4 ms·

I am a graduate student, first-year, so I have a lot of problem sets to work on. I've been thinking about tau every so often since the last time this was posted

by scythe 15y ago

I am a graduate student, first-year, so I have a lot of problem sets to work on. I've been thinking about tau every so often since the last time this was posted here, and so every time I work a problem, I think to myself: would this be clearer if I used tau? And the answer is usually "no". Sometimes 2 is next to pi because of the natural reason appealed to in the tauifesto, and other times it's there for some other reason, other times it isn't there at all; there's usually a huge constant factor out front anyway, and often a single equation goes across the whole sheet of paper, so even making it one character shorter doesn't accomplish much. Not that I haven't learned notation that really helps. Gaussian units and Einstein notation are a godsend. If you could standardize introductory physics courses using these, I think it could help significantly, especially when people struggle to work out the curl of a cross product or some-such strenuous vector calculation. All hail the Levi-Civita tensor! But we haven't even been able to agree on this. Furthermore, as a test grader of nearly-a-year, the work done by undergraduates is nigh-inscrutable in the maximally acceptable way already, and so the tau "rebellion" promoted here would just make my job harder. It's not too much of an issue: only the technically-inclined, who already do pretty clear work, are likely to use it; still, telling people "just start doing your work like this, let them figure it out!" means that I have to know if that scribble is a tau or a T or what-have-you, two hundred times. Pi is a very recognizable character. Plus, tau is the letter I reach for whenever I need to introduce an adjusted time of some sort, such as proper time; it's also the natural temperature in stat mech (though thermodynamic beta is itself more natural and usually better), it's torque, it's a common time constant, &c. People generally avoid pi as a character which does not represent 3.14159, though it is the prime counting function and an adjusted momentum, in which case it usually has a half-arrow, so you can tell what it's doing. We don't avoid tau at all; it's everywhere. So the tau-switch, as notational improvements go -- and math has had many over the years -- seems like a relatively large-pain, small-gain deal. Many things must change, since pi is all over the place, but few are greatly improved. So I don't see much reason to use tau in my work, or for my students to use it in theirs. If it works for you, though, good!

haberman 15y ago
To me it has almost nothing to do with "making it one character shorter," it's about making it easier to see fundamental relationships intuitively. It's not a notational improvement as much as a model improvement. It's sort of like how angles can be measured using degrees or radians, but the derivative of sine is cosine only when using radians. It's not the notation of radians that is better, it's the model.
loup-vaillant 15y ago
The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice. We should perform experiments, but I strongly suspect that using Tau (or 2Pi as if it where a single symbol) would significantly reduce confusion in the heads of the pupils. That would count as a "big advantage" in my book. Now does Tau have an actual disadvantage besides clashing with time constants and such? If not, we should keep in mind that switching is a one-time cost while the cost of using a worse notation is unbounded (proportional to the number of uses, actually). I have to reckon however that making an effort to switch away from Pi probably shouldn't be our first priority. I don't know Gaussian units nor Einstein notation, but if they do "really help" then we probably should take care of that first.
- kiba 15y ago
  The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice. If you can live long enough, you could literally forget your trigonometry. After all, humans are not exactly the pantheon of long term memory or even...reliable memory.
  - loup-vaillant 15y ago
    If we ever come to that, I want a brain update.
cygx 15y ago
Einstein notation is nice when you actually have to compute something, but it doesn't help in understanding the geometry of the problem. In fact, it might be more of a hindrance because it encourages thoughts along the general line of I don't need to care about the geometry of the problem as long as I know how to calculate the stuff I'm interested in, or as a more specific example It doesn't matter if there's an upper or lower index as I can always contract with the metric tensor. A personal pet peeve is when tensors are introduced as entities with given transformation laws (transforms like a vector in each component, etc) without ever mentioning specific geometric meanings. Thing is, classical tensor calculus makes everything look the same - even things that aren't. I prefer the 'modern' coordinate-free notation of differential geometry (which has been around since at least the 60s), and it's easy to introduce Einstein notation on top of it...
- scythe 15y ago
  For me, Einstein notation had the major advantage in electrostatics of reeling in the complexity of really long integrals that show up in boundary-value problems in electrostatics as well as being able to rederive formulas like "curl of curl is div-grad minus laplacian" by hand in seconds (which I always forget -- I took Calc ). I had only a relatively cursory introduction to general relativity last semester; I have at best a vague understanding of Christoffel symbols, to give you an idea. So if Einstein notation can at some level become a way to fling symbols around and forget you're doing physics, I guess I haven't gotten there yet. I would like to think, though, that were it introduced alongside vector calculus instead of several years later, people might connect the adscripts with their meaning more easily. Usually, if I want to understand the geometry of a problem, though, I find the best tool is a diagram, if at all possible.
  - cygx 15y ago
    For historical reasons, many areas of physics come with heir own notation, eg introductory courses on mechanics and electrodynamics are often done using vector notation you know from school with some additional differential operators thrown in, thermodynamics uses differentials, analytical mechanics and general relativity use index notation and quantum mechanics uses bras and kets. Specialization sometimes makes sense, but it's non-obvious (at least it wasn't to me) that when checking if a force field is conserved by computing it's rotation, you're doing the same thing as when computing the derivative of a differential to see if it belongs to a conserved thermodynamical potential, or that the difference between a bra and a ket is the same as between a covector (lower index in Einstein notation) and a vector (upper index) - things look so different that it's hard to see when they are the same. Another example is the relation between Newtonian and Lagrangian mechanics. In the lectures I took, it was presented as if Lagrangian mechanics is somehow special because you have an invariant formulation using generalized coordinates, wheres Newtonian mechanics was only ever done in Euclidean or Minkowski space. It turns out that Newtonian mechanics is as invariant and general as Lagrangian mechanics (however, it's possible to further generalize Lagrangian mechanics, whereas as far as I can tell, you're pretty stuck with second-order system when doing Newtonian mechanics): The Euler-Lagrange-equations are Newtonian equations and the differential of the Lagrange function dL is just a funny way to write down a force field - ie the main difference between Newtonian and Lagrangian formulation is that you require your force to be derived from a generalized potential (more formally: every hyper-regular Lagrangian system is a Newtonian system, any Newtonian system where the force maps to a closed form under the isomorphism T* TM ~ TT* M is locally Lagrangian). In my opinion, lectures on theoretical physics are somewhat broken, and that's a more serious problem than the non-issue of whether to use τ or 2π… PS: Please don't get me started on Christoffel symbols if you're not prepared for another rant ;)
archgoon 15y ago
Let me guess, Biology grad?

