The Tau Manifesto - With Michael Hartl
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- Опубліковано 16 чер 2024
- The Tau Manifesto is dedicated to one of the most important numbers in mathematics, perhaps the most important: the circle constant relating the circumference of a circle to its linear dimension. Originally published on Tau Day, 2010, it introduced the idea of using τ for the circle constant C/r, which has now become a mathematical and cultural phenomenon. This is a video version of the manifesto and is the result of a collaboration between me and Michael Hartl, the author of The Tau Manifesto. I did the animations, he did most of the writing, and as you can see in the video, we both did the narrating.
Here's a link to the written form of The Tau Manifesto: tauday.com/
Discord: / discord
Patreon: / sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
LoganMP
Richard Penner
Rosario
Credit for the image of Conway at the beginning goes to Thane Plambeck via Wikimedia commons, under the conditions of the CC BY 2.0 license (creativecommons.org/licenses/....
Sections:
00:00 Introduction
00:54 An Immodest Proposal
02:51 A Powerful Enemy
03:51 The Number Tau
04:46 Circles and Angles
08:32 The Circle Functions
10:16 Euler's Identity
14:09 Circular Area
17:28 Reasons for the Letter
19:00 Ambiguous Notation
22:25 Social Proof
11:47 The identification of τ with "one turn" makes Euler's identity sound almost like a tau-tology.
As a kid, I got constantly confused by what pi was, and always thought it was the Circumference over Radius. So to now hear that I was thinking in *Tau* this entire time... I cannot believe it has taken society this long to finally get the info to me.
yeah... its "societie's" fault for you not being able to seek out UBER trite and simple and irrelevant info. clearly there's a conspiracy for pi. wana know about it? buy Hartl's empty book of triteness!
surprising to see you here
@@MithicSpirit is it? 👀
You weren't wrong. The mathematical community was wrong 😂
@@joefuentes2977 well that's the thing! I realised that it made sense in my head, but I was just calling it "pi", when I was thinking in tau
This is especially true in physics. The number of theoretical physics formulae that contains (2pi)^n is ridiculous, and often the equations are much more beautiful (and understandable) when written with tau
Sure, but you’d also get plenty of instances of tau/2 as well, like Fourier series and transforms or the stationary-state solution of a quantum particle in a potential well. Physics formulas honestly have arguments for both, depending on what you’re looking at - some equations look significantly nicer when you use pi, and some look significantly nicer when you use tau.
@@yourewrong9028 Fourier series and transforms and the inverse Laplace transforms are written in terms of 2pi. Maybe you are refering to sine and cosine transforms, which in some conventions they are written in terms of pi. The most fundamental physical formulas are written in terms of tau=2pi and 2tau=4pi, which are the perimeter of a circle and area of a sphere respectively.
@@alberto.hijano Exactly. Physics fundamentally arises from symmetries, and of the most fundamental of these is rotations. This is why tau is the fundamental constant - it is the phase of a single full rotation. Why would half a rotation ever be fundamental. Fourier transforms are a functional that translate functions between coordinate and frequency dimensions. If you translate from coordinate to frequency then to coordinate, you pick up a factor of unit rotation i.e. tau. Therefore in order to formally define an inverse transform, the overall factor of 1/tau is required. The appearance of this is again fundamentally from the connection between frequencies and rotations.
Unless there is a meaning to a half of the gravitational acceleration constant, I’m sold!
There's a funny coincidence in the symbols. Tau has one 'leg', and corresponds to one turn. Pi has two legs, and is half a turn. So either symbol represents one turn divided by the number of legs. Imagine the pi symbol but with 3 legs - call it 'tri'. It represents a third of a turn (120 degrees), and so on.
I've jokingly proposed this notationally too! Can't see it being practical but have found it very funny to do.
@@Quargos If anything it's just a reminder which is which for someone first learning it.
what about a symbol with no leg?
@@uamru2933 Infinity!
Read the Manifesto: Bob Palais's original proposal used a pi with three legs instead of tau. :-)
You could also state that e^(i*Tau/2) = -1 + 0i shows an interrelationship between *_six_* fundamental numbers: e, i, Tau, 0, 1, and 2! Moreover, it *_also_* uses six fundamental algebraic symbols: +, -, *, /, ^, and =. Furthermore, the combination of ^ and the (1/2) in the exponent is one way to define another algebraic symbol, the square root!
And finally, as explained already in the video, it directly connects polar and rectangular forms of complex numbers.
Indeed, if you want to go really crazy and 'fully express' the polar form, with its modulus of 1 (being the square root of 1 squared),
[ (1^2 + 0^2) * e^(i*Tau) ]^(1/2) = -1*1 + 0*i
then you are exhibiting *_eight_* fundamental numbers: e, i, Tau, 0, 1, -1, 2, and 1/2, as well as many of the algebraic/field axioms/theorems such as existence of additive inverses (-1), multiplicative inverses (1/2), multiplicative identity (multiplying by 1), additive identity (adding 0), multiplying by 0 gives 0 (0 times i), multiplying by (-1) gives the additive inverse (-1 times 1) etc. Even Pythagoras' Theorem shows up in the calculation of the modulus.
