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The Pi Manifesto

thepimanifesto.com

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Re: The Pi Manifesto

#41

Post by sesqu »

MostAwesomeDude wrote: post #36
ianterrell wrote: post #22

1. This is a good discussion to have. Those who are dismissive show, in my opinion, a lack of intellectual curiosity. Elegance for the sake of elegance is a worthwhile goal. 2. From a pragmatist point of view, you're right, it doesn't matter. You continue reading and writing PHP and using π. They both get the job done. You don't have to participate any further. 3. There will be 2s floating around some equations forev…

The area of the traditional unit circle is π, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of π is related to the integrals used to derive arc length. Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a…

Why do you refer to the area? Why not the diameter or the circumference? If you care about the area, why not use the unit circle with area 1?

A quarter arc of a circle is π/2 radians, regardless of area.

Re: The Pi Manifesto

#42

Post by mikeash »

ianterrell wrote: post #22

1. This is a good discussion to have. Those who are dismissive show, in my opinion, a lack of intellectual curiosity. Elegance for the sake of elegance is a worthwhile goal. 2. From a pragmatist point of view, you're right, it doesn't matter. You continue reading and writing PHP and using π. They both get the job done. You don't have to participate any further. 3. There will be 2s floating around some equations forev…

I consider myself to have intellectual curiosity but don't see this discussion to be particularly compelling. I don't think there is anything about intellectual curiosity which requires one to consider every question that's raised no matter what. There simply isn't enough time to consider everything, so prioritization is always necessary.

Re: The Pi Manifesto

#43

Post by ianterrell »

mikeash wrote: post #42
ianterrell wrote: post #22

1. This is a good discussion to have. Those who are dismissive show, in my opinion, a lack of intellectual curiosity. Elegance for the sake of elegance is a worthwhile goal. 2. From a pragmatist point of view, you're right, it doesn't matter. You continue reading and writing PHP and using π. They both get the job done. You don't have to participate any further. 3. There will be 2s floating around some equations forev…

I consider myself to have intellectual curiosity but don't see this discussion to be particularly compelling. I don't think there is anything about intellectual curiosity which requires one to consider every question that's raised no matter what. There simply isn't enough time to consider everything, so prioritization is always necessary.

I agree with the sentiment expressed in your last sentence, but it's hard for me to imagine that one wouldn't give priority to something so fundamental as the circle constant (which touches so much in math, science, engineering, software, etc). That's doubly true considering the time investment for reading about the arguments is rather small, 15 minutes or so. And it's triply true considering that very many other people have vetted the argument for you, enough that mainstream media are writing articles on the topic.

So no, you don't have to consider every question that's raised "no matter what" to be considered intellectually curious, but this is hardly such a trivial, general case as that implies.

Re: The Pi Manifesto

#44

Post by edw »

sesqu wrote: post #37
edw wrote: post #16

Earlier quoted context omitted.

I would add that this stupid tau thing speaks to the conspiratorialist instinct common to many HN readers. You see, the self-evident truth of tau's superiority has been masterfully obscured by powerful dark forces in an attempt to protect their crude economic self interest, and if you don't agree, you're obviously either part of the conspiracy or one of the sheeple that's been snowed by it.

I have had that thought, but find it incredibly hard to entertain. Do you genuinely believe it, or is it just something that explains why others' opinions might differ?

I should have put quotes around everything following the first sentence and attritubted it to a hypothetical (and straw-man) tinfoil hat-wearing tau advocate.

Re: The Pi Manifesto

#46

Post by sesqu »

amalcon wrote: post #38
sesqu wrote: post #34

Earlier quoted context omitted.

I don't think in radians, I think in cycles (radians/tau). I used to think in degrees, and could never get the hang of radians. Unfortunately, I cannot imagine the mathematical world moving to cycles, ever. The most I can hope for is that teachers introduce others to a single intermediary constant (tau) rather than two (2,pi).

The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians. I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived…

The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot, or the integral of a centered quadratic function), but not ones considered fundamental.

Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.

