had a big 50% off in the window… I went in and decided to buy a dress, obviously :))) and looked on the label: old price £180, new price £120. I went to shop assistant and asked if all the prices are reduced by 50% or just some items. She so cheerful said that all of them. So, I thought it was a mistake on the label. Decided to buy the dress after all :P
At the till the lady really smiling and happy told me the dress is £120 and started talking with me randomly. She even asked my what I study at the university and told her I am doing math :P I asked about the old price. She said £180. I looked at her like a small curios child and asked: so after 50% off the dress is £120? She said: yes. When I asked how much is 50% from 180 she blocked. Her face turned red and she used a calculator to find out. After that she panicked and I ended up helping her calculate the 50% off for all the shop because apparently I was faster than her with a calculator.
I felt sorry for her that day, but after a couple of hours of doing this I got a new friend and the dress for free ^_^

Like

Reading the Comics, March 10, 2015: Shapes Of Things Edition

If there’s a theme running through today’s collection of mathematics-themed comic strips it’s shapes: I have good reason to talk about a way of viewing circles and spheres and even squares and boxes; and then both Euclid and men’s ties get some attention.

Eric the Circle (March 5), this one by “regina342”, does a bit of shape-name-calling. I trust that it’s not controversial that a rectangle is also a parallelogram, but people might be a bit put off by describing a circle as a sphere, what with circles being two-dimensional figures and spheres three-dimensional ones. For ordinary purposes of geometry that’s a fair enough distinction. Let me now make this complicated.

• Kurt Struble 5:28 am on Wednesday, 11 March, 2015 Permalink | Reply

i have a question. if feeling impish describes how you would feel by proving that two points on a line can be shown to be a sphere … which i THINK you are implying you can do … then .,..

if by some formula or definition (i’m not sure which) D, you can describe a forest of trees by the density of trees within a certain area … AND … by another formula or definition B, you can, determine the number of board feet contained within the density of trees within a certain area …

… can it be said that, since you already know the number of board feet needed to build a two bedroom home, you can use the relationship between D and B to determine the number of two bedroom homes that could be built within any area populated by the same density of trees?

wull … assuming this is true then, is proving that two points on a line can be a sphere (you little devil) the same thing as not being able to see the forest for the number of two family homes that could be built?

Like

• Joseph Nebus 8:10 pm on Thursday, 12 March, 2015 Permalink | Reply

Well, now, indeed, I am putting forth that two points on a line can be, in the right context, interpreted as a sphere, or at least the way a sphere happens to exist in a one-dimensional space.

And, now, I agree with you almost all the way about describing the forest and the number of two-family homes it could be made into, if you have the density of trees and the number of board feet needed to build a home. But I think the setup falls a little short of what’s needed because it doesn’t make clear that the density of trees is really the density of usable board-feet in that area, and I’m not sure that we have a clear idea of how much area there is in the forest. If we take those as given, though, then yes: we’ve got missing forests for two-family homes.

Like

• Kurt Struble 11:04 pm on Thursday, 12 March, 2015 Permalink | Reply

well … first of all thanks for reading my earlier posts … the ones about rain, ice, the moon, universe, infinity, black holes, etc.

well … first of all my comment was attempt to draw a comic analogy between my interpretation of YOUR interpretation that two points on a straight line could be interpreted as a sphere, at least in one dimensional space.

the really humorous aspect of the statement for me being that, you could make the statement with a straight face knowing that most of the people who read your blog would accept and understand exactly what you meant by being impish (you little devil).

The mere thought that making this interpretation would be ‘’impish’’ is so far from my lexicon that, despite the fact that the people in your milieu probably completely understood your meaning … the statement took on a massive amount (like … the amount of energy it would take to completely fill up a black hole … which i realize is a really stupid thing to say but maybe funny?) of absurdist humor, to me!

I mean, i had never in my life heard … or indeed THOUGHT that i’d ever hear … such a statement made about such a subject. The absurdity of the statement … the sheer unexpected aspect of it … (please interpret ‘’absurdist’’ as an example of writing something good) caused me laugh last night, so hard that i woke my wife up who had been sleeping for at least an hour.

In fact, the statement was ALMOST as funny as a comment you made recently about pulling some really interesting artwork … the ‘medium’ being i believe, ice … ) out from beneath your swimming pool heater, which also sent me into paroxysms of laughter.

For me … humor … is the result of seeing or hearing something totally unexpected.