You could go on and on, but that's kinda the point: It's kind of arbitrary which is 'most beautiful', containing the 'most fundamental' elements in it.
@@ebog4841 Ok but tau is more MEANINGFUL than pi :)
@@JwalinBhatt Oh, I'm not disagreeing with Tau being better than Pi, only referring to the 'beauty' of Euler's Identity.
@@robharwood3538 All cool then 😀
@@JwalinBhattif you ask an ancient Greek mathematician, they would tell you both are absolutely meaningless 😂
@@joefuentes2977I don't see why one would do that. And it seems quite impossible, anyway, considering that the ancient Greeks are all dead.
As a game developer I nearly always define TAU when working with geometry. This gets rid of all 2 * PI mentions in the code, and in code the pi constant is typically invoked from a library, so it's never just 2 * PI but 2 * SomeMathLibrary.PI. Having TAU in place of it makes it universally more readable and just all around better. And in my line of work I've never had a conflict with fellow developers most of which are also tau fans. So I really trust this shift to tau to be a slowly changing game of a very silent peer-pressure.
I come from a C ,C++ background so your post makes sense.
Tell me you watched freyas' videos without telling me you watched freyas' videos :P
@@_justnick how? what? I did but it's unrelated wth
it could be that this comment is older than her video, but I can't check for it, maybe it was subconscious and you're right.
@@milanstevic8424 i was half joking, lol. She's really into tau. It's not older, she just uses tau overall for everything, including her math for programmers playlist.
@@_justnick She does yes, I count her as my own tribe when it comes to game dev, having seen her libraries, we're pretty much likeminded. You can find some of my tutorials on Unity forums I'm orionsyndrome there.
I have recently started to call pi the semicircle constant. Multiplying the zero by i in the tau version of Euler's formula is a more recent update, right? I don't remember having seen that, last time I read the tau manifesto. Before, it was just a laconic: "...Fine: e^(i*tau) = 1 + 0", right? Writing it as "e^(i*tau) = 1 + 0*i" makes a lot of sense and is a good observation!
All you needed to do was show me the circle/angle illustration and I'm convinced!
This entire thing feels just like what I've been thinking for a long time. Thanks a ton for making this! I hope that tau becomes more recognized.
This video is great. Definitely sold me, both for 6.28... being the circle constant, and for notation tau. I think the only problem this video didn't address is how to deal with appealing to tradition. I.e; I'd happily use tau personally, but I don't know what I'd do when faced with someone challenging my usage in a public setting or in a research paper, persay. "Do I really love tau enough to always expend the energy to clarify my usage to others' resistance?" I don't know. Of course, the more people use it, the less of a problem that is, but what makes a good manifesto a great manifesto is how to address social aspects like this, I think.
In any case, videos like these are an excellent step,
happy tau day!
I've always wondered why we use the diameter of a circle only when talking about pi, but use the radius for everything practical. Seems a massive oversight that's just been going on forever because of convention. Kind of like linear algebra instead of Clifford algebra. Great video!
Me too
June 28 is also Perfect Number Day.
What do you call an animated math manifesto?
A Manimfesto
... A manifestau? ^^
@@SupGaillacThat's the name for a manifesto about tau. The name for an animated math manifesto about tau, of course, is "manimfestau".
This shows why π is actually the _semicircle_ constant.
wearing my official Tau shirt that I bought probably near on a decade ago, whenever the first run of them was, as I watch this lovely animation. Happy Tau day from Australia!
Oh my, happy Tau Day! I didn't even notice, shame on me! I haven't completely adopted Tau in my everyday math work for now, but the concept of "one turn" and the "area integral" thing have stuck with me since I read the original manifesto. And also I vaguely remember that there was some discussion of a third constant λ=τ/4, which has relevance for the calculation of surface area & volume of higher dimensional spheres? (where Pi just doesn't fit into the equation whatsoever) I gotta reread this sometime. And go back to evangelizing.
One thing I learned when looking at tau is that surface and volumes of higher-dimensional spheres/balls are best calculated using a two-term recurrence, which -- needless to say -- involves tau. This saved me a great deal of time in places where I'd previously used the Gamma function.
I have collected some quotations about the τ/4 concept here: en.wikipedia.org/wiki/User:Waldyrious/Tau/Right_angle - perhaps it may help you refresh? Btw, any suggestions or corrections to that page are most welcome!
12:43 exp(iτ) = 1 + 0ι
This explicitly says the full meaning of the expression calling attention to the sometimes invisible 0ι.
