## A Leap Day 2016 Mathematics A To Z: Riemann Sphere

To my surprise nobody requested any terms beginning with `R’ for this A To Z. So I take this free day to pick on a concept I’d imagine nobody saw coming.

## Riemann Sphere.

We need to start with the complex plane. This is just, well, a plane. All the points on the plane correspond to a complex-valued number. That’s a real number plus a real number times i. And i is one of those numbers which, squared, equals -1. It’s like the real number line, only in two directions at once.

Take that plane. Now put a sphere on it. The sphere has radius one-half. And it sits on top of the plane. Its lowest point, the south pole, sits on the origin. That’s whatever point corresponds to the number 0 + 0i, or as humans know it, “zero”.

We’re going to do something amazing with this. We’re going to make a projection, something that maps every point on the sphere to every point on the plane, and vice-versa. In other words, we can match every complex-valued number to one point on the sphere. And every point on the sphere to one complex-valued number. Here’s how.

Imagine sitting at the north pole. And imagine that you can see through the sphere. Pick any point on the plane. Look directly at it. Shine a laser beam, if that helps you pick the point out. The laser beam is going to go into the sphere — you’re squatting down to better look through the sphere — and come out somewhere on the sphere, before going on to the point in the plane. The point where the laser beam emerges? That’s the mapping of the point on the plane to the sphere.

There’s one point with an obvious match. The south pole is going to match zero. They touch, after all. Other points … it’s less obvious. But some are easy enough to work out. The equator of the sphere, for instance, is going to match all the points a distance of 1 from the origin. So it’ll have the point matching the number 1 on it. It’ll also have the point matching the number -1, and the point matching i, and the point matching -i. And some other numbers.

All the numbers that are less than 1 from the origin, in fact, will have matches somewhere in the southern hemisphere. If you don’t see why that is, draw some sketches and think about it. You’ll convince yourself. If you write down what convinced you and sprinkle the word “continuity” in here and there, you’ll convince a mathematician. (WARNING! Don’t actually try getting through your Intro to Complex Analysis class doing this. But this is what you’ll be doing.)

What about the numbers more than 1 from the origin? … Well, they all match to points on the northern hemisphere. And tell me that doesn’t stagger you. It’s one thing to match the southern hemisphere to all the points in a circle of radius 1 away from the origin. But we can match everything outside that little circle to the northern hemisphere. And it all fits in!

Not amazed enough? How about this: draw a circle on the plane. Then look at the points on the Riemann sphere that match it. That set of points? It’s also a circle. A line on the plane? That’s also a line on the sphere. (Well, it’s a geodesic. It’s the thing that looks like a line, on spheres.)

How about this? Take a pair of intersecting lines or circles in the plane. Look at what they map to. That mapping, squashed as it might be to the northern hemisphere of the sphere? The projection of the lines or circles will intersect at the same angles as the original. As much as space gets stretched out (near the south pole) or squashed down (near the north pole), angles stay intact.

OK, but besides being stunning, what good is all this?

Well, one is that it’s a good thing to learn on. Geometry gets interested in things that look, at least in places, like planes, but aren’t necessarily. These spheres are, and the way a sphere matches a plane is obvious. We can learn the tools for geometry on the Möbius strip or the Klein bottle or other exotic creations by the tools we prove out on this.

And then physics comes in, being all weird. Much of quantum mechanics makes sense if you imagine it as things on the sphere. (I admit I don’t know exactly how. I went to grad school in mathematics, not in physics, and I didn’t get to the physics side of mathematics much at that time.) The strange ways distance can get mushed up or stretched out have echoes in relativity. They’ll continue having these echoes in other efforts to explain physics as geometry, the way that string theory will.

Also important is that the sphere has a top, the north pole. That point matches … well, what? It’s got to be something infinitely far away from the origin. And this make sense. We can use this projection to make a logically coherent, sensible description of things “approaching infinity”, the way we want to when we first learn about infinitely big things. Wrapping all the complex-valued numbers to this ball makes the vast manageable.

It’s also good numerical practice. Computer simulations have problems with infinitely large things, for the obvious reason. We have a couple of tools to handle this. One is to model a really big but not infinitely large space and hope we aren’t breaking anything. One is to create a “tiling”, making the space we are able to simulate repeat itself in a perfect grid forever and ever. But recasting the problem from the infinitely large plane onto the sphere can also work. This requires some ingenuity, to be sure we do the recasting correctly, but that’s all right. If we need to run a simulation over all of space, we can often get away with doing a simulation on a sphere. And isn’t that also grand?

The Riemann named here is Bernhard Riemann, yet another of those absurdly prolific 19th century mathematicians, especially considering how young he was when he died. His name is all over the fundamentals of analysis and geometry. When you take Introduction to Calculus you get introduced pretty quickly to the Riemann Sum, which is how we first learn how to calculate integrals. It’s that guy. General relativity, and much of modern physics, is based on advanced geometries that again fall back on principles Riemann noticed or set out or described so well that we still think of them as he discovered.