Both of your statements reached the highest level of humor since … how could i (a total and complete layman) EVER consider a person ‘impish’ because they could interpret two points on a straight line as a sphere but ONLY in one dimensional space? (hahahahahah

AND that you had found some interesting art work beneath your swimming pool heater?!!! (i’ve got a smile on my face as i write this … but hope i don’t start laughing because i want to finish this comment soon …

so anyway … as a result of my interpretation of your statement, about spheres while being impish, i decided to make up a situation that was absurd but, that i thought was fairly logical … except that, at the end i gave it an ‘absurd’ twist based on the statement … “you can’t see the forest for the trees” … which i thought was absurd enough to be pretty funny … wull … it made me laugh anyway …

i don’t think you got the joke … but that’s ok since … the fact that you proceeded to tell me the falsities of my ‘’suppositions’’ and why my conclusions were not necessarily valid … this caused even greater paroxysms of laughter.

as to your conclusion about the variation of the density of wood invalidating my conclusion … i would disagree with you completely since given a large enough sample, perhaps in a specific geographic area (which i KNOW i didn’t mention … ) the variations would even out so that the number of two bedroom homes COULD be determined.

so there … ! This is just another example of you NOT being able to see the forest for the trees … !! i enjoy your blog and will return … ks

Like

• Joseph Nebus 12:07 am on Monday, 16 March, 2015 Permalink | Reply

Aw, well, now, I did expect that you were being whimsical with writing about forests and trees, but I also didn’t want you to think I was just passing over your comment, and when I thought about it I could think of something interesting to say about it, and maybe useful in thinking about how to think about problems.

I like to think one of my good traits is being unafraid to look like I didn’t get the joke. It gives other people something amusing to respond to, if nothing else, and makes me look like a better sport than I worry I am.

Like

• Kurt Struble 1:16 pm on Friday, 13 March, 2015 Permalink | Reply

on jeeze … i didn’t see the forest for the trees!! ” … yes: we’ve got missing forest for two family homes.” ha ha ha ha … i guess if you assume that the ‘exercise’ applies to the real world (what is ‘real world’?) then there WOULD be missing forests … pardon my uppityness … .

Anyway, not being part of the mathematics/physics demographic, would you say that overall, this group is obsessed with definition? i wonder if i’d get along better with groups of “scientists” than people outside of this field … (i know there are other fields concerned with definitions … ) since i tend to zero in on words that people use, within statements, that i don’t think apply to the situation which eventually tends to piss people off if taken too far … .

i have learned to overlook the use of what i think could be a more specific word, but the word used, still bangs around inside my mind.

what happens is, my brain starts thinking of absurd usages for the word while i’m stifling myself from saying something.

this tendency to think of absurd uses for words is a great asset when it comes to writing. the result is, i love being a hermit (from time to time) when i want to do a lot of writing since i can focus in on word usage and not bug the shit out of people. i wonder if mathematicians think/feel the same way about numbers?

these kind of ridiculous”theories” that i pass on to people usually leave them scratching their heads wondering what the fuck i’m talking about. do mathematicians overall, tend to hang out in cliques of like minded people for the same reason?

i’m being serious and speaking the truth about such matters, from my own perspective … but i think it’s absurdly hilarious at the same time. please forgive me if it seems like i’m being patronizing because my intent is NOT to be patronizing.

my latest ‘obsession’ revolves around my thoughts on a much broader scope of POLARITY in the world around us.

my attempts to explain these thoughts to my wife the other day, resulted in her getting pissed off at me which lead me to try to use that circumstance as an example of how polarity can ratchet up in the real world which … made matters worse.

the last comment or question i would like to make is … is there any reason why the Great Salt Lake couldn’t be made into a ‘super dynamo’ since salt water is a necessary ingrediant for making electricity?

i figure that, since plus and minus are mathematic terms maybe you’d have an opinion.

or, you could table this discussion since i’m sure you have much more important issues to deal with. (another example of polarity)

in any case … don’t waste your time if you have limited amounts of it .,.. and thank you so much for reading my blog … ks

Like

• Joseph Nebus 12:17 am on Monday, 16 March, 2015 Permalink | Reply

I’ve been doing a fair bit of thinking about this because I think it’s an interesting question: are mathematicians (and people in related fields) obsessed with definitions? I think I’ve come to conclude that while mathematicians (and that demographic) tend to think more, and maybe more critically, about definitions than average folks do, it’s not exactly because they’re obsessed with definitions, but because they’re interested in what they can do with definitions.