Pi and tau are both equally valid *_IF_* you're actually being consistent with the definition, and for example switch to "diametrians" as your angle measure in the case of pi.
The issue is that no one actually _wants_ to use pi correctly since most problems involve some kind of vector rotating around its tail, which only really makes sense to think of as a radius.
So instead, people end up mixing two different unit systems and sprinkling conversion factors all over the place.
The only reason they don't notice what is going on is that 2 is a much more inconspicuous conversion factor than something like 360/2pi.
It has nothing to do with conversion factors. Radian measure is radian measure and defining tau as 2pi is not a conversion factor. This is like saying 4 is a conversion factor of 2 since 2*2 =4. A conversation factor has 'units,' nut just a numerical value
@@pyropulseIXXI You've misunderstood what they're saying here, and likely arguing against a point was not made.
First of all: "A conversation factor has 'units,' nut just a numerical value" - This is already contradicted as mentioned above by the 360/2pi conversion factor, which is arguably unitless.
The point though is that if working in terms of the diameter of a circle, pi would make sense, we'd have pi "diametrians" as suggested above, in a full turn. But that's not the case, we work in terms of the radius, while using a constant defined by the circle's diameter. Hence creating this conversion factor of 2, which is a lot easier to just kind of ignore than a messier conversion factor, like say 57.296. (Which is the value of the 360/tau conversion factor for degrees radians as mentioned above.)
@@Quargos 360/2pi converts to double degrees, you oaf; they are dimensionless but not unitless
@@pyropulseIXXIWhat are you even talking about?
@@Aurora-oe2qp herp derp
Congratulations, you have convinced me that indeed tau would be the more natural choice over pi. As for my usage of tau vs pi... well, honestly I don't do enough math for it to matter lol
I'm more of a 90 degrees man. It's the "jump into a new dimension" that captivates me.
Yeah but a full turn is clearly a more fundamental thing than a quarter turn, is it not? And sure, there are some very few cases where a full turn is actually not the most fundamental rotation, with a double turn instead being it, but that's quantum mechanics and it's weird.
Minor nitpick: the double cover aspect of spinors is not a quantum phenomenon. It's just due to the fact that the best way of representing rotations (rotors in geometric algebra) ends up rotating the vector twice, so you need to use half the angle you want in the spinor.
We also mustn’t forget that the legend himself, Euler used π to denote 6.28… not 3.14…
if π is for circle, then π is 2π
Ummmm, ua-cam.com/video/bcPTiiiYDs8/v-deo.html
Who is the idiot that tried thought "im going to come up with something better than Euler" ?
I think I saw that in a 3b1b video. Iirc, pi wasn’t a universal constant back then, so Euler always defined it to be whatever was most convenient for the problem at hand, be that one turn, a half turn, or even a quarter turn.
Edit: here’s the video I mentioned ua-cam.com/video/bcPTiiiYDs8/v-deo.html
@@isaiah0xA455 Yes, that is the link above.
@@isaiah0xA455 Oh wow so it got erroneously assigned to 3.14 afterwards because of someone being biased? I'm just realizing, of course Euler meant for pi symbol to only be a placeholder for whatever he needed at the moment, turn-wise, mostly because he was mainly experimenting/toying with it. The idea that we now must do exactly as the textbook says, or else people will get angry / reprimand or expel us, is completely unfun and non-Euler. Such solemn status quo, no wonder we don't have people like Euler any more.
I can't be the only one who has often mistakenly used pi/8 instead of pi/4 when I wanted an 8th of a turn / 45 degree angle. Same with pi/2, but I seem to be most error prone with small angles like tau/8, tau/16 etc, That's despite years of conditioning at university and "experience" using pi. If people are anything like me, I bet pi has caused real and significant physical losses and costs because of similar mistakes.
I thought that "the powerful enemy" section was just a joke bit before the history section, but after reading the comment section, that joke went meta.
I fully agree that tau is the superior circle constant.
However the main thing preventing its widespread adoption is the existence of pi. There is no need for both to exist concurrently. In an ideal world tau would replace pi or pi would be redefined to 6.28... but alas social inertia is what it is.
Also would like to add that my main work involves modelling plasma in a torus and the amount of hours i've wasted searching for a bug, only for it to turn out to be a factor of 2 missing from a 2*pi is astonishing! I would like to move to a tau based approach but this has other issues such as other people auditing my work and expending more effort when translating maths from papers to the computer.
Dang... I remember translating algorithms out of papers a mere 30 years old and I had to work out translating not just the symbols but almost the entire mathematical and metamathematical framework. It's even worse going back 50-60 years. Numerical algorithms must have it easy compared to programming language semantics!
Props for submitting this in a contest ran by a math guy commonly featuring pi mascots in his videos.
... and who has stated a liking for tau.
