The Summer 2017 Mathematics A To Z: N-Sphere/N-Ball


Today’s glossary entry is a request from Elke Stangl, author of the Elkemental Force blog, which among other things has made me realize how much there is interesting to say about heat pumps. Well, you never know what’s interesting before you give it serious thought.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

N-Sphere/N-Ball.

I’ll start with space. Mathematics uses a lot of spaces. They’re inspired by geometry, by the thing that fills up our room. Sometimes we make them different by simplifying them, by thinking of the surface of a table, or what geometry looks like along a thread. Sometimes we make them bigger, imagining a space with more directions than we have. Sometimes we make them very abstract. We realize that we can think of polynomials, or functions, or shapes as if they were points in space. We can describe things that work like distance and direction and angle that work for these more abstract things.

What are useful things we know about space? Many things. Whole books full of things. Let me pick one of them. Start with a point. Suppose we have a sense of distance, of how far one thing is from one another. Then we can have an idea of the neighborhood. We can talk about some chunk of space that’s near our starting point.

So let’s agree on a space, and on some point in that space. You give me a distance. I give back to you — well, two obvious choices. One of them is all the points in that space that are exactly that distance from our agreed-on point. We know what this is, at least in the two kinds of space we grow up comfortable with. In three-dimensional space, this is a sphere. A shell, at least, centered around whatever that first point was. In two-dimensional space, on our desktop, it’s a circle. We know it can look a little weird: if we started out in a one-dimensional space, there’d be only two points, one on either side of the original center point. But it won’t look too weird. Imagine a four-dimensional space. Then we can speak of a hypersphere. And we can imagine that as being somehow a ball that’s extremely spherical. Maybe it pokes out of the rendering we try making of it, like a cartoon character falling out of the movie screen. We can imagine a five-dimensional space, or a ten-dimensional one, or something with even more dimensions. And we can conclude there’s a sphere for even that much space. Well, let it.

What are spheres good for? Well, they’re nice familiar shapes. Even if they’re in a weird number of dimensions. They’re useful, too. A lot of what we do in calculus, and in analysis, is about dealing with difficult points. Points where a function is discontinuous. Points where the function doesn’t have a value. One of calculus’s reliable tricks, though, is that we can swap information about the edge of things for information about the interior. We can replace a point with a sphere and find our work is easier.

The other thing I could give you. It’s a ball. That’s all the points that aren’t more than your distance away from our point. It’s the inside, the whole planet rather than just the surface of the Earth.

And here’s an ambiguity. Is the surface a part of the ball? Should we include the edge, or do we just want the inside? And that depends on what we want to do. Either might be right. If we don’t need the edge, then we have an open set (stick around for Friday). This gives us the open ball. If we do need the edge, then we have a closed set, and so, the closed ball.

Balls are so useful. Take a chunk of space that you find interesting for whatever reason. We can represent that space as the joining together (the “union”) of a bunch of balls. Probably not all the same size, but that’s all right. We might need infinitely many of these balls to get the chunk precisely right, or as close to right as can be. But that’s all right. We can still do it. Most anything we want to analyze is easier to prove on any one of these balls. And since we can describe the complicated shape as this combination of balls, then we can know things about the whole complicated shape. It’s much the way we can know things about polygons by breaking them into triangles, and showing things are true about triangles.

Sphere or ball, whatever you like. We can describe how many dimensions of space the thing occupies with the prefix. The 3-ball is everything close enough to a point that’s in a three-dimensional space. The 2-ball is everything close enough in a two-dimensional space. The 10-ball is everything close enough to a point in a ten-dimensional space. The 3-sphere is … oh, all right. Here we have a little squabble. People doing geometry prefer this to be the sphere in three dimensions. People doing topology prefer this to be the sphere whose surface has three dimensions, that is, the sphere in four dimensions. Usually which you mean will be clear from context: are you reading a geometry or a topology paper? If you’re not sure, oh, look for anything hinting at the number of spatial dimensions. If nothing gives you a hint maybe it doesn’t matter.