The big interest that mathematicians have, I think, is in finding interesting things which are true to say about something. But what makes something interesting to say? It’s probably got to be something which is implied by the system you’ve set up, but which isn’t obvious from the original setup. But if it’s not obvious, then it’s got to be something that can be deduced from the setup, and it has to be something which can be judged against the rules of the setup and said to be either true or false.

And that’s where definitions come in: if you don’t have a fairly good idea of, say, what a Therblig Number is, you can’t really say whether “2,038” is a Therblig Number, or whether it isn’t. And if nobody knows what a Therblig Number is (certainly I don’t, and I made up the name), it’s not going to be interesting to say whether it is or isn’t. Your idea of what the definition is might not be precise, and it might need revision as you find it implies things you don’t want it to, but you have an idea there is this thing called a Therblig Number and that it has some traits you find interesting enough to label, and that’s why definitions — or, more generally, working out what the properties of a well-defined problem are — end up being of interest to mathematician types.

As for the Great Salt Lake, I’m afraid I don’t actually know that salt water is necessary for making electricity, or how using it for electricity would affect the lake’s other uses, so I don’t think I know enough about the problem to venture an opinion.

Like

• Kurt Struble 5:39 am on Monday, 16 March, 2015 Permalink | Reply

you bring up some really interesting points which i am going to LOVE to delve into with, having to do with art and science … (and i don’t mean art V.S. science) i don’t have time to delve at the moment though since it’s going on 1:00 a.m. here. but i WILL and i’m looking forward to it. i have reservations though, … you might say i am an over analytical person … so for every statement you make i’ll come up with a couple of conclusions which can branch off and soon i am lost in space … whoops .. i’m going off on a tangent right now … and i don’t want to … maybe one way to overcome this problem is to look at how i think … in terms of vectors … now …. i always think of vectors as straight lines … and i think if vectors are straight lines then they will intersect with other vectors … but, intersecting vectors are not ”tangents’ ‘ (which is the way i think … ) (jeeze, this gets complicated since i don’t know your lexicon so my use of ‘tangent’ and your usage is probably completely different than mine) … so while two vectors … after they intersect … even though they go off in two separate directions … this is not the same as a tangent … which ….. TO ME … a tangent would be a vector that suddenly splits … so that the vector goes off in two directions … … maybe this ‘species’ of vector has a definition i’m not aware of …. the point i was tying to make is … my mind … thinks in terms of split vectors … let’s call it a Therblig Vector, so discussions can get pretty tedious … i think the beauty of science is that you are looking for Truth … but by scientific definition, (i’m going out on a real long limb here … ) TRUTH is a single statement … E = mc2 …

there’s great beauty i n the language you use i order to find this ‘truth” because it’s so ‘precise’ … but based on the way i think … everything is ‘tangental” .. thoughts splitting and resplitting … to me, discoveries can be made by this constant re-splitting … re-splitting … going deeper and deeper into these splits …. so that there are many many discoveries made along the way as opposed to having an idea that there’s something then trying to find the language to confirm that the thing exists … vectors continually splitting gives an infinite places to go and i suppose an infinite amount of discoveries to make along the way … ………. i think there’s some truth to this since structurally, the brain is comprised of branching ‘vectors’ … as are … if you look .. at the structure of trees … their limbs … basically there are splits and splits and splits … reaching out collecting information … while at the same time the roots of the tree … as a reflection of the tree’s limbs .. are doing the same thing underground …

so i started out saying i had to go to bed and it’s now getting close to two o’clock and it’s all because i was being over analytical looking at the application of vectors and ”tangents” (by my definition) as they apply to how we think and whether it is best to journey through time seeking TRUTHS as opposed to seeking TRUTH … hey … please forgive me … i think i’m going off half cocked here … making all these statements … my final comment is …. that, maybe the definition of ART is … a random discovery made while searching for the right word or the right color … a search that’s almost random in nature waiting for something to ‘fall into place ” by following the right ‘split’ … or tangent …

it’s the idea that maybe this is the difference between pure art and pure science … DON’T GET ME WRONG … I’M NOT SAYING THAT WITHIN MATH. OR SCIENE IN GENERAL there isn’t creativity … i guess i’m writing more about the different languages and different approaches … even though i know there is plenty of overlap … i’m really sorry if i’ve confused you … i’m not going to proof read this … so i know it’s probably confusing but maybe there are a couple of grains of corn that we can harvest into a nicely organized corn cob that we both can look at and think is beautiful for the same reasons and for different reasons … jeeze … i had no idea where this whole thing was going … and didn’t even think i’d continue and … here i am … i hope you slept well … thanks … for even considering my words … ks

Like

• ivasallay 2:34 pm on Thursday, 12 March, 2015 Permalink | Reply

I liked that the equation on the blackboard EQUALED z, an often used variable. Dark Side of the Horse often seems to have good math comics, doesn’t it?
I can think of times when that Frank and Ernest strip would be quite good to show in a classroom.