Having done some Geometric Algebra, we should have one turn of a circle as Constant/2. That way two turns gets you back to the start. See physics too. :-)
12.566370614359172953850573533118 is the _real_ circle constant!
as a warm up towards getting to world to adopt tau how about starting with the easier task of getting to US to switch to metric
ok yea wow we're fuked
What I would suggest is to use both tau and pi, but to favor tau, and reserve pi for contexts where the letter tau is already well established to mean something else, such as when talking about torque in mechanics, as a way to avoid confusion. This sort of double-notation kind of thing is already done when it comes to the unit imaginary number "i" - electrical engineers typically write "j" instead to not confuse it with alternating or variable electric current (written with a small "i", unlike DC which is written with a capital "I").
Ok, I'm in. It's so unbelievably clear that the correct circle constant is Tau, everybody that truly understands maths knows this. How do we get it into schools? I predict this will go down in history as one of the biggest advancements in science ever. Up there with realising we are not the centre of the universe.
3:30 What's even worse: They only remember the first digits of decimal π. Why not hexadecimal, octal, binary, seximal or niftimal?
hello fellow Jan Misali fan
@@ataraxianAscendant Oh, how did you know 😳
"niftimal" is just a normal word everyone can use!
hexadecimal sucks. Not even a superior highly composite number or a colossally abundant number. Dozenal is clearly superior, as evidenced by it's broad usage in history. BTW, the base 60 systems are essentially dozenal, but you count 12 in groups of 5. Base 60 could work too but I do like the consistency of pure dozenal.
@@Aurora-oe2qp I just included hexadecimal since it's very popular. It's useful for programming. But which number system you use doesn't matter for the point I'm making.
"They only remember the first digits of decimal π."
How do you know?
Once you've 3b1b on board, it's only a matter of time. I'm convinced. I've always felt that angle units should be full turns, i.e. 1 unit = tau radians. It doesn't usually save me any time to know that the ratio with the circumference is already in there; as a coder, I'm mostly passing angles off to trig functions anyway, and it would be easier to type 1/4 than Math.PI/2.
Is this the big project you were talking about? Thought it'd be about GA but im definitely not disappointed. This is great!
This is actually not the big project! Tau Day just snuck up on me. I was going to make a community post about it but forgot.
@@sudgylacmoe THANK YOU this isn't the ACTUAL project! thank god, too. whewwwwwwwww!
@@sudgylacmoe what are your thoughts on π as the ratio of the AREA of the circle to its radius squared?
@@ValkyRiver As mentioned in the section at 14:09 in the video, the factor of one half belongs in the area of a circle.
23:15 If τ is supported by πthon, why isn't it calley τthon?
Python supports both pi and tau.
Tau is obviouisly the correct letter. Write 2 Pi. Now simplify the two legs of Pi with the factor of 2 so that only one leg remains. The result is Tau ;-)
By this argument, wouldn't it make more sense to define tau = pi/2 because the glyph looks half of the one of pi? 😬
@@13ohmsWe're interpreting the top stroke of the letter π as a fraction bar.
The video is great! I would also like to point out that the online mathematics engine Desmos already support the use of tau. just type tau and it will replace it by the character τ
I can't believe you ended the video with 'There's no stopping tau now' when 'Don't stop me _tau_ ' was _right there_ .
But that sounds like, "Don't stop me, tau," as if the number tau is trying to halt this individual's journey.
Happy τ day!
I completely agree, except that to conserve scarce Greek letters, I think the symbol should be called 2pi.
get a load of this guy
We could name all the formulas in the same way for the same purpose 😅
How the tables have tau'd.
Great video, will surely be using tau over pi, much easier to use
what about hyperbolic angles?
And this video was published on Tau day! 28 June or 6/28
i have no words left, because you said all of them!
this needs more recognition. lets get Tau the glory it deserves!
its also easier to write than pi
The sarcasm! 😂💀💀💀
12:06
Suppose the circle constant was discovered just two weeks ago; we certainly would choose τ to be the circle constant instead of π, but we don't have to abolish π. Personally, I will be using both. It does make the classical formulae more comprehensible, and it has a aesthetic value too, however.. tradition is on the side of π, and in some formulae π is better. 🥧
16:59 Correction: The circular sector is not subtended by the angle. The angle is subtended by the arc boundary of the sector.
Happy tau day!
I can't say I've ever had trouble with pi or anything like that, but I can't deny that tau is almost always much easier to read. I'll definitely be celebrating tau day from now on
Nice username.
@@isavenewspapers8890 Thankies
I'll say that for me it's never the pi itself that gives me trouble, but rather the associated 2 😮💨
One day, tau will win !!!!
τ > π
Lol literally
Tao is twice the number pi ever was.
SICK burn- all the "pi enthusiast cultists" will NEVER recover. (NOT CLICKBAIT)
VERY mathematics
SUCH mathematics
MANY interesting
wow.
Happy Tau day friends!