Either way, we do want to talk about the family of shapes without committing ourselves to any particular number of dimensions. And so that’s why we fall back on ‘N’. ‘N’ is a good name for “the number of dimensions we’re working in”, and so we use it. Then we have the N-sphere and the N-ball, a sphere-like shape, or a ball-like shape, that’s in however much space we need for the problem.

I mentioned something early on that I bet you paid no attention to. That was that we need a space, and a point inside the space, and some idea of distance. One of the surprising things mathematics teaches us about distance is … there’s a lot of ideas of distance out there. We have what I’ll call an instinctive idea of distance. It’s the one that matches what holding a ruler up to stuff tells us. But we don’t have to have that.

I sense the grumbling already. Yes, sure, we can define distance by some screwball idea, but do we ever need it? To which the mathematician answers, well, what if you’re trying to figure out how far away something in midtown Manhattan is? Where you can only walk along streets or avenues and we pretend Broadway doesn’t exist? Huh? How about that? Oh, fine, the skeptic might answer. Grant that there can be weird cases where the straight-line ruler distance is less enlightening than some other scheme is.

Well, there are. There exists a whole universe of different ideas of distance. There’s a handful of useful ones. The ordinary straight-line ruler one, the Euclidean distance, you get in a method so familiar it’s worth saying what you do. You find the coordinates of your two given points. Take the pairs of corresponding coordinates: the x-coordinates of the two points, the y-coordinates of the two points, the z-coordinates, and so on. Find the differences between corresponding coordinates. Take the absolute value of those differences. Square all those absolute-value differences. Add up all those squares. Take the square root of that. Fine enough.

There’s a lot of novelty acts. For example, do that same thing, only instead of raising the differences to the second power, raise them to the 26th power. When you get the sum, instead of the square root, take the 26th root. There. That’s a legitimate distance. No, you will never need this, but your analysis professor might give you it as a homework problem sometime.

Some are useful, though. Raising to the first power, and then eventually taking the first root, gives us something useful. Yes, raising to a first power and taking a first root isn’t doing anything. We just say we’re doing that for the sake of consistency. Raising to an infinitely large power, and then taking an infinitely great root, inspires angry glares. But we can make that idea rigorous. When we do it gives us something useful.

And here’s a new, amazing thing. We can still make “spheres” for these other distances. On a two-dimensional space, the “sphere” with this first-power-based distance will look like a diamond. The “sphere” with this infinite-power-based distance will look like a square. On a three-dimensional space the “sphere” with the first-power-based distance looks like a … well, more complicated, three-dimensional diamond. The “sphere” with the infinite-power-based distance looks like a box. The “balls” in all these cases look like what you expect from knowing the spheres.

As with the ordinary ideas of spheres and balls these shapes let us understand space. Spheres offer a natural path to understanding difficult points. Balls offer a natural path to understanding complicated shapes. The different ideas of distance change how we represent these, and how complicated they are, but not the fact that we can do it. And it allows us to start thinking of what spheres and balls for more abstract spaces, universes made of polynomials or formed of trig functions, might be. They’re difficult to visualize. But we have the grammar that lets us speak about them now.

And for a postscript: I also wrote about spheres and balls as part of my Set Tour a couple years ago. Here’s the essay about the N-sphere, although I didn’t exactly call it that. And here’s the essay about the N-ball, again not quite called that.

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Reading the Comics, March 30, 2016: Official At-Bat Edition


Comic Strip Master Command slowed down the pace at which the newspaper comics were to talk mathematical subjects. All right, that’s their prerogative. But it leaves me here, at Thursday, with slightly too few comics for my tastes. On the other hand, if I don’t run with what I have, I might not have anything to post for the 31st of March, and it would be a shame to go this whole month with something posted every day only to spoil it on the 31st. This is a pretty juvenile reason to do a thing, so here we are. Enjoy, please.

