Calculus

So where is the big magic and mystery of calculus? I submit that in the fundamental equation F=ma the difficult variable is the "a." Forces are related to a mysterious and abstract idea of acceleration. What exactly is an acceleration?

The standard story is "a first derivative is a rate of change and a second derivative is a rate of change of a rate of change." This is not the right way to think about it in general: this story only works well in one dimension, and it only works because of an accident of the one dimensional case. This is one of the reasons calculus seems much harder than it needs to be, because one must spend great effort learning one story, and then a similar amount of effort learning that it is wrong. If one masters the one dimensional case, the next step is to embark on another a different, difficult subject, "multivariate calculus" --- and one discovers, not to put too fine a point on it, that multivariate calculus basically doesn't work at all. The idea of "a rate of change of a rate of change" just doesn't generalize --- instead it dissolves into a disastrous mess of partial derivatives. To clean up the mess, there is yet another subject to tackle: differential geometry. This also comes in multiple different historical flavors, another mess the student is made to wade through. Finally there is the "advanced" version that physicists use, filled with fancily named objects: groups, fiber bundles, dizzily high dimensional spaces, and truly frightening abstraction. Only a vanishingly small number of students ever make it that far.

I was quite annoyed to realize, after burning six years of my life wading through all this stuff, that it basically added to what I learned in introductory calculus only one simple --- but tremendously powerful --- idea. I shouldn't say "added": the more accurate word is "corrected." The problem is that the idea of a "rate of change of a rate of change" leaves something important out. Suppose we think of a "rate of change" as a little vector, representing the velocity of a moving particle. Now ask the question: how can that "unit of change" itself change? In one dimension, it has limited options: it can get faster or slower. But in three dimensions, it has more possibilities. Not only can it get faster or slower, it can also rotate --- like an airplane, we can think of it having three more degrees of freedom: pitch, yaw and roll.

The key point here is that the "space of changes" of our little "unit of change" is in itself a new space, which, notably, might not even be the same dimension as the original space, nor is it even necessarily flat. It happens in one dimension that the "space of changes" of a vector constrained to a line has the same dimension and shape as the line itself. But that is merely an accident of one dimension, and an unfortunate one, because it allows the narrative to silently sweep this "space of changes" under the rug.

The principal contribution the modern mathematician's conception makes to this subject matter is one single powerful idea: the observation that it is useful to become conscious of a space of changes --- to give a name to both a tranformation and to the space of all possible transformations. In other words, if we are observing an airplane that has, at a certain moment, say, tacked to the left with a certain pitch, yaw and roll parameter, it is useful not just to see this action as a transformation of the airplane's state, but to look at the action as an object in itself, with its own name, that has a location of its own in a space of all possible related actions. That is to say, we want to talk precisely about the "change of the change" and think consciously about the space in which these second level changes are objects rather than transformations. In other words, we want verbs to turn into nouns, transitions to turn into objects, so that the evanescent moment of "something changing" can become something concrete we can look at, stand back from, manipulate, and measure in relation to other similar objects. If we can do that, we can use this consciousness to cut the Gordian knot of partial derivatives which blights the teaching of calculus and differential geometry.

Why is it so important to become conscious of this space? Well, let us return to the question "what is an acceleration, really?" The standard story makes it out to be "the same kind of thing" as a first derivative -- that is to say, a double application of the "rate of change" operator. But this story breaks down in multivariate calculus, when it becomes clear that ordinary derivatives and partial derivates cannot be doubly-applied to capture the idea of an acceleration in general. Is there any operation you can doubly-apply to get an acceleration object in every case?

Yes there is, but you have to think carefully about what you are trying to do. I submit that the right way to tell the story about the operation that is performed (twice!) is this: we want to make the problem of understanding motion in the world tractable by observing that it helps to approximate a complex, curved world with a linear space. For example, even though you know the earth is round, you follow a flat road atlas without fear it will mislead you. If you were flying a plane from here to China, you might have to correct the route a flat map would suggest to you, but for most of us, who aren't traveling that far from home, a flat road atlas is a perfectly good approximation. That is how you should think about the operation one is doing when one takes a derivative: it is approximating a complex surface with a flat tangent space in order to make route understanding and planning easier.