• #### elkement (Elke Stangl) 7:16 am on Sunday, 10 April, 2016 Permalink | Reply

Of course I jump onto the ‘quantum physics on a sphere’ stuff. Again, I can’t resist posting a link but I am pretty sure you will like it: Scott Aaronson explains what quantum physics actually is – from a math / computer science perspective, why complex numbers are used etc. – in the most intriguing and concisest way I have ever seen:

http://www.scottaaronson.com/democritus/lec9.html

Quote: “Why did God go with the complex numbers and not the real numbers?
Years ago, at Berkeley, I was hanging out with some math grad students — I fell in with the wrong crowd — and I asked them that exact question. The mathematicians just snickered. “Give us a break — the complex numbers are algebraically closed!” To them it wasn’t a mystery at all.”

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• #### Joseph Nebus 3:08 am on Friday, 15 April, 2016 Permalink | Reply

That is a fantastic review and thank you for it. I hadn’t seen it before. I like the review of it.

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## y-axis.

It’s easy to tell where you are on a line. At least it is if you have a couple tools. One is a reference point. Another is the ability to say how far away things are. Then if you say something is a specific distance from the reference point you can pin down its location to one of at most two points. If we add to the distance some idea of direction we can pin that down to at most one point. Real numbers give us a good sense of distance. Positive and negative numbers fit the idea of orientation pretty well.

To tell where you are on a plane, though, that gets tricky. A reference point and a sense of how far things are help. Knowing something is a set distance from the reference point tells you something about its position. But there’s still an infinite number of possible places the thing could be, unless it’s at the reference point.

The classic way to solve this is to divide space into a couple directions. René Descartes made his name for himself — well, with many things. But one of them, in mathematics, was to describe the positions of things by components. One component describes how far something is in one direction from the reference point. The next component describes how far the thing is in another direction.

This sort of scheme we see as laying down axes. One, conventionally taken to be the horizontal or left-right axis, we call the x-axis. The other direction — one perpendicular, or orthogonal, to the x-axis — we call the y-axis. Usually this gets drawn as the vertical axis, the one running up and down the sheet of paper. That’s not required; it’s just convention.

We surely call it the x-axis in echo of the use of x as the name for a number whose value we don’t know right away. (That, too, is a convention Descartes gave us.) x carries with it connotations of the unknown, the sought-after, the mysterious thing to be understood. The next axis we name y because … well, that’s a letter near x and we don’t much need it for anything else, I suppose. If we need another direction yet, if we want something in space rather than a plane, then the third axis we dub the z-axis. It’s perpendicular to the x- and the y-axis directions.

These aren’t the only names for these directions, though. It’s common and often convenient to describe positions of things using vector notation. A vector describes the relative distance and orientation of things. It’s compact symbolically. It lets one think of the position of things as a single variable, a single concept. Then we can talk about a position being a certain distance in the direction of the x-axis plus a certain distance in the direction of the y-axis. And, if need be, plus some distance in the direction of the z-axis.

The direction of the x-axis is often written as $\hat{i}$, and the direction of the y-axis as $\hat{j}$. The direction of the z-axis if needed gets written $\hat{k}$. The circumflex there indicates two things. First is that the thing underneath it is a vector. Second is that it’s a vector one unit long. A vector might have any length, including zero. It’s convenient to make some mention when it’s a nice one unit long.

Another popular notation is to write the direction of the x-axis as the vector $\hat{e}_1$, and the y-axis as the vector $\hat{e}_2$, and so on. This method offers several advantages. One is that we can talk about the vector $\hat{e}_j$, that is, some particular direction without pinning down just which one. That’s the equivalent of writing “x” or “y” for a number we don’t want to commit ourselves to just yet. Another is that we can talk about axes going off in two, or three, or four, or more directions without having to pin down how many there are. And then we don’t have to think of what to call them. x- and y- and z-axes make sense. w-axis sounds a little odd but some might accept it. v-axis? u-axis? Nobody wants that, trust me.

Sometimes people start the numbering from $\hat{e}_0$ so that the y-axis is the direction $\hat{e}_1$. Usually it’s either clear from context or else it doesn’t matter.

## N-tuple.

We use numbers to represent things we want to think about. Sometimes the numbers represent real-world things: the area of our backyard, the number of pets we have, the time until we have to go back to work. Sometimes the numbers mean something more abstract: an index of all the stuff we’re tracking, or how its importance compares to other things we worry about.