Tom Thaves’s Frank and Ernest for the 25th of March is a students-grumbling joke. I’m not sure what to make of the argument “arithmetic might be education, but that algebra stuff is indoctrination”. I imagine it reflects the feeling that the rules of arithmetic are all these nice straightforward things, and then algebra’s rules seem a bewildering set of near-gibberish. I can understand people looking at the quadratic formula, being told it has something to do with parabolas and an axis, throwing up their hands, and declaring it all this crazy game they’ll never play.

What people are forgetting in this is that everything sounds like this crazy gibberish game at first. The confusion you felt when first trying to factor a quadratic polynomial? It’s the same confusion you felt when first doing long division. And when you first multiplied a three-digit by a two-digit number. And when you had to subtract with borrowing. It’s also the same confusion you have when you first hear the first European settlement of Manhattan was driven by the Netherlands’ war for independence from Spain. Learning is changing the baffling confusion of life into an understandable pattern.

Which is not to deny that we could do a better job motivating stuff. You have no idea how many drafts of the Dedekind Domain essay I threw out because there were just too many words describing conditions and not why any of them mattered. I’m lazy; I don’t like scrapping that much text. And I’m still not quite happy with Normal Groups.

Jeff Mallet’s Frazz for the 27th is an easier joke to explain. It’s also one whose appeal I really understand. There is a compelling beauty to the notation and the symbols of higher mathematics. I remember when a kid I peered at one of my parents’ calculus textbooks. The reference page of common integrals was enchanting. It wasn’t the only thing that drove me towards mathematics. But the aesthetic beauty is there.

And it’s not just mathematicians and mathematics-based fields that see it. The arts editor for my undergraduate school’s unread leftist weekly newspaper asked me to work out a problem, any problem, to include as graphic arts. I was happy to. (I was the managing editor for the paper at the time.) I even had a great problem, from the final exam in my freshman Classical Mechanics course. The problem was to derive the equivalent of Kepler’s Laws of Motion with a different force law. Instead of the inverse-square attraction of gravity we used the exponential-decay-style interactions of the weak force. It was a brilliant exam question, frankly, and made for a page of symbols that maybe nobody understood but that I’ll bet everyone thought pretty.

John Forgetta and L A Rose’s The Meaning of Lila for the 27th is probably a rerun. The strip mostly is, although a few new or updated comics are fit into the rotation. It’s an example of a census joke, in which you classify away the whole population of the world. I remember first seeing it, as a kid, in a church bulletin. That one worked out how the entire working population of the United States was actually only two people and that’s why you’re always so tired. You could probably use the logic of this sort of joke to teach Venn diagrams. The logic that produces a funny low count relies on counting people several times, once for each of many categories they might fit in.

Mark Anderson’s Andertoons for the 30th made me giggle. I suppose there’s an essay to be written about whether we need mathematics, and what we need it for. But wouldn’t that just take away from the fun of it?

The Set Tour, Stage 1: Intervals


I keep writing about functions. I’m not exactly sure how. I keep meaning to get to other things but find interesting stuff to say about domains and ranges and the like. These domains and ranges have to be sets. There are some sets that come up all the time in domains and ranges. I thought I’d share some of the common ones. The first family of sets is known as “intervals”.

[ 0, 1 ]

This means all the real numbers from 0 to 1. Written with straight brackets like that means to include the matching point there — that is, 0 is in the domain, and so is 1. We don’t always want to include the ending points in a domain; if we want to omit them, we write parentheses instead. So (0, 1) would mean all the real numbers bigger than zero and smaller than one, but neither zero nor one. We can also include one but not both endpoints: [0, 1) is fine. It offends copy editors, by having its open bracket and its closed parenthesis be unmatched, but its meaning is clear enough. It’s the real numbers from zero to one, with zero allowed but one ruled out. We can include or omit either or both endpoints, and we have to keep that straight. But for most of our work it doesn’t matter what we choose, as long as we stay consistent. It changes proofs a bit, but in routine ways.