Like

• Joseph Nebus 8:03 pm on Thursday, 12 March, 2015 Permalink | Reply

Yes, letting the variable be z is one of those little touches of craftsmanship that makes Dark Side of the Horse stand out in these mathematics roundups. I don’t know Samson’s biography. It’s easy to suppose she or he might have a mathematics-inclined background, although it’s just as easy to suppose she agrees with the notion that having the irrelevant details check out makes the overall joke stronger.

Like

When 2 plus 2 Equals 5, plus Another Unsettling Equation

I just wanted to note for folks who don’t read The Straight Dope — the first two books of which were unimaginably important to the teenage me, hundreds of pages of neat stuff to know delivered in a powerful style, that overwhelmed even The People’s Almanac 2 if you can imagine — that the Straight Dope Science Advisory board tried to take on the question of Does 2 + 2 equal 5 for very large values of 2?

Straight Dope Staffer Dex takes the question a bit more literally than I have ever interpreted the joke to be. I’ve basically read it as just justifying a nonsense result with a nonsense explanation, fitting in the spectrum of comic answers somewhere between King Lear’s understanding of why there are seven stars in the Pleiades and classic 1940s style double-talk. But Dex uses the equation to point out how rounding and estimation, essential steps in translating between the real world and the mathematical representation of the world, can produce results which are correct at every step but wrong in the whole, which is worth considering.

Also, in a bit of reading I’m doing and which I might rip off^W^W use as inspiration for some posts around here the (British) author dropped in an equation meant to be unsettling and, yeah, this unsettles me. Let me know what you think:

$3 \mbox{ feet } + 2 \mbox{ tons } = 36 \mbox{ inches } + 2440 \mbox{ pounds }$

I should say it’s not like I’m going to have nightmares about that, but it feels off anyway.

• abyssbrain 1:42 am on Sunday, 8 March, 2015 Permalink | Reply

Then there’s also the classic of Abbott and Castello “proving” that 13 x 7 = 28 :)

Like

• Joseph Nebus 11:37 pm on Monday, 9 March, 2015 Permalink | Reply

You know, I’m not familiar with that sketch offhand.

Like

• abyssbrain 12:24 am on Tuesday, 10 March, 2015 Permalink | Reply

Actually, it’s a part of a film.

Like

• Joseph Nebus 7:50 pm on Thursday, 12 March, 2015 Permalink | Reply

Oh, that’s a great routine, and I hadn’t seen it before. Thank you. (It’s surely from their TV show, though?)

Liked by 1 person

• abyssbrain 1:07 am on Friday, 13 March, 2015 Permalink | Reply

If I remembered correctly, that particular video was from one of their films, but they had also done this sketch once on their tv show.

Like

• Joseph Nebus 3:25 am on Saturday, 14 March, 2015 Permalink | Reply

That makes sense. It’d be uncharacteristic for them to use a good bit only the once, especially since it could be years between anyone in the audience seeing a movie, a radio program where they did it, and a TV show using the same bit.

Liked by 1 person

• Matthew Wright 1:55 am on Sunday, 8 March, 2015 Permalink | Reply

To me British Imperial measures like the long ton (which is only half a smoot longer than a short ton) pretty much sum up the problem the British also have making reliable cars. And landing things on Mars, when one half of the team is using Imperial and the other half metric.

Like

• Joseph Nebus 11:48 pm on Monday, 9 March, 2015 Permalink | Reply

As best I can tell the short ton is an invention of the Americans, so the British aren’t directly at fault for the long ton/short ton divide. Granted that 2240 pounds is a superficially weird number of pounds to put into any unit, but that is at least a nice convenient twenty hundredweights, which admittedly moves the problem back to why a hundredweight is a hundred pounds. In that case it’s because a hundredweight was a nice convenient eight stone, which had been twelve and a half pounds avoirdupois, until King Edward III yielded to the convenience of the wool trade and increase the stone to fourteen pounds (making a sack of cloth, 28 stone, more conveniently measured without cheating on available scales and also a nice (nearly) round 500 Florentian libbrae, and the rest followed from there.) Which is to admit that it’s daft, but every step made sense at the time, which is the best we can ever hope for.