I think the hardest problem Tao has to overcome is that people like celebrating by eating pie.
eat 2 pies in June, then
@@eldersprig Aww heck yeah.
I think they’ll like to eat 2 pies instead of just one! So I don’t see an issue there.
I'm all for Tau. But I might be a bit biased since June 28th is my birthday 😂
Congrats on going one more tau around the sun!
Happy birthday! (belated)
In ideal world pi should be equal to 6.283.. Instead of creating new constant and also because pie is circular.
As A Chinese, very humble to learn from teacher up. I also don't know this.
21:11 Correction: a_0 = 2τ*ε_0*hBar^2 / m_e * e^2
where ε_0 = 8.8541878128e-12 [Farads/meter] electric permittivity of space,
hBar = 1.05457181764616e_34 [Joules/second] Planck's reduced constant,
m_e = 9.10938370157334e-31 [kg] the mass of the electron,
e = 1.60217663400000e-19. [Coulombs] elementary charge
without the ε_0 the units of the formula presented are [m^4 kg C^-2 s^-2] (awkward) not [meters]
Would be really nice to see how Tau works for all the various sums and products usually used as identities involving Pi, such as the sum 1/i^2 = Pi^2/6 or the alternating sum +/- 1/i = Pi/4. Etc. Do these likewise give 'nice' versions involving Tau?
Perhaps a follow up video?
Just substitute Tau/2 for pi. So Tau²/24 or Tau/8
@@canaDavid1 That's not necessarily all you have to do. These identities might not be useful in that form. You could go on and mutate both sides of the equation after the substitution to make both sides either aesthetically more pleasing or more likely to show up in actual applications.
I have been a 2pi supporter ever since I read the pi is wrong article, but I have never used it in my academic life due to the confusion that it might lead to. My main concern is that there is no consensus on which symbol to use to represent 2pi. One thing I do not like about the tau notation is that, as mentioned, the tau symbol already represents several magnitudes in science (proper time, many characteristic times, Pauli matrices in spaces other than spin...), while pi is almost exclusively used for the circumference to diameter ratio. In addition, the argument about being easy to remember due to being related to the word turn is a rather anglocentric argument. I prefer the three legged pi symbol proposed by Bob Palais, which is a symbol without any other meaning and it can be related to the Greek word περίτονος (perítonos), which is related to one turn. It has the additional benefit of being a duplicated pi symbol, so it would be easier for the broad audience to identify it with 2pi.
The only issue I've had with using τ instead of π is that in my college homework sometimes the professors wouldn't see the bolded, boxed, and colored statement at the beginning saying that τ = 2π and saying I'm off by a factor of two (which was easy to get corrected by just pointing it out to them). I've never had issues with conflicting notation. And honestly I personally don't care about turn and τ starting with the same letter. The main point to me is that we need a letter that is easy to understand and use, and at this point τ has gained a lot of momentum so it's best to just use τ.
22:22 This is different. There are PLs, where functions and values (constants, variables, etc.) don't live in the same namespace. (Lisp-1 vs. Lisp-2)
pi is also sometimes used for the canonical momentum density
and projection operators, and a buncha other things i can't member right now.
The point is that what you just said- IS MATHEMATICS (albeit- applied to physics)
the argument about "OMG pi is so inferior buy my manifesto book durrr" ISN'T EVEN ABOUT MATHEMATICS AT ALL.
@ebog4841 Nor is "e^πi + 1 = 0" including "all five most important constants". It's purely numerological. Why should this equation include "all five most important constants" anyways? And are these really the five most important constants? I don't think it's fair to claim that Michael Hartl is just doing this to make money either. This video is free and the manifesto is freely available pretty much anywhere on the Internet. That's more than can be said of most maths articles, is it not? To actually adress if τ is good, well, yes it is. It's a turn. A turn is surely more fundamental than a half-turn, no? It does clean up the fractions of the circle a lot and 2π shows up a lot. Even when π shows up, that's because there's a factor of a half in the equation or it has to do with a semicircle or half-rotation.
@@ebog4841 You are saying they want us to buy their book… On a free presentation pn the books contents.
@@ebog4841You have a very forward approach to being inflammatory on the internet, it seems.
Blasphemy, pure Blasphemy!
Edit: Well, after the video i am now kind if convinced, i guess i will start learning the digits of tau now...
GO BUY HARTL'S SHITTY BOOK THEN
@@ebog4841The fuck is your problem?
I want to apply to join the Tau cult! Unleash the Tau-ism on the world!
2:19 the prime counting function doesn't seem weird to me, because it has the (x) so it's easy to remember it's a function, and the π alone is just π, so to me this case isn't notationally ambiguous.
Every other usage of τ seems more readable, except for A = πr^2 = (τr^2)/2 because the version with π seems more clean, but the version with τ has that pattern with other formulas so having both formulas is more pedagogical.