Zero to one is a popular interval. Negative 1 to 1 is another popular interval. They’re nice round numbers. And the intervals between -π and π, or between 0 and 2π, are also popular. Those match nicely with trigonometric functions such as sine and tangent. You can make an interval that runs from any number to any other number — [ a, b ] if you don’t want to pin down just what numbers you mean. But [ 0, 1 ] and [ -1, 1 ] are popular choices among mathematicians. If you can prove something interesting about a function with a domain that’s either of these intervals, you can then normally prove it’s true on whatever a function with a domain that’s whatever interval you want. (I can’t think of an exception offhand, but mathematics is vast and my statement sweeping. There may be trouble.)

Suppose we start out with a function named f that has its domain the interval [ -16, 48 ]. I’m not saying anything about its range or its rule because they don’t matter. You can make them anything you like, if you need them to be anything. (This is exactly the way we might, in high school algebra, talk about the number named `x’ without ever caring whether we find out what number that actually is.) But from this start we can talk about a related function named g, which has as its domain [ -1, 1 ]. A rule for g is g: x \mapsto f\left(32\cdot x + 16\right) ; that is, g(0) is whatever number you get from f(16), for example. So if we can prove something’s true about g on this standard domain, we can almost certainly show the same thing is true about f on the other domain. This is considered a kind of mapping, or as a composition of functions. It’s a composition because you can see it as taking your original number called x, seeing what one function does with it — in this case, multiplying it by 32 and adding 16 — and then seeing what a second function — the one named f — does with that.

It’s easy to go the other way around. If we know what g is on the domain [ -1, 1 ] then we can define a related function f on the domain [ -16, 48 ]. We can say f: x \mapsto g\left(\frac{1}{32}\cdot\left(x - 16\right)\right). And again, if we can show something’s true on this standard domain, we can almost certainly show the same thing’s true on the other domain.

I gave examples, mappings, that are simple. They’re linear. They don’t have to be. We just have to match everything in one interval to everything in another. For example, we can match the domain (1, ∞) — all the numbers bigger than 1 — to the domain (0, 1). Let’s again call f the function with domain (1, ∞). Then we can say g is the function with domain (0, 1) and defined by the rule g: x \mapsto f\left(\frac{1}{x}\right) . That’s a nonlinear mapping.

Linear mappings are easier to deal with than nonlinear mappings. Usually, mathematically, if something is divided into “linear” and “nonlinear” the linear version is easier. Sometimes a nonlinear mapping is the best one to use to match a function on some convenient domain to a function on some other one. The hard part is often a matter of showing that something true for one function on a common domain like (0, 1) will also be true for the other domain, (a, b).

However, showing that the truth holds can often be done without knowing much about your specific function. You maybe need to know what kind of function it is, that it’s continuous or bounded or something like that. But the actual specific rule? Not so important. You can prove that the truth holds ahead of time. Or find an analysis textbook or paper or something where someone else has proven that. So while a domain might be any interval, often in practice you don’t need to work with more than a couple nice familiar ones.

One Way We Write Functions


During the Summer A To Z I talked a bit about functions. Mathematically we see these as a collection of three things: a set of things which we call the domain, a set of things which we call the range, and a rule that matches things in the domain to something in the range. The domain and the range can be the same set, or they can be different ones. The definition is quite flexible. What I want to talk about here is how to write them down.

We can describe each of these sets in words, and often will when speaking or when describing a line of argument. But when we want to work, we start using shorthand names, often single letters. For the sets of the domain and range these are usually capital letters. I haven’t noticed much of a preference for which letters to use. D for domain and R for range have a hard-to-resist logic if we don’t really care what the sets are.

There are some sets that are used as domains or ranges a lot, and those have common shorthands. The set of real numbers is often written as R — bold, in print, or written with a double vertical stroke on the R if you’re doing this by hand. The set of whole numbers, integers, gets written as I (for integer) or J (again for integer; the letters I and J weren’t perceived as truly different things until recently) or Z (for Zahlen, German for “counting number”). There are a lot of others and don’t worry about them.