Often we’ll want to group together several numbers. Each of these numbers may measure a different kind of thing, but we want to keep straight what kind of thing it is. For example, we might want to keep track of how many people are in each house on the block. The houses have an obvious index number — the street number — and the number of people in each house is just what it says. So instead of just keeping track of, say, “32” and “34” and “36”, and “3” and “2” and “3”, we would keep track of pairs: “32, 3”, and “34, 2”, and “36, 3”. These are called ordered pairs.

They’re not called ordered because the numbers are in order. They’re called ordered because the order in which the numbers are recorded contains information about what the numbers mean. In this case, the first number is the street address, and the second number is the count of people in the house, and woe to our data set if we get that mixed up.

And there’s no reason the ordering has to stop at pairs of numbers. You can have ordered triplets of numbers — (32, 3, 2), say, giving the house number, the number of people in the house, and the number of bathrooms. Or you can have ordered quadruplets — (32, 3, 2, 6), say, house number, number of people, bathroom count, room count. And so on.

An n-tuple is an ordered set of some collection of numbers. How many? We don’t care, or we don’t care to say right now. There are two popular ways to pronounce it. One is to say it the way you say “multiple” only with the first syllable changed to “enn”. Others say it about the same, but with a long u vowel, so, “enn-too-pull”. I believe everyone worries that everyone else says it the other way and that they sound like they’re the weird ones.

You might care to specify what your n is for your n-tuple. In that case you can plug in a value for that n right in the symbol: a 3-tuple is an ordered triplet. A 4-tuple is that ordered quadruplet. A 26-tuple seems like rather a lot but I’ll trust that you know what you’re trying to study. A 1-tuple is just a number. We might use that if we’re trying to make our notation consistent with something else in the discussion.

If you’re familiar with vectors you might ask: so, an n-tuple is just a vector? It’s not quite. A vector is an n-tuple, but in the same way a square is a rectangle. It has to meet some extra requirements. To be a vector we have to be able to add corresponding numbers together and get something meaningful out of it. The ordered pair (32, 3) representing “32 blocks north and 3 blocks east” can be a vector. (32, 3) plus (34, 2) can give us us (66, 5). This makes sense because we can say, “32 blocks north, 3 blocks east, 34 more blocks north, 2 more blocks east gives us 66 blocks north, 5 blocks east.” At least it makes sense if we don’t run out of city. But to add together (32, 3) plus (34, 2) meaning “house number 32 with 3 people plus house number 34 with 2 people gives us house number 66 with 5 people”? That’s not good, whatever town you’re in.

I think the commonest use of n-tuples is to talk about vectors, though. Vectors are such useful things.

• #### howardat58 3:29 pm on Wednesday, 24 June, 2015 Permalink | Reply

Now I’m waiting for V

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• #### Joseph Nebus 3:38 am on Thursday, 25 June, 2015 Permalink | Reply

I haven’t written it yet. I’m open to suggestions or to bribes, if there’s enough money in the Vandermonde Identity community.

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## Measure.

Before painting a room you should spackle the walls. This fills up small holes and cracks. My father is notorious for using enough spackle to appreciably diminish the room’s volume. (So says my mother. My father disagrees.) I put spackle on as if I were paying for it myself, using so little my father has sometimes asked when I’m going to put any on. I’ll get to mathematics in the next paragraph.

One of the natural things to wonder about a set — a collection of things — is how big it is. The “measure” of a set is how we describe how big a set is. If we’re looking at a set that’s a line segment within a longer line, the measure pretty much matches our idea of length. If we’re looking at a shape on the plane, the measure matches our idea of area. A solid in space we expect has a measure that’s like the volume.

We might say the cracks and holes in a wall are as big as the amount of spackle it takes to fill them. Specifically, we mean it’s the least bit of spackle needed to fill them. And similarly we describe the measure of a set in terms of how much it takes to cover it. We even call this “covering”.

We use the tool of “cover sets”. These are sets with a measure — a length, a volume, a hypervolume, whatever — that we know. If we look at regular old normal space, these cover sets are typically circles or spheres or similar nice, round sets. They’re familiar. They’re easy to work with. We don’t have to worry about how to orient them, the way we might if we had square or triangular covering sets. These covering sets can be as small or as large as you need. And we suppose that we have some standard reference. This is a covering set with measure 1, this with measure 1/2, this with measure 24, this with measure 1/72.04, and so on. (If you want to know what units these measures are in, they’re “units of measure”. What we’re interested in is unchanged whether we measure in “inches” or “square kilometers” or “cubic parsecs” or something else. It’s just longer to say.)

You can imagine this as a game. I give you a set; you try to cover it. You can cover it with circles (or spheres, or whatever fits the space we’re in) that are big, or small, or whatever size you like. You can use as many as you like. You can cover more than just the things in the set I gave you. The only absolute rule is you must not miss anything, even one point, in the set I give you. Find the smallest total area of the covering circles you use. That smallest total area that covers the whole set is the measure of that set.