So in the end I like both, so I'll began using τ where it better fits which is a lot of places.
Also if I recall correctly the gaussian integral is √π which would be (√2√τ)/2 = √τ/√2 so this time π wins.
Edit: Some formulas look weird with τ and I guess it's just a matter of time, so I also appreciate π for being so iconical.
I advocate a new symbol: a three-legged pi, just like the aliens used in Tom Swift and His Flying Lab, although its meaning was never deduced.
All of you advocating for three leggedness just don’t understand why tau is better than that. Actually it would be best to come up with some very well suited geometric shape or something like that. Problem is people need to use the constant in their papers. So you have to use some of the existing letters in any language that exists in our computers (in all or the most of the fonts)! And that’s just more realistic if know the world you live in.
In regard to the section at 16:35, I wonder if e=mc^2 has a role to play in this. It is so close but so far from fitting into that (1/2)kx^2 formula. Where would the half come from in that situation, and if not, then I suppose the pattern isn't as ubiquitous as I thought.
Just thinking about it from the perspective of what an integral is, it wouldn't seem to make much sense for the pattern to hold. (1/2)kx^2 comes from integrating kx with respect to the variable x. However, the speed of light is not a variable, so you can't integrate with respect to it.
The full formula is actually E^2 = (pc)^2 + (mc^2)^2, which has more to do with the Pythagorean theorem. E=mc^2 is just the special case when the object isn't moving (relative to you) and therefore has no momentum p (in your reference frame).
@@okuno54 There are some subtle issues surrounding "rest mass" vs "relativistic mass" or "invariant mass". E=m(r)c^2 in all frames, and E^2 = (pc)^2 + (m(0)c^2)^2 in all frames. - en.wikipedia.org/wiki/Mass_in_special_relativity - In the past I think the idea of "relativistic mass" was given more weight in introductory Special Relativity instruction (pun not actually intended). More recently, the use of the invariant mass only has become more prevalent.
I really prefer using tau, but the only situation where it just doesn't seem to fit as nicely is with the Gamma function / generalised factorial. (-1/2)! = sqrt(pi) is just such a nice result, and tau adds in a factor of sqrt(2) into all the half-integer factorials. To y'all who use tau more than I do, how does one remedy this? I haven't been able to come up with a easy fix.
When deriving these results, you find that this is yet another result of one half cancelling with the 2 in 2π, and the simplicity is just accidental. In general, the reason I prefer using τ is not that it makes equations simpler (although it does in most cases), but that it just makes more sense.
But then again, you already have 1/2 floating all over the place there!
Gamma(1/2) = Pi^(1/2)
But why not illuminate the hidden 1/2 within Pi, by writing it as
Gamma(1/2) = ((1/2)Tau)^(1/2)
@@sudgylacmoe Completely agree that it is useful because it makes sense. It is as a result of the 1/2 tau r^2 area formula, which is why pi shows up here without its fellow 2. But it does unfortunately result in much less-nice formulae when dealing with half-integer factorials. It's just a shame tau doesn't make everything nicer.
@@sudgylacmoe I had to unsub, because using tau or 2pi literally does not matter and you are just making ad hoc justifications to use pi (if tau was the expected convention, I would say the same as trying to change to 'pi)
The definition of radian measure doesn't change
Saying it is "accidental" is stupid. If you see a case of 3pi, but call it omega, I'm sure you would see how stupid this entire thing is
Tau or 2pi..... they are literally the same thing, so nothing changes. This is not mathematics; math starts from axioms and deduces things; this has nothing to do with any of that and is just idiots arguing over nothing
This is why anyone that engages with this nonsense I simply ignore
@@pyropulseIXXI"This is why anyone that engages with this nonsense I simply ignore"
You have posted a large number of comments in this comment section engaging in highly heated debates with other people over the matter.
I think there's really no conflict with that. Use tau whenever you want. You just have to use it and that's it. Add its definition before any proof and you're done.
If tau is better than pi, just use it and time will put it where it belongs.
Do you think you'd say the same thing if instead of starting out with pi, we started out with pi/2, and instead of tau advocates, we had pi advocates?
If we started out with tau, using it for centuries, would you find it reasonable for someone to advocate to have a special symbol for tau/2?
I think the only reason to advocate for pi is because of its historical precedent, and nothing else.
@@metachirality Of course. Everybody knows pi and very few knows tau.
If the only problem is having to define it from pi, it's not the end of the world.
But if you use tau enough, and is truly useful, in the end it will prevail without pi.
I just finished relearning trigonometry today before going back to calculus. I can't deny, it would have been easier if tao is the convention.
Wasn’t the π and τ both invented by Euler?
(That was long time ago I saw the video that said that, so I may forgot and saying the wrong thing)
π was invented by Euler, but τ is very recent.