The rules for a function are generally described by a lowercase letter. It’s most commonly f, with g and h pressed into service if f won’t do. Subscrips are common also: f1, f2, fj, fn, and so on. Again, any name is allowed, as long as you’re consistent about it. But f and g and h are used as “names of functions” so often that it’s what the reader will expect they mean even without being told.

One common shorthand for saying that a function named “f” has the domain “D” and the range “R” is to use an arrow. Write out “f: D –> R”. The function name comes first, before the colon; then the domain, and an arrow, and the range. There are other notations but this is the one I see most often. This is often read aloud as “f maps D into R”. The activity of the verb “map” — well, it’s kind of action-y — suggests motion to my mind. Functions are commonly used to describe how a system changes over time. This seems mnemonic to me, as arrows suggest flow and motion. We often use the language of flowing things even for problems that don’t have anything to do with moving objects or any sense of time.

There’s another part of function-defining that has to be done, though. Most often we’re interested in domains and ranges that are both numbers, or at least collections of numbers. And we want to describe matching something in the domain with something in the range based on a formula. If “x” is a number in the domain then, say, “x2 – 4x + 4” is the corresponding number in the range.

One way to write down this rule is the way we get in introductory algebra class, and to write something like “f(x) = x2 – 4x + 4”. The “x” is, here, a dummy variable. We will never care about pinning it down to any particular number. If we write “f(3)” we mean to evaluate whatever’s on the right hand of the equals sign, using 3, the thing in parentheses, wherever “x” appears in the rule definition. In this case that would be the number 32 – 4*3 + 4 which it happens is 1. If we write “f(1 – t)” we would evaluate “(1 – t)2 – 4(1 – t) + 4” which we might want to leave as is, or might want to simplify somehow. It depends what we’re up to.

But we can also use an arrow notation, and write the same rule as “f: x –> x2 – 4x + 4”. My feeling is this notation makes it clearer that the definition isn’t itself something to solve, and that the definition doesn’t care what value x is. It should suggest how we can substitute anything for x and should do so throughout the expression to the right of the arrow.

Wikipedia asserts that when writing the rule this way there should be a vertical stroke on the left side of the arrow. This is probably a good rule, since “f: D –> R” and “f: x –> x2 – 4x + 4” are talking about different things. I’m not sure the rule is consistently followed, though. I suspect that in most contexts it’s clear what is meant.

Reading the Comics, August 29, 2015: Unthemed Edition


I can’t think of any particular thematic link through the past week’s mathematical comic strips. This happens sometimes. I’ll make do. They’re all Gocomics.com strips this time around, too, so I haven’t included the strips. The URLs ought to be reasonably stable.

J C Duffy’s Lug Nuts (August 23) is a cute illustration of the first, second, third, and fourth dimensions. The wall-of-text might be a bit off-putting, especially the last panel. It’s worth the reading. Indeed, you almost don’t need the cartoon if you read the text.

Tom Toles’s Randolph Itch, 2 am (August 24) is an explanation of pie charts. This might be the best stilly joke of the week. I may just be an easy touch for a pie-in-the-face.

Charlie Podrebarac’s Cow Town (August 26) is about the first day of mathematics camp. It’s also every graduate students’ thesis defense anxiety dream. The zero with a slash through it popping out of Jim Smith’s mouth is known as the null sign. That comes to us from set theory, where it describes “a set that has no elements”. Null sets have many interesting properties considering they haven’t got any things. And that’s important for set theory. The symbol was introduced to mathematics in 1939 by Nicholas Bourbaki, the renowned mathematician who never existed. He was important to the course of 20th century mathematics.

Eric the Circle (August 26), this one by ‘Arys’, is a Venn diagram joke. It makes me realize the Eric the Circle project does less with Venn diagrams than I expected.

John Graziano’s Ripley’s Believe It Or Not (August 26) talks of a Akira Haraguchi. If we believe this, then, in 2006 he recited 111,700 digits of pi from memory. It’s an impressive stunt and one that makes me wonder who did the checking that he got them all right. The fact-checkers never get their names in Graziano’s Ripley’s.