Generally, measure matches pretty well the intuitive feel we might have for length or area or volume. And the idea extends to things that don’t really have areas. For example, we can study the probability of events by thinking of the space of all possible outcomes of an experiment, like all the ways twenty coins might come up. We find the measure of the set of outcomes we’re interested in, like all the sets that have ten tails. The probability of the outcome we’re interested in is the measure of the set we’re interested in divided by the measure of the set of all possible outcomes. (There’s more work to do to make this quite true. In an advanced probability course we do this work. Please trust me that we could do it if we had to. Also you see why we stride briskly past the discussion of units. What unit would make sense for measuring “the space of all possible outcomes of an experiment” anyway?)

But there are surprises. For example, there’s the Cantor set. The easiest way to make the Cantor set is to start with a line of length 1 — of measure 1 — and take out the middle third. This produces two line segments of length, measure, 1/3 each. Take out the middle third of each of those segments. This leaves four segments each of length 1/9. Take out the middle third of each of those four segments, producing eight segments, and so on. If you do this infinitely many times you’ll create a set that has no measure; it fills no volume, it has no length. And yet you can prove there are just as many points in this set as there are in a real normal space. Somehow merely having a lot of points doesn’t mean they fill space.

Measure is useful not just because it can give us paradoxes like that. We often want to say how big sets, or subsets, of whatever we’re interested in are. And using measure lets us adapt things like calculus to become more powerful. We’re able to say what the integral is for functions that are much more discontinuous, more chopped up, than ones that high school or freshman calculus can treat, for example. The idea of measure takes length and area and such and makes it more abstract, giving it great power and applicability.

• #### sarcasticgoat 3:31 pm on Monday, 22 June, 2015 Permalink | Reply

Living with black mold on walls, thoughts?

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• #### Joseph Nebus 4:41 pm on Monday, 22 June, 2015 Permalink | Reply

See if you can get any rent from it. Unfortunately the mold may be aware the only way to get rid of it is to burn the house down and move to another time zone, so it tends to make lowball offers. Make sure you have a flame source and an accelerant in view while negotiating.

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• #### sarcasticgoat 5:10 pm on Monday, 22 June, 2015 Permalink | Reply

Thanks, I’ll have a chat Head-Spore, the intimidating three inch big mold spore who taunts me as I walk to the bathroom. :)

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• #### Joseph Nebus 4:49 am on Tuesday, 23 June, 2015 Permalink | Reply

Best of luck …

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• #### howardat58 7:14 pm on Monday, 22 June, 2015 Permalink | Reply

I love the Cantor middle third set. It really is mind blowing on first encounter.

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• #### John Friedrich 12:29 am on Tuesday, 23 June, 2015 Permalink | Reply

Of further interest, the Hausdorff dimension of the Cantor set is ln2/ln3, which proves that it is a fractal.

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• #### Joseph Nebus 4:56 am on Tuesday, 23 June, 2015 Permalink | Reply

Yeah, that’s a neat trait. I might get around to dimensions if I do another a-to-z run, or maybe as an independent discussion.

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• #### Joseph Nebus 4:55 am on Tuesday, 23 June, 2015 Permalink | Reply

It is mind-blowing, and it’s one of those sets that just keeps giving strangeness.

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• #### Aquileana 2:08 pm on Wednesday, 24 June, 2015 Permalink | Reply

5,280 is such an interesting number!… I appreciate that you share all about its twists and meanings with us… Also congratulations on your stats on Twitter, Joseph. All my best wishes. Aquileana :D

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• #### Joseph Nebus 3:35 am on Thursday, 25 June, 2015 Permalink | Reply

I’m glad that you liked. Thank you.

Now on to 6,076! (Feet in a nautical mile. Approximately.)

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## Bijection.

To explain this second term in my mathematical A to Z challenge I have to describe yet another term. That’s function. A non-mathematician’s idea a function is something like “a line with a bunch of x’s in it, and maybe also a cosine or something”. That’s fair enough, although it’s a bit like defining chemistry as “mixing together colored, bubbling liquids until something explodes”.

By a function a mathematician means a rule describing how to pair up things found in one set, called the domain, with the things found in another set, called the range. The domain and the range can be collections of anything. They can be counting numbers, real numbers, letters, shoes, even collections of numbers or sets of shoes. They can be the same kinds of thing. They can be different kinds of thing.

• #### Ken Dowell 7:16 pm on Wednesday, 27 May, 2015 Permalink | Reply

Have to admit I have never heard the term. Thought it was a double ejection.

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• #### Joseph Nebus 10:49 pm on Thursday, 28 May, 2015 Permalink | Reply

It’s not a common word. I don’t think anyone besides an upper-level mathematics undergraduate or a grad student would need to use it.