Re: The Pi Manifesto

#47

Post by mikeash »

ianterrell wrote: post #43
mikeash wrote: post #42

Earlier quoted context omitted.

I consider myself to have intellectual curiosity but don't see this discussion to be particularly compelling. I don't think there is anything about intellectual curiosity which requires one to consider every question that's raised no matter what. There simply isn't enough time to consider everything, so prioritization is always necessary.

I agree with the sentiment expressed in your last sentence, but it's hard for me to imagine that one wouldn't give priority to something so fundamental as the circle constant (which touches so much in math, science, engineering, software, etc). That's doubly true considering the time investment for reading about the arguments is rather small, 15 minutes or so. And it's triply true considering that very many other peo…

I guess the pertinent question is whether this stimulates your intellectual curiosity because you find it to be an interesting question, or at least because it raises interesting questions, or whether it stimulates your curiosity because you think it's exceedingly important and making this change will have big consequences.

I think the dismissive types see people obsessing over it as the latter, and this is, at least in my opinion, way more than is called for. There's nothing wrong with considering the consequences of such a change just for some mathematical fun. (After all, who here hasn't drawn out a complete system of units based on furlongs, fortnights, etc. as the fundamental units, rather than mks or cgs? What's that? None of you? Oh....) But the original manifesto seems to be overstating its case severely in terms of the consequences of such a change, and I think that's what others are reacting to.

Ignoring the boring costs of switching the installed base of mathematicians, software, etc. from one constant to the other, the benefit of a switch seems to be that certain things become easier and other certain things become harder. At best, there's a small net gain. Hardly seems worth the discussion when considered in that light.

If you're considering it as an interesting exercise to extract meaning from equations, well, go for it! But that seems to fit more under the banner of "What would happen if we switched this?" rather than "It would provide enormous benefit if everybody switched this!" as the original manifesto seemed to be saying.

Re: The Pi Manifesto

#48

Post by ianterrell »

mikeash wrote: post #47
ianterrell wrote: post #43

Earlier quoted context omitted.

I agree with the sentiment expressed in your last sentence, but it's hard for me to imagine that one wouldn't give priority to something so fundamental as the circle constant (which touches so much in math, science, engineering, software, etc). That's doubly true considering the time investment for reading about the arguments is rather small, 15 minutes or so. And it's triply true considering that very many other peo…

I guess the pertinent question is whether this stimulates your intellectual curiosity because you find it to be an interesting question, or at least because it raises interesting questions, or whether it stimulates your curiosity because you think it's exceedingly important and making this change will have big consequences. I think the dismissive types see people obsessing over it as the latter, and this is, at least…

I won't belabor the point any further than this, but the argument really is that it would provide enormous benefit. It's not just a "what would happen?", and it's not that τ "extracts" meaning from equations. It makes the intrinsic meanings of equations drastically more clear. For instance, what does sin(x) mean? What does Euler's equation mean? They're both eminently simple concepts, but they remain obscured by π.

It's exactly like refactoring code: sure, it does the same thing, but now it's more compact, more concise, more clear, more elegant; the parts of the system all fit together better, and people coming onto the project will be able to learn it faster. If you don't care about those things, then you won't refactor your code, or see the point of τ.

Re: The Pi Manifesto

#49

Post by lotharbot »

fexl wrote: post #4

When you ask hard-core Tauists what the area of a unit circle is, do they actually answer "tau over two"? Or do they just say "pi"? By the way, this whole discussion reminds me of what W.V.O. Quine called "mathematosis".

It's the integral of tau * r * dr from 0 to 1, which is tau/2.

Re: The Pi Manifesto

#50

Post by MostAwesomeDude »

sesqu wrote: post #41
MostAwesomeDude wrote: post #36

Earlier quoted context omitted.

The area of the traditional unit circle is π, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of π is related to the integrals used to derive arc length. Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a…

Why do you refer to the area? Why not the diameter or the circumference? If you care about the area, why not use the unit circle with area 1? A quarter arc of a circle is π/2 radians, regardless of area.

(To answer your question, the unit circle, with radius one, has area π.)

A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3

So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with.

Maybe keeping the unit circle at its current size is a better idea.

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