Mark Parisi’s Off The Mark (August 27, rerun from 1987) mentions Monty Hall. This is worth mentioning in these parts mostly as a matter of courtesy. The Monty Hall Problem is a fine and imagination-catching probability question. It represents a scenario that never happened on the game show Let’s Make A Deal, though.

Jeff Stahler’s Moderately Confused (August 28) is a word problem joke. I do wonder if the presence of battery percentage indicators on electronic devices has helped people get a better feeling for percentages. I suppose only vaguely. The devices can be too strangely nonlinear to relate percentages of charge to anything like device lifespan. I’m thinking here of my cell phone, which will sit in my messenger bag for three weeks dropping slowly from 100% to 50%, and then die for want of electrons after thirty minutes of talking with my father. I imagine you have similar experiences, not necessarily with my father.

Thom Bluemel’s Birdbrains (August 29) is a caveman-mathematics joke. This one’s based on calendars, which have always been mathematical puzzles.

A Summer 2015 Mathematics A To Z: unbounded


Unbounded.

Something is unbounded if it is not bounded. To summon a joke from my college newspaper days, all things considered, this wasn’t too tough a case for Inspector Bazalo.

Admittedly that doesn’t tell us much until we know what “bounded” means. But that means nearly what you might expect from common everyday English. A set of numbers is bounded if you can identify a value that the set never gets larger than, or smaller than. Specifically it’s bounded above if there’s some number that nothing in the set is bigger than. It’s bounded below if there’s some number that nothing in the set is smaller than. If someone just says bounded, they might mean that the set is bounded above and below simultaneously. Or she might mean there’s just an upper or a lower bound. The context should make it clear. If she says something is unbounded, she means that it’s not bounded below, or it’s not bounded above, or it’s not bounded on both sides.

We speak of a function being unbounded if its smallest possible range is unbounded. For example, think of a function with domain of all the real numbers. Give it the rule “match every number in the domain with its square”. In high school algebra you’d write this “f(x) = x2”. Then the range has to be the real numbers from 0 up to … well, just keep going up. It’s unbounded above, although it is bounded below. 0 or any negative number is a valid lower bound.

That’s a fairly obvious example, though. Functions can be more intricate and still be unbounded. For example, consider a function whose domain is all the counting numbers — 1, 2, 3, and so on. (This domain is an unbounded set.) Let the rule be that you match every number in the domain with one divided by its sine. That is, “f(x) = 1 / sin(x)”. There’s no highest, or lowest, number in this set. Pick any possible bound and you can find at least one x for which f(x) is bigger, or smaller.

Regions of space can be bounded or unbounded, too. A region of space is what it sounds like, some blotch on the map. The blotch doesn’t have to be contiguous. If it’s possible to draw a circle that the whole region fits within, then the region is bounded. If it’s impossible to do this, then the region is unbounded. I write blotches on maps and circles as if I’m necessarily talking about two-dimensional spaces. That’s a good way to get a feeling for bounded and unbounded regions. It appeals to our sense of drawing stuff out on paper and of looking at maps. But there’s no reason it has to be two-dimensional. The same ideas apply for one-dimensional spaces and three-dimensional ones. They also apply for higher dimensions. Just change “circles” to “spheres” or “hyperspheres” and the idea carries over.

You might remember the talk about measure, and how it gives an idea of how big a set is. And in that case you might expect an unbounded region has to have an infinitely large measure. After all, imagine a rectangle that’s one unit wide, starts at the left side of your paper, and goes off forever to the right. That’s obviously got infinitely large area. But it’s not so. You can have regions that are unbounded, but have finite — even zero — measure.

It’s often possible to swap a bounded set (function, region) for an unbounded one, or vice-versa. For example, if your set was the range of “1 / sin(x)”, you might match that up with “sin(x)”, its reciprocal. That’s obviously bounded. It’s less obvious how you might make a bounded set out of the range of “x2”. One way would be to match it with the function whose rule is “1 / (x2 + 1)”, which is bounded, above and below. As with duals, this is a way we can turn one problem into another, that we might be able to solve more easily.