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• #### sheldonk2014 10:23 pm on Wednesday, 27 May, 2015 Permalink | Reply

When numbers are used in the comics do they have a reason they use those specific ones

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• #### Joseph Nebus 10:55 pm on Thursday, 28 May, 2015 Permalink | Reply

I think the only fair answer is “sometimes”. Occasionally a particular number will be part of the joke. Or an in-joke. For example, the number 47 turns up a lot in Star Trek because of a joke on the writing staff.

But sometimes the cartoonist feels that, if there’s got to be some mathematical content, then it should be correct. Bill Amend of FoxTrot — a physics major, I feel the need to admit — does this a lot. (He’s also talked about the challenges of writing mathematics comics, and how that relates to teaching.) Some do that because it’s fun. Some because they feel it makes the joke stronger if the mathematics talk is authentic. Some, surely, because they learned this stuff so why not put it to use?

And there are other cartoonists who just pick numbers because they look pretty, or they sound funny, or for other aesthetic reasons like that. And that’s fine also: comic strips are a form of art and aesthetic grounds have to count.

But overall there just isn’t a universal rule.

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• #### baffledbaboon 4:00 pm on Thursday, 28 May, 2015 Permalink | Reply

At first glance you’d think the word “Bijection” would mean “to be ejected twice”

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• #### Joseph Nebus 10:58 pm on Thursday, 28 May, 2015 Permalink | Reply

It’s not far off. There’s a related term, “injection”, which would have made sense to put first if I were not doing this in the a-to-z order. (If I weren’t sticking with alphabetical order I’d have defined “function”, then “injection”, then “bijection”.) In an injective function everything in the domain is matched to something in the range, but it’s possible something in the range is missed.

In a bijective function, everything in the domain matches exactly one thing in the range. So you can read it as two rules, one that matches everything in the domain to something in the range, and another rule that matches everything in the range to something in the domain. So in a way you can see it as a double injection.

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• #### Michelle H 11:08 pm on Sunday, 31 May, 2015 Permalink | Reply

Makes me think that ‘bijection’ could be a metaphor for a set of ‘perfect’ social grafts (i.e. old-fashion marriage), especially in engineered situations like those that favour monogamy, don’t allow divorce, and only recognize heterosexuality. Hmm, a Jane Austen novel? (Sorry… I recognized your sense of humour and couldn’t refrain.)

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• #### Joseph Nebus 8:56 pm on Monday, 1 June, 2015 Permalink | Reply

No need to apologize. Never fear that.

I am interested in the metaphors that could be made out of bijection. Functions are all about pairing up things, one from the domain and one from the range. This does suggest parallels to social groupings, although I’m not sure how far the metaphor could be pushed before it stops being enlightening.

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• #### Michelle H 9:42 pm on Monday, 1 June, 2015 Permalink | Reply

Sometimes metaphor is more entertaining than enlightening.

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• #### Joseph Nebus 8:02 am on Tuesday, 2 June, 2015 Permalink | Reply

This is so, although I really love when a metaphor can serve both roles well.

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• #### Michelle H 1:23 pm on Tuesday, 2 June, 2015 Permalink | Reply

In the history of each metaphor, its birth probably brought shock and likely both amusement and disgust. Slowly, it aged to cliché and perhaps ceased to be recognized even as cliché (table’s leg). ‘Bijection’ is a really difficult term for metaphor; I have been thinking on it since reading your post. Putty and country dances might be the best, although one assumes too much experience with Jane Austen.

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• #### Joseph Nebus 10:36 pm on Friday, 5 June, 2015 Permalink | Reply

Bijection is indeed a difficult term to find a metaphor for. But I do appreciate your trying and enjoy your describing of the history of a metaphor. Thank you.

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• #### Michelle H 10:08 pm on Monday, 8 June, 2015 Permalink | Reply

This might work: transposing music, when a tune is moved from one key to another, everything in the domain has a match in the range (that is to say, every note in the original key must be rewritten into the new one).

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• #### Joseph Nebus 9:24 pm on Tuesday, 9 June, 2015 Permalink | Reply

Oh, I think you have it, if not in transpositions then at least arranging a musical piece for a different instrument. A bijective function, among other things, lets you go from the range back to the domain. And some transpositions and arrangements let you reconstruct the original music perfectly.

But there are some functions, and some arrangements, that don’t allow that. For example, a song arranged for carousel organs will typically have a narrower range of notes available, and maybe less flexibility in time signatures. A skilled arranger can convert the song so that it sounds right on the new instrument, but you couldn’t reconstruct the original song perfectly from the carousel-organ version. The function matching the original song to the transposed/arranged version is not a bijective one.

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• #### Michelle H 9:58 pm on Tuesday, 9 June, 2015 Permalink | Reply

Cool. I was at the piano when the thought settled, but wondered if there would be difficulty transposing among instruments. Indeed, the range is not equal to all instruments. However, this discussion has helped illuminate the movement in bijection, as a kind of reciprocity. There must be a connection here to ratios and proportions, then?

