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A trick to eliminate 2π (sometimes)

marci.gunyho.com

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Re: A trick to eliminate 2π (sometimes)

#11

Post by ctoth »

Unrelated to the content but complaining about the website is a popular thing to do here so I'd like to share my experience, as a blind user.

Here is what my screen reader sees for this page:

Recently, I came up with a trick that can get rid of

in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so:

That’s all. You just take the derivative and then divide by

. I call this derivative operator with the bar on the upper

the reduced derivative.Now, why is this interesting? To start off, we’ll note that the unique function

... The unfortunate truth is a ton of math content on the web reads like this. It has crippled me as a blind user who would like to appreciate math for over a decade. In university I was forced to pursue a degree other than CS because the math program used software which produced output like this and refused to change.

There has been technical progress, and many sites are starting to work better--Wikimedia most fantastically, but this old bugbear made me want to speak up and beg people to try and review their math content with a screen reader before publishing (I think MathJax has some built-in accessibility now?).

Re: A trick to eliminate 2π (sometimes)

#12

Post by movpasd »

NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere.

Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!

Re: A trick to eliminate 2π (sometimes)

#13

Post by H8crilA »

Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius?

It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant?

e^(π*i) = -1

e^(τ*i) = 1

Re: A trick to eliminate 2π (sometimes)

#14

Post by ajkjk »

I've thought of this as well, and sorta agree. But I think the real source of the 2pi-s is a bit more subtle? Which is basically that anywhere 2pi shows up is a quantity that is supposed to have different 'units' than the rest of the equation it's in, but for whatever reason we have erased all the units so we keep finding quantities multiplied together with a conversion factor of 2pi. Now one way to handle that is to write Tau or something in its place... but another is to, somehow, erase all the 2pis entirely but keep track of the 'units' on everything.

In fact it is very hard to find places in math where 2pi shows up 'on its own', added to other quantities that are not also in angular units of some sort. That is, most of the pis show up in calculations that involved pi, or circles, in some way (often quite sneakily). Of course it shows up on its own in the circumference/area of a circle, but you can express the area in terms of the circumference, so really it's just about the circumference that has a special value. And I wonder about whether it's possible to just... pick a different value for the circumference, an arbitrary symbol with no value, and then expressing every other use of pi in terms of that one without ever being forced to pick its value.

(Of course when you tie a string around a circle and measure it and it comes out to 2 pi r, yeah, you're forced to pick the correct value. Oh well.)

Re: A trick to eliminate 2π (sometimes)

#15

Post by gorkish »

NotYourLawyer wrote: post #5

Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.

This is more or less "the point" for those of us who argue for tau instead of pi.

I should note that using this "trick" of prescaling rotations by 2pi so that they are in the 0..1 range is de rigueur in computer graphics programming.

Re: A trick to eliminate 2π (sometimes)

#16

Post by jovial_cavalier »

ctoth wrote: post #11

Unrelated to the content but complaining about the website is a popular thing to do here so I'd like to share my experience, as a blind user. Here is what my screen reader sees for this page: Recently, I came up with a trick that can get rid of in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so: That’s all. You just take the…

I'm curious how blind people normally engage with math. For me, engaging with math almost always means conjuring up a visual representation in my mind. Failing that, an equation.

Since visualization is so fundamental to doing math, and since mathematical symbols and equations are a written language for which there is no spoken analog, I really can't imagine engaging with math without my eyes. Even reading equations aloud verbatim is not reliable. "X plus B squared" can mean (x + b)^2 or x + b^2

Re: A trick to eliminate 2π (sometimes)

#17

Post by enriquto »

You can also eliminate the constant in some of the integral formulas by using đx instead of dx. I'm surprised the author does not propose this.

However, some constants will still remain. Most conspicuously, the 2π constant in the very definiton of the Fourier transform. I once took a personal crusade to eliminate all such constants in the elementary Fourier formulas (plancherel-parseval, convolution theorems, commutation with derivatives), and it turns out to be possible by using the Lebesgue measure divided by sqrt(2π) in all the integrals. Thus it may seem that defining đx=dx/sqrt(2π) can be a better choice.

Re: A trick to eliminate 2π (sometimes)

#18

Post by ohwellhere »

garbagecoder wrote: post #10
orangecat wrote: post #2

Interesting. It's probably not worth defining a new constant for exp(2π), but this is a further demonstration of the Tau Manifesto's argument that 2π is much more of a fundamental value than π.

Fundamental sounds like a value judgment. Pi is transcendental. 2 isn't. That's really the distinction. Unless there were other finite factors in pi, that is the number that's always going to have to be approximated in computation.

"Tau is transcendental. 0.5 isn't."

It's definitely a value judgment. But the value judgment is: Is the radius or the diameter more fundamental to a circle?

Re: A trick to eliminate 2π (sometimes)

#19

Post by curtis3389 »

ctoth wrote: post #11

Unrelated to the content but complaining about the website is a popular thing to do here so I'd like to share my experience, as a blind user. Here is what my screen reader sees for this page: Recently, I came up with a trick that can get rid of in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so: That’s all. You just take the…

> review their math content with a screen reader before publishing

I second this for anything you put on the web. I opened the webapp I work on with a screen reader, and it was an incredibly valuable experience. You get to see your site from a different perspective, and various issues stand out like a sore thumb, and I honestly found fixing the accessibility issues extremely satisfying.

Re: A trick to eliminate 2π (sometimes)

#20

Post by tgv »

H8crilA wrote: post #13

Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1

τ = 0, got it.
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