But wait! you might ask: I'm not so naive as to think the earth is flat! That belief went out of style millennia ago. Perhaps it makes for a useful local approximation, but what do I do when I travel beyond my local area? People do, after all, fly to China, and they do expect to arrive in the right place. If the pilot arrived in Thailand rather than Shanghai because he flew a straight line on a map, forgetting to correct for the curvature of the earth, the passengers would be quite annoyed indeed. We expect our mathematics to handle routes that venture out of the range of a local flat approximation; we expect it to tell us what a "straight line on a sphere" really means

Notice that if a pilot is following a "straight line on a sphere," he is not really flying the plane straight. He will be constantly turning it a little bit. In reality, he is probably applying small corrections periodically, but in our mathematical fantasy world we imagine that he is in every tiniest instant of time applying the tiniest possible correction, in exactly the amount necessary to keep him on course.

What kind of thing is this tiny course correction? In particular, how can we understand it as result of a "double application" of some operator, so as to understand the essence of the meaning of "rate of change of the rate of change"? This question gets to the mystery of the nature of acceleration. In order to stay on the route to Shanghai in our mathematical fantasy world, our pilot has to apply an "acceleration" --- in the mathematical sense --- to the plane at every moment. What kind of object is this "acceleration"?

I was setting up for this question when I emphasized how useful it is to think of "the change of a change" --- i.e. in this case, a pitch, roll or yaw of the plane --- as an object, and to think of these objects all living together in some space, a space which might be higher-dimensional and interestingly curved. Now that we are facing the problem of understanding our necessary "tiny course correction," let us slow down and consider carefully what we are trying to do. We are trying to pick out some object which is related to all the objects in the space of all possible pitches, rolls, and yaws the plane can do. But it cannot be any particular one of them, because, remember, we are living in our mathematical fantasy world where we are not correcting periodically, but somehow constantly and instantaneously. Where can we find this object we are applying to keep our plane on course in this ideal mathematical world? In particular, in what way can we find it by applying some kind of "derivative" operator a second time?

The key idea is that the operation we need to apply the second time is the same operation we applied when we made a flat map of a curved earth --- but this time the operation is not being applied in our real world, it is being applied in this "pitch, yaw, roll" space. That is to say, we want to imagine some being living in this other world, the world where all the "changes of changes" are concrete places --- just ordinary locations in an ordinary neighborhood --- and we want to imagine him doing the same trick we did when we made ourselves a local road atlas. He wants to simplify his world by pretending it is flat, locally. In particular, the locality he wants to map is the neighborhood of all the very small changes, the small movements we can make to the airplane, because these are the movements we care about when we make the small course corrections that keeps our plane headed towards Shanghai rather than Bangkok. He wants to make a flat map of this particular neighborhood, and he wants a procedure to find on this map the right correction to get him to Shanghai. The magic idea is that, just the same way we can use a heading on a map of our real space to get us on a course to a certain location, we can use a heading on this map in the space of "changes of changes" to get the overall change we are seeking. If we want the plane to end up with an overall 30 degree rotation, for instance, we can use our local map of small changes near the identity rotation to find a heading that will get us the right overall change when applied consistently over the course of our trip.

By the way, if you want to check that this story is, in fact, standard (in certain mathematical circles), then I should tell you that the key words which are used when these things are normally talked about. A space of (composable, reversible) actions or changes is called a group. One that is continuous, like the changes of velocity of an airplane, is called a Lie group, after Sophus Lie, who first studied them. The map of "the local neighborhood of smallest changes" is called the Lie algebra. There is a standard construction that makes a Lie algebra out of any Lie group. It is a deep and beautiful theorem that all the local neighborhoods in a space of changes look alike, so one map suffices for every neighborhood, and they stitch together into a complete atlas in a canonical way. As a consequence, it is always possible to pick a small local change-of-changes and keep applying it continually to grow it into a large change. This transformation is called the "exponential map". The hypersphere, viewed as a Lie group, also has a standard name: the Special Unitary group of dimension two, or SU2. The space of all "pitches, yaws and rolls" is called the Special Orthogonal group of dimension three, or SO3. There is another very nice theorem which proves that these two groups can be identified if we add in an extra binary choice for "spin" (in the subatomic particle sense of the word). To keep this story simple and focused I won't talk much about spin, except to say that one of the particularly interesting properties of the hypersphere is that it has three different mathematical identities: as a unitary group, an orthogonal group and a spin group. Four identities, in fact, if you note that it is the quaternion group too. A great deal of interesting mathematics converges on the hypersphere.