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• #### Joseph Nebus 7:34 pm on Thursday, 11 June, 2015 Permalink

There might be a connection to ratios and proportions. I admit at this point I know so little music theory that I’d be wary of making a stupid mistake. I’ll make stupid mistakes, sure, but I do try to limit them to subjects I should know better about.

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• #### Michelle H 9:59 pm on Thursday, 11 June, 2015 Permalink

I thank you for your many exchanges on the topic. My pet idea lately has been excess as discussed in cultural theory, and bijection feels like a strong example of efficiency and quite different, so it attracted me. Thanks again and have a great upcoming weekend.

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• #### Joseph Nebus 4:16 am on Saturday, 13 June, 2015 Permalink

You’re quite welcome. I’m glad the topic’s so inspired you.

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## Reading the Comics, September 28, 2014: Punning On A Sunday Edition

I honestly don’t intend this blog to become nothing but talk about the comic strips, but then something like this Sunday happens where Comic Strip Master Command decided to send out math joke priority orders and what am I to do? And here I had a wonderful bit about the natural logarithm of 2 that I meant to start writing sometime soon. Anyway, for whatever reason, there’s a lot of punning going on this time around; I don’t pretend to explain that.

Jason Poland’s Robbie and Bobby (September 25) puns off of a “meth lab explosion” in a joke that I’ve seen passed around Twitter and the like but not in a comic strip, possibly because I don’t tend to read web comics until they get absorbed into the Gocomics.com collective.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (September 26) shows how an infinity pool offers the chance to finally, finally, do a one-point perspective drawing just like the art instruction book says.

Bill Watterson’s Calvin and Hobbes (September 27, rerun) wrapped up the latest round of Calvin not learning arithmetic with a gag about needing to know the difference between the numbers of things and the values of things. It also surely helps the confusion that the (United States) dime is a tiny coin, much smaller in size than the penny or nickel that it far out-values. I’m glad I don’t have to teach coin values to kids.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 27) mentions Lagrange points. These are mathematically (and physically) very interesting because they come about from what might be the first interesting physics problem. If you have two objects in the universe, attracting one another gravitationally, then you can describe their behavior perfectly and using just freshman or even high school calculus. For that matter, describing their behavior is practically what Isaac Newton invented his calculus to do.

Add in a third body, though, and you’ve suddenly created a problem that just can’t be done by freshman calculus, or really, done perfectly by anything but really exotic methods. You’re left with approximations, analytic or numerical. (Karl Fritiof Sundman proved in 1912 that one could create an infinite series solution, but it’s not a usable solution. To get a desired accuracy requires so many terms and so much calculation that you’re better off not using it. This almost sounds like the classical joke about mathematicians, coming up with solutions that are perfect but unusable. It is the most extreme case of a possible-but-not-practical solution I’m aware of, if stories I’ve heard about its convergence rate are accurate. I haven’t tried to follow the technique myself.)

But just because you can’t solve every problem of a type doesn’t mean you can’t solve some of them, and the ones you do solve might be useful anyway. Joseph-Louis Lagrange did that, studying the problem of one large body — like a sun, or a planet — and one middle-sized body — a planet, or a moon — and one tiny body — like an asteroid, or a satellite. If the middle-sized body is orbiting the large body in a nice circular orbit, then, there are five special points, dubbed the Lagrange points. A satellite that’s at one of those points (with the right speed) will keep on orbiting at the same rotational speed that the middle body takes around the large body; that is, the system will turn as if the large, middle, and tiny bodies were fixed in place, relative to each other.

Two of these spots, dubbed numbers 4 and 5, are stable: if your tiny body is not quite in the right location that’s all right, because it’ll stay nearby, much in the same way that if you roll a ball into a pit it’ll stay in the pit. But three of these spots, numbers 1, 2, and 3, are unstable: if your tiny body is not quite on those spots, it’ll fall away, in much the same way if you set a ball on the peak of the roof it’ll roll off one way or another.

When Lagrange noticed these points there wasn’t any particular reason to think of them as anything but a neat mathematical construct. But the points do exist, and they can be stable even if the medium body doesn’t have a perfectly circular orbit, or even if there are other planets in the universe, which throws off the nice simple calculations yet. Something like 1700 asteroids are known to exist in the number 4 and 5 Lagrange points for the Sun and Jupiter, and there are a handful known for Saturn and Neptune, and apparently at least five known for Mars. For Earth apparently there’s just the one known to exist, catchily named 2010 TK7, discovered in October 2010, although I’d be surprised if that were the only one. They’re just small.

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt, from the 23rd of August, 1953 (rerun the 28th of September, 2014).