I very much agree with Tau being superior to Pi, but I have one question that's been bothering me recently. Why use radians at all? You mentioned that Tau radians is perfect because it's equal to exactly one turn. Well, that's basically skipping over the most obvious question. Why not use a TURN to measure a turn? Why measure angles as Tau/2, Tau/4, etc., when we can measure them as half a turn, quarter of a turn, etc. It's almost so obvious that they don't even look like units. Why are we measuring circumferences using radii instead of using circumferences themselves? Is the only answer that it makes the Taylor Series expansions and the derivatives look nicer? Is there no truly geometric reason?
I find the sine and the cosine to be most naturally defined as being functions representing oscillations, and in that context, they are most naturally defined using radians (as the solution to the differential equation y'' = -y). This means that almost everywhere in calculus, analysis, and physics, it's better to use radians than anything else. Even in geometry it's often better to use radians, such as when representing rotations using bivector exponentials.
Also, using τ is basically the same thing as your turn idea! You can just think of τ as being one turn and then you can forget about it's numerical value.
I think the answer to your questions is that you still have to think about the turns (any angles) as angles. Which means you still have to have a letter for it, otherwise we may get confused (if there are just ratios).
The most general application of π and τ would have to be the polar blow-up, where you switch between cartesian coordinates and polar coordinates.
This transformation comes in many varieties, it *explains the substitution of sine and/or cosine in integrals like ∫√(1-x²)dx.
You can also see eⁱˣ by thinking about it like it simulates this blow-up.
Since this blow-up represents π and τ in general, you can look whether half rotations (π) or full fotations (τ) are more useful.
Usually, τ wins out, but there are some cases in which half rotations are useful. They are both more useful that 2τ and ½π.
I don't think π should be discarted, as it has enough interesting places to be its own little constant.
*When given a circle with diameter 1 though the origin, you can look at a turning line though the origin, and take the other intersection with the circle as P.
Now, you can describe sine and cosine as the measured lenght of chord OP.
I think this is really cool.
Будучи школьником у меня всегда вызывало недоумение - почему константа Пи это длина окружности на диаметр, а не на радиус... Отношение на радиус просто напрашивается элементарной логикой, так как никто не рисует окружность используя диаметр и эта двойка реально все время мешает... Эти внутренние противоречия подтвредила именно тригонометрия: мне очень не нравилось то, что полный оборот 2Пи радиан, это действительно вызывает кучу неудобств и путаницу в пересчете на 360 градусов. В видео это все отлично сформулировано, тут меня даже убеждать не надо. И я был бы рад чтобы мои дети были знакомы с Тау уже в школе.
i have my reasons for believing that τ/4 = π/2 is actually a much better circle constant. going to try to make a video on it
This is all very interesting, but it seems to matter about as much as whether toast is cut corner to corner vs site to side. Why is tau used anyway. Why not do what Planck did an chuck a bar on one of the limbs of pi, i.e. pi-bar? Edit ... and then you covered exactly this 2 seconds after I wrote the comment!
It's because it'll look weird, h with a bar looks normal because h has an ascender so it ok for a stroke to go though. π in the other hand doesn't have an ascender or descender, so it might look weird.
The main motivation is to make angle measurements more intuitive, especially for people first learning trigonometry.
Got the t-shirt. No, I mean I have the t-shirt.
What about from zero to geo?
I'm still working on it! But τ day was coming up sooner.
e is probably the most important constant in math, but τ is a close second. Although, τi is the constant that actually comes up more naturally. Like in the identity e^z = e^(z+iτ) or the equation ∮[γ] 1/z dz = iτn.
nah, 1 and 0 are much more important, as they are the multiplicative and additive identities, respectively, of the regular numbers.
@@Aurora-oe2qp That's what I get for deciding not to clarify that I was excluding rational constants.
"Tau is right, and Pi is wrong. We have already won the coup. There is no point in resisting."
euler originally had pi as 6.28 (tau) but changed it to 3.14 for some reason.
Apparently, he just used π to mean "some angle", like how we'd use θ today.
For the same reason, shouldn't we advocate in favor of switching the signs of charge?
I feel like that’s a bit too rigid of a convention. The people who actually are concerned about charge don’t really think twice about it at this point
If you said that the direction of flow of electrons and current should match, I’d be on board
From The Tau Manifesto:
"It is true that some conventions, though unfortunate, are effectively irreversible. For example, Benjamin Franklin’s choice for the signs of electric charges leads to the most familiar example of electric current (namely, free electrons in metals) being positive when the charge carriers are negative, and vice versa-thereby cursing beginning physics students with confusing negative signs ever since. To change this convention would require rewriting all the textbooks (and burning the old ones) since it is impossible to tell at a glance which convention is being used. In contrast, while redefining π is effectively impossible, we can switch from π to τ on the fly by using the conversion
π ⟷ 1/2*τ
It’s purely a matter of mechanical substitution, completely robust and indeed fully reversible. The switch from π to τ can therefore happen incrementally; unlike a redefinition, it need not happen all at once."