These aren't very pretty names, but they are standard, for what it is worth. Wikipedia will tell you all about them, though I think my way of telling the story is more fun.

So I have answered my question: what is an acceleration? It is a heading on a local map of the neighborhood near the identity of the space of "changes of changes." We can understand that acceleration is generated by "the same operation" that we previously used to find the velocity, by noticing that in both cases we approximated a complex curved space with a simpler, local, flat map, and then used that map to plot a course. The subtle point, however, is that when we do this operation the second time we are doing it in a different space. We are not in our own world anymore, we are in the space of "changes of changes." This subtle point is lost in the account of single variable calculus because of the pure accident that in that case the space of "changes of changes" looks too much like our own space. It just happens to be the same dimension, and it just happens to be flat, so you can get away with not mentioning it. But I think you lose a tremendous amount of real understanding when you make this simplification. The result is fruitless confusion for young people which soaks up some of the most valuable time in their lives in a painful gauntlet of partial derivatives and a frustrating archaeological dig through layers of obsolete historical conception and notation. I feel strongly that this is deeply unfair -- in this high stakes game, kids deserve to hear the complete, correct story from the beginning.

This, to my eyes, is why the hypersphere is so enormously useful and important as a teaching tool. I have a slightly different emphasis from most people who talk about the appeal of the four-dimensional sphere in teaching: they talk about how it increases awareness of higher dimensions, or of symmetry, or of the general idea of a polytope. These things are nice, but the thing I really care about is that it is the smallest and most relevant non-degenerate example of a space of "changes of changes." It is the space where a pitch, roll, or yaw becomes a concrete and physical object, and where you can stand back and look at the whole collection of all possible combinations of pitching, rolling, and yawing (and an extra choice of "spin", which I can explain, but won't now) all sitting together in their right places, and see how they are naturally arranged. Seeing their arrangement is key, because it is a deeply important fact that this kind of space is not, in general, flat. (If I just told you that the space of all possible pitches, yaws and rolls is not a flat space, would that be immediately obvious? I don't think so.) It is the space where the idea of "picking a heading" to get to a certain desired change makes sense: where a "thirty degree rotation about a given axis" is a place it is possible to set your sights on and navigate towards. That, in turn, is important because it allows you to understand exactly what you are talking about when you say, for instance, that applying a force on a spinning string causes the object at the end of the string to accelerate in such a way that it will to rotate through a thirty degree arc in a certain given time. In this way, the hypersphere is a concrete embodiment of the mathematics that is necessary to understand the language of physics.

I also care abouut the regular polytopes in four dimensional space because I am of the strong opinion if one wishes --- in the interest of creating a easier more kid-friendly, narrative --- to simplify the story, the right simplification to introduce is not to focus on the (rather degenerate) one dimensional case, but instead, to introduce the discrete analogue to this story. By discrete, by the way, I mean discrete in the "space of changes" --- for instance, a world where a hypothetical airplane is not allowed to move in arbitrary ways, but perhaps is restricted to, say, only multiples of ninety degree turns in every direction. This restricted space of changes is represented by a regular polytope in the four sphere. In this setting it is still possible to talk about "the smallest possible rotation" and show how all other rotations are generated by these smallest rotations. In this way, the ideas that underly the derivative and exponential map could be introduced in a simple and concrete setting, and generalized to the continuous case later. For instance, if we show how a Logo-style turtle could generate a path around a tetrahedron by making one-third turns each time it steps over the edge, it isn't that hard to continue our story by suggesting to kids that if the tetrahedron is smoothed out into a cone, the turtle could learn to walk on it by making many more steps with a much tinier uniform turn at each step. In this way it might be possible to introduce the concept of derivative and exponential map in a way that proceeded gently from the concrete to the abstract, and left a child the possibility to bail out into a concrete world again any time that the abstractions got overwhelming