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt (September 28, originally run August 23, 1953) has been on the Sunday strips now running a tale about a mathematics professor, Peter Peddle, who’s threatening to revolutionize Big Ben Bolt’s boxing world by reducing it to mathematical abstraction; past Sunday strips have even shown the rather stereotypically meek-looking professor overwhelming much larger boxers. The mathematics described here is nonsense, of course, but it’d be asking a bit of the comic strip writers to have a plausible mathematical description of the perfect boxer, after all.

But it’s hard for me anyway to not notice that the professor’s approach is really hard to gainsay. The past generation of baseball, particularly, has been revolutionized by a very mathematical, very rigorous bit of study, looking at questions like how many pitches can a pitcher actually throw before he loses control, and where a batter is likely to hit based on past performance (of this batter and of batters in general), and how likely is this player to have a better or a worse season if he’s signed on for another year, and how likely is it he’ll have a better enough season than some cheaper or more promising player? Baseball is extremely well structured to ask these kinds of questions, with football almost as good for it — else there wouldn’t be fantasy football leagues — and while I am ignorant of modern boxing, I would be surprised if a lot of modern boxing strategy weren’t being studied in Professor Peddle’s spirit.

Eric the Circle (September 28), this one by Griffinetsabine, goes to the Shapes Singles Bar for a geometry pun.

Bill Amend’s FoxTrot (September 28) (and not a rerun; the strip is new runs on Sundays) jumps on the Internet Instructional Video bandwagon that I’m sure exists somewhere, with child prodigy Jason Fox having the idea that he could make mathematics instruction popular enough to earn millions of dollars. His instincts are probably right, too: instructional videos that feature someone who looks cheerful and to be having fun and maybe a little crazy — well, let’s say eccentric — are probably the ones that will be most watched, at least. It’s fun to see people who are enjoying themselves, and the odder they act the better up to a point. I kind of hate to point out, though, that Jason Fox in the comic strip is supposed to be ten years old, implying that (this year, anyway) he was born nine years after Bob Ross died. I know that nothing ever really goes away anymore, but, would this be a pop culture reference that makes sense to Jason?

Tom Thaves’s Frank and Ernest (September 28) sets up the idea of Euclid as a playwright, offering a string of geometry puns.

Jef Mallet’s Frazz (September 28) wonders about why trains show up so often in story problems. I’m not sure that they do, actually — haven’t planes and cars taken their place here, too? — although the reasons aren’t that obscure. Questions about the distance between things changing over time let you test a good bit of arithmetic and algebra while being naturally about stuff it’s reasonable to imagine wanting to know. What more does the homework-assigner want?

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 28) pops back up again with the prospect of blowing one’s mind, and it is legitimately one of those amazing things, that $e^{i \pi} = -1$. It is a remarkable relationship between a string of numbers each of which are mind-blowing in their ways — negative 1, and pi, and the base of the natural logarithms e, and dear old i (which, multiplied by itself, is equal to negative 1) — and here they are all bundled together in one, quite true, relationship. I do have to wonder, though, whether anyone who would in a social situation like this understand being told “e raised to the i times pi power equals negative one”, without the framing of “we’re talking now about exponentials raised to imaginary powers”, wouldn’t have already encountered this and had some of the mind-blowing potential worn off.

## Reblog: The Math That Saved Apollo 13

GCDXY here presents images from the Apollo 13 flight checklist. This is itself a re-representing of images that Gizmodo posted when Apollo 13 Commander James Lovell sold the checklist last year, but I’m just coming across this now. And it nicely combines the mathematics and the space history interests I so enjoy.

The particular calculations done here were shown in one of many, many, outstanding scenes in the movie Apollo 13. However, the movie presents the calculations as being done on slide rule, when the computations needed are mostly addition and subtraction. It is possible to use slide rules to do addition and subtraction, but that’s really the hard way to do it; slide rules are for multiplication, division, and raising numbers to powers.

But considerable calculation for Apollo (and Gemini, and Mercury) was done without electronic computers, and the movie would have missed out on presenting an important point if it didn’t have the scene. So the movie achieved that strange state of conveying something true about what happened by showing it in a way it all but certainly did not.

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Two hours after a service module’s oxygen tank explosion on Apollo 13, Commander James Lovell did calculations that helped put the ship back on course so that they could return back to Earth. They needed to establish the right course to use the Moon’s gravity to get back home. Check out the article on Gizmodo from November 2011.

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## Reblog: Kant & Leibniz on Space and Implications in Geometry

Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.

Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.

Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.

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Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…

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• #### BunnyHugger 7:42 pm on Thursday, 9 August, 2012 Permalink | Reply

As Joseph knows, I have a framed portrait of Kant in my dining room (I have to stop calling things that; now it’s our dining room) that was taken from a set of prints from the Moscow Observatory — I believe from the 1940s — celebrating people who had made contributions to astronomy. I like that I have a souvenir recognizing Kant as an astronomer.