Now, the Imperial/Metric problem with the space probe is interesting because while the difference in units is the proximate cause of the vehicle’s loss, it’s not the real cause. There were hints, from earlier maneuvers, that something was wrong in the way thrusts were being calculated or executed, but those weren’t followed up on. Had they been, a correction would’ve been straightforward. It’s a lesson in the importance of having good project management, and that project management has to include people signaling clearly when they suspect there’s problems and exploring adequately whether these suspicions are well-founded.

Like

• elkement 2:55 pm on Sunday, 8 March, 2015 Permalink | Reply

It was not until recently that I learned how ‘ton’ is used in engineering (related to air conditioning). I learned a lot of – maybe a ton of – new units when trying to respond to questions in the comments section on my blog :-)

Like

• Joseph Nebus 11:54 pm on Monday, 9 March, 2015 Permalink | Reply

I did not know there were custom uses of the ‘ton’ for engineering purposes until just now, and I’m fascinated to see how many different “big mass of the thing we’re measuring” get called tons, now. (Panama Canal Net Ton? Who ordered that?)

Liked by 1 person

• elkement 10:57 am on Tuesday, 10 March, 2015 Permalink | Reply

I hope I understood it correctly finally – but I was baffled about “ton” being used as a unit for (heating or cooling) *power*, rather than weight.

Like

• Joseph Nebus 7:55 pm on Thursday, 12 March, 2015 Permalink | Reply

Yeah, thinking of ton as a unit of power is weird, although I suppose it’s not inherently stranger than describing a distance by the amount of time it’d take to get there. It’s just less familiar.

Liked by 1 person

Looking At Things Four-Dimensionally

I’d like to close out the month by pointing to 4D Visualization, a web site set up by … well, I’m not sure the person, but the contact e-mail address is 4d ( at ) eusebeia.dyndns.org for whatever that’s worth. (Worse, I can not remember what site led me to it; if you’re out there, referent, please say so so I can thank you properly. In the meantime, thank you.) The author takes eleven chapters to discuss ways to visualize four-dimensional structures, and does quite a nice job at it. The ways we visualize three-dimensional structures are used heavily for analogies, and the illustrations — static and animated — build what feels like an intuitive bridge to me, at least.

Eusebeia (if I may use that as a name) goes through cross-sections, which are generally simple to render but which tax the imagination to put together1, and projections, and the subtleties in rendering two-dimensional images of three-dimensional projections of four-dimensional structures so that they’re sensible. It’s all quite good and I’m just sorry that my belief in the promise “More chapters coming soon!” clashes with the notice, “Last updated 13 Oct 2008”.

The main page is still being updated regularly, including a Polytope Of The Month feature. A polytope is what people call a polygon or polyhedron if they don’t want their discussion to carry the connotation of being about a two- or three-dimensional figure. It’s kind of the way someone in celestial mechanics talking about the orbit of an object around another might say periapsis and apoapsis, instead of perigee and apogee or perihelion and aphelion, although as far as I can tell people in celestial mechanics are only that precise if they suspect someone pedantic is watching them. I’m not well-versed enough to say how much polytope is used compared to polyhedron.

Anyway, for those looking for the chance to poke around higher dimensions, consider giving this a try; it’s a good read.

[1: I knew that a three-dimensional cube has, on the right slice, a hexagonal cross-section. It’s something I discovered while fiddling around with the problem of charged particles on a conductive-particule sphere, believe it or not. ]

• elkement 7:55 pm on Sunday, 1 June, 2014 Permalink | Reply

Fascinating – thanks!
Another WordPress fellow blogger has recently published a tutorial: How to create your own tesseract (3D glasses required): http://geneticfractals.wordpress.com/2014/03/04/fourth-dimension-for-skeptics/

Like

• Joseph Nebus 3:50 am on Wednesday, 4 June, 2014 Permalink | Reply

Ooh, thank you. That’s a really neat page. And I think I have the 3D glasses for it, somewhere.

Like

c
Compose new post
j
Next post/Next comment
k
Previous post/Previous comment
r