One of the reasons I believe so emphatically that the right way to simplify the story of calculus is to restrict to discrete rotations, rather than restrict to one dimension, is that this discrete story is not just a toy for children! Discrete differential geometry is a flourishing subject (see http://ddg.cs.columbia.edu/ for example): understanding this subject well enough can get one a job right now at Disney or Pixar programming simulations for computer graphics. (However --- truth in advertising --- it is a new subject, so some aspects of it are still research problems; it isn't completely clear what's the best way to discretize to give the models the desired properties.) Perhaps you can understand that it makes sense that people in the real world care about discrete approximations to our continuous story, because, as I mentioned before, it is a fiction that an airplane makes a constant smooth infinitesimal correction to stay on course. In reality the pilot makes periodic adjustments intermittently. The interaction between the fictional world of continuous infinitesimal adjustment and the actual world where those adjustments are finite and discontinuous is a burning problem that occupies the minds and careers of many adults in the real world.

So it is quite fair to introduce kids to these concepts early: even if the ideas are introduced as a "toy," it is the kind of toy that can be justified to teenagers as also much more than a toy. I think it is disrespectful to eighteen year olds to hold them for years in a classroom calculating the trajectories of go-carts and thrown balls, because these truly are toy problems that almost no one in the real world thinks about much. But it is not disrespectful to give them "toy" discretizations of three dimensional problems, because without too much extra work a restless teenager can transform an understanding of such "toys" into knowledge that can land him a real job and a real adult life.

I have suggested to various teachers that images of the hypersphere be incorporated into the teaching of calculus and physics in the way I suggest, but I have encountered the objection that it is too strange a space, too hard to wrap one's head around, to be suitable for the classroom. I agree, it is somewhat hard to learn to visualize something projected from four dimensions, and hard to get used to the curvature of the space. I submit, however, that this difficulty is the inherent, rather than gratuitous, complexity of the subject of calculus and differential geometry. The space in which "changes of changes" live IS higher dimensional, and it IS curved, and that makes it hard to think about. However, you cannot avoid wrapping your mind around this problem and still claim to really understand the subject. If you try to simplify away these difficulties you are simplifying away the essence of the subject, and you inevitably pay for it when you try to make your way into the advanced material and get beaten back by a blizzard of incomprehensible partial derivatives.

To wrap up this essay I want to quickly sketch a vision of a mobile app with accompanying physical toy that could be used, as I have suggested, to introduce polytopes in the hypersphere and their connection to the fundamental ideas of differential geometry. They key thing I would want the mobile app to accomplish is to make clear the connection between the points of the hypersphere and the rotations they represent in real space, and show how those choices can be used to control trajectories. For instance, using the regular polytope that represents all the rotations that are multiples of ninety degrees, one could imagine controlling a logo-style turtle restricted to a surface made of stacks of cubes. By picking faces of the 4D polytope a kid could turn the turtle to make it walk on the surface. One could also set the turtle on a programmed trajectory --- say, walking around a cube --- and show in the 4D polytope how this corresponded to a uniform motion (the turtle makes the same change at each step). Then you could introduce the idea of acceleration --- a change to a change --- that alters this programmed trajectory in the way you would expect if a force had kicked the turtle off its path, and show how this corresponds to a change in the choice of 4D point. There are more game ideas along these lines; this is just a sketch of the kind of thing that might be both fun and educational. It could also be built at two levels, a "child's level" where the emphasis is to make a fun toy, and a "teenager's level" where the toy could be re-explained as a route into an understanding of adult problems.

If I had the chance to make a companion physical toy, I would want one that emphasized the structures that are mathematically important. As I mentioned, there is an important idea of a "smallest rotation of a given type" that generates a family of rotations. The whole object then is divided into internal structure generated by these families. These internal structures are a nice way to pave the way to the continuous ideas of tangent space (Lie algebra). In addition, as I mentioned earlier, the exponential map works because of the uniformity of the space. There is more to say about the flavors of these internal structures, but the short story is that it would be nice to have a physical toy that could be taken apart and put back together in a way that emphasized both the structure and the uniformity of the space. In this way, a thing that was a fun toy to a ten year old could be explained again to a teenager as a thing that was much more than a toy.