It used to bother me that people call the nebular hypothesis “the Laplace theory” when Kant’s work on it was earlier. (I have also heard it called “the Kant-Laplace theory,” but, I think, usually by philosophers.) However, then Wikipedia told me that Kant himself may have gotten the rudiments of it from Swedenborg, and no one ever calls it “the Swedenborg-Kant-Laplace” theory, as far as I’ve heard.

I hope that DeVita’s article called back ideas for you (regarding both Leibniz and Kant) that I tried to explain to you in our cabin on the Amsterdam-Newcastle ferry while fighting off seasickness.

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## Wednesday, June 6, 1962 – Food Contract, Boilerplate Purchase

The Manned Spacecraft Center has awarded to the Whirlpool Corporation Research Laboratories of Saint Joseph, Michigan, a contract to provide the food and waste management systems for Project Gemini. Whirlpool is to provide the water dispenser, food storage, and waste storage devices. The food and the zero-gravity feeding devices, however, are to be provided by the United States Army Quartermaster Corps Food and Container Institute, of Chicago. The Life Systems Division of the Manned Spacecraft Center is responsible for directing the program.
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## Friday, June 1, 1962 – Operations Coordination Meeting, Astronaut Applications Close

Today’s was the first spacecraft operations coordination meeting. Presented at it was a list of all the aerospace ground equipment required for Gemini spacecraft handling and checkout before flight.

June 1 was also the nominal closing date for applications to be a new astronaut. Applications were opened April 18. The plan is to select between five and ten new astronauts to augment the Mercury 7.

## Tuesday, May 29, 1962 – Ejection Seat Plans

The development testing plans for the Gemini spacecraft ejection seat were settled in a meeting between representatives of McDonnell, Weber Aircraft, the Gemini Procurement Office, Life Systems Division, Gemini Project Office, and the US Naval Ordnance Test Station at China Lake, California.
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## Thursday, May 24, 1962 – Parachute testing starts

At the Naval Parachute Facility in El Centro, California, North American completed a successful drop test of the emergency parachute recovery system, using a half-scale test vehicle.
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## Wednesday, May 23, 1962 – Paraglider Wing Wind Tunnel Test

The Ames Research Center has begun the first wind tunnel test of the inflatable paraglider wing, using a half-scale model of the wing intended to bring Gemini flights (after the first one) to a touchdown on the ground. This is the first large-scale paraglider wing in the full-scale test facility. The objective of the test program, to run over two months, are to understand the basic aerodynamic and loads data for the wing and spacecraft system, and to identify potential aerodynamic and design problems.
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## Thursday, May 17, 1962 – Retrorocket and Parachute Decisions

The meeting about the retrograde rocket motors has concluded the design should be changed to provide about three times the thrust level. This will allow retrorocket aborts at altitudes as low as between 72,000 and 75,000 feet. The meeting was between representatives of McDonnell and the Gemini Project Office.
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## Wednesday, May 16, 1962 – Retrorockets, Parachutes, and Interface Group

Representatives of the Gemini Project Office and of McDonnell are meeting to discuss retrograde rockets for the Gemini spacecraft. These rockets are currently to be provided by Thiokol.
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## Tuesday, May 15, 1962 – Ejection seat in review; rocket catapult contract; new liaison

The first ejection seat design review has been completed. The two-day conference at McDonnell in Saint Louis was attended by representatives of McDonnell, Northrop Ventura (formerly Radioplane), Weber Aircraft, and the Manned Spacecraft Center. This is the first of a series of ejection seat design meetings planned from March 29.
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## Monday, May 14, 1962 – Titan II Statement of Work Given

The Manned Spacecraft Center has issued its final Statement of Work for the Air Force Space Systems Division. Space Systems Division is, in this context, contractor to NASA procuring Titan II launch vehicles, as modified for the manned program’s needs. The statement, itemizing the tasks Space Systems Division is expected to do or provide, and on what schedule, and how acceptable performance will be measured, and so on, began being prepared by the Manned Spacecraft Center on January 3. The initial budgeting and planning were completed by the end of March. Though final the plan is subject to amendment.

## Saturday, May 12, 1962 – Project Gemini Cost Estimates Growing

A current estimate of Project Gemini costs shows considerable increases from the projections of December 1961. The spacecraft cost, estimated at $240.5 million, is now projected at$391.6 million. Titan II costs, expected five months ago to be about $113.0 million, have risen to$161.8 million. The Atlas-Agena budget has risen from $88.0 million to$106.3 million, despite this part of the program’s slowing down. Support development, including the paraglider program, has increased from $29.0 to$36.8 million. There is a bright spot on the budgetary front: the estimate of operations cost has declined from $59.0 to$47.8 million.
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