by scythe 15y ago

haberman 15y ago

To me it has almost nothing to do with "making it one character shorter," it's about making it easier to see fundamental relationships intuitively. It's not a notational improvement as much as a model improvement. It's sort of like how angles can be measured using degrees or radians, but the derivative of sine is cosine only when using radians. It's not the notation of radians that is better, it's the model.

loup-vaillant 15y ago

The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice. We should perform experiments, but I strongly suspect that using Tau (or 2Pi as if it where a single symbol) would significantly reduce confusion in the heads of the pupils. That would count as a "big advantage" in my book. Now does Tau have an actual disadvantage besides clashing with time constants and such? If not, we should keep in mind that switching is a one-time cost while the cost of using a worse notation is unbounded (proportional to the number of uses, actually). I have to reckon however that making an effort to switch away from Pi probably shouldn't be our first priority. I don't know Gaussian units nor Einstein notation, but if they do "really help" then we probably should take care of that first.

kiba 15y ago
The biggest advantage of Tau over Pi probably lies in learning trigonometry for the first time. That's not something you can judge for yourself, as you can't learn something for the first time twice. If you can live long enough, you could literally forget your trigonometry. After all, humans are not exactly the pantheon of long term memory or even...reliable memory.
- loup-vaillant 15y ago
  If we ever come to that, I want a brain update.

cygx 15y ago

Einstein notation is nice when you actually have to compute something, but it doesn't help in understanding the geometry of the problem. In fact, it might be more of a hindrance because it encourages thoughts along the general line of I don't need to care about the geometry of the problem as long as I know how to calculate the stuff I'm interested in, or as a more specific example It doesn't matter if there's an upper or lower index as I can always contract with the metric tensor. A personal pet peeve is when tensors are introduced as entities with given transformation laws (transforms like a vector in each component, etc) without ever mentioning specific geometric meanings. Thing is, classical tensor calculus makes everything look the same - even things that aren't. I prefer the 'modern' coordinate-free notation of differential geometry (which has been around since at least the 60s), and it's easy to introduce Einstein notation on top of it...

scythe 15y ago
For me, Einstein notation had the major advantage in electrostatics of reeling in the complexity of really long integrals that show up in boundary-value problems in electrostatics as well as being able to rederive formulas like "curl of curl is div-grad minus laplacian" by hand in seconds (which I always forget -- I took Calc ). I had only a relatively cursory introduction to general relativity last semester; I have at best a vague understanding of Christoffel symbols, to give you an idea. So if Einstein notation can at some level become a way to fling symbols around and forget you're doing physics, I guess I haven't gotten there yet. I would like to think, though, that were it introduced alongside vector calculus instead of several years later, people might connect the adscripts with their meaning more easily. Usually, if I want to understand the geometry of a problem, though, I find the best tool is a diagram, if at all possible.
- cygx 15y ago
  For historical reasons, many areas of physics come with heir own notation, eg introductory courses on mechanics and electrodynamics are often done using vector notation you know from school with some additional differential operators thrown in, thermodynamics uses differentials, analytical mechanics and general relativity use index notation and quantum mechanics uses bras and kets. Specialization sometimes makes sense, but it's non-obvious (at least it wasn't to me) that when checking if a force field is conserved by computing it's rotation, you're doing the same thing as when computing the derivative of a differential to see if it belongs to a conserved thermodynamical potential, or that the difference between a bra and a ket is the same as between a covector (lower index in Einstein notation) and a vector (upper index) - things look so different that it's hard to see when they are the same. Another example is the relation between Newtonian and Lagrangian mechanics. In the lectures I took, it was presented as if Lagrangian mechanics is somehow special because you have an invariant formulation using generalized coordinates, wheres Newtonian mechanics was only ever done in Euclidean or Minkowski space. It turns out that Newtonian mechanics is as invariant and general as Lagrangian mechanics (however, it's possible to further generalize Lagrangian mechanics, whereas as far as I can tell, you're pretty stuck with second-order system when doing Newtonian mechanics): The Euler-Lagrange-equations are Newtonian equations and the differential of the Lagrange function dL is just a funny way to write down a force field - ie the main difference between Newtonian and Lagrangian formulation is that you require your force to be derived from a generalized potential (more formally: every hyper-regular Lagrangian system is a Newtonian system, any Newtonian system where the force maps to a closed form under the isomorphism T* TM ~ TT* M is locally Lagrangian). In my opinion, lectures on theoretical physics are somewhat broken, and that's a more serious problem than the non-issue of whether to use τ or 2π… PS: Please don't get me started on Christoffel symbols if you're not prepared for another rant ;)

archgoon 15y ago

Let me guess, Biology grad?