I'd like to see how formulas involving the gamma function (like the zeta reflection formula) would look like if BOTH the pi to tau correction AND the gamma correction (fixing gamma so that gamma(n) = n!) are applied.
I'm a believer (now)
I hate pi because every formula relies on the radius. Area is defined by radius, but pi is for diameter.
In Arabic, people have it bad. They don't have a word for radius; they call it half a diameter (nisf qutr). That is according to an Iraqi I know.
Basically, all along, it's not the problem of pi or tao, but instead radius or diameter?
This might be a tad cinical, but isn't it more of a hassle to change everything we have from pi to tau? Don't get me wrong, I absolutely think tau is the correct constant, but pi is, for lack of a better term, it is good enough.
For example, for someone who is first learning about angles, yes the obvious choice would to use tau, but changing everything we have both in maths and physics. This would be quite the undertaking, no matter the moment of the change. The amount of work necessary for the change would be immense, but I think the extra clarity is not enough to be worth it.
It would not be hard at all, time consuming maybe, but not hard. We would just have to get used to it.
I think there's a reason pi is more popular - because the majority of the first few people who thought about it (Archimedes comes to mind) preferred pi to 2pi.
The "tau manifesto" seems like posthoc analysis.
pi obviously make sense since we think about area relative to radius squared and a dodecagon has area 3R², a square has area 2R² = 4r².
additionally a full period being -pi to pi is useful.
Generally speaking people's brains work better with numbers between 2 to 4, that's why pi and e are in that range.
pi² being about 10 is much nicer than tau² being 40ish
pi/4 is commonly found in math for instance the ratio between area of circle and square or 0.5! = √(pi/4), using tau introduces a square root of 2 factor there.
Sure tau is 2pi which is fine, it's a factor of 2.
4:36 obviously, they all use the entire circle and there's nothing wrong with associating 2pi to a full circle.
Why is a measly little factor of 2 so scary?
7:45 I don't really see the difference, it's just a factor of 2.
9:49 I don't think a period is necessarily the unit.
for example it goes from minimum to maximum and vice versa in pi (half a period)
13:15 all these points are just acting like meaning disappears because a factor of 2 is so scary to you.
14:06 it doesn't reveal anything other than a weird obsession with a full turn.
There's absolutely nothing wrong with a half turn, for instance I might think dividing by 12 is scary, so viewing 30° as pi/6 might work better than trying to do 360/12
As fun as Tau is, I actually feel like it is, in some ways, the wrong direction. Tau is fundamental to circles, yes, but what about the right-angle constant, h=pi/2=tau/4=arccot(0)=2arctan(1)=arcsin(1)? That's fundamental to coordinates, and has the very nice property that exp(ih)=i, which I think illuminates the nature of the complex exponential better than either the pi or tau formulas.
There's actually a whole discussion about this in the manifesto itself! It ultimately concludes that the "recurrence factor" is 4h, i.e. tau.
Pi-nauts to arms!
Pi-turd
@@jacquesfaba55 Not very nice.
@@dominiquefortin5345Indeed.
22:54 That's a stupid definition of Pau. It should be √2•π
One problem you havent addressed is engineering. Its much easier to calculate the diameters of stuff than the redius
Er.. divide by 2?
That _is_ true. Probably why the use of diameter was more prevalent in antiquity.
@@hamishallan4723 Similar to you tau guys being unable to multiply by 2?
@@grantofat6438 I replied to a claim about “one problem you haven’t addressed” to point out how trivial that problem is. Obviously the factor of two is easy to address in either direction, but there are many other reasons to prefer Tau over Pi, as enumerated in this video.
@@grantofat6438The sheer blatantness of your lack of willingness to engage in constructive conversation is genuinely impressive.
One of my favorite literally genders is the manifesto
I'm not entirely sure what this means.
But of course earths gravitational acceleration g≈π²
... meters per second squared.
Well, pi and tau are just greek letters. Does it matter which one to use? No. But, convention matters. Like, you can think in terms of both tau and pi, or even better, pi/2! Which im now calling pitau, a pi with 3 legs, and is the measure of a quarter circle, and the most important angle, the 90 degree angle...
Yeah, no one will ever use pitau :(, its just an unconventional term, therefore, not really practical.
This is a comment I made in another video: "I've been fully onboard with the whole tau thing even before I heard of the tau manifesto. However, I have one disagreement -- the symbol. I don't like using yet another Greek letter to represent a number that is SO important to math and science. It needs something more universal and fundamental, something not bound to language (i.e. "tau for turn" in English).... something more like the ten basic Arabic numeral digits we use. My proposal is ⊙ because of the symbology of its importance to circles/oscillations; it's also the symbol for the Sun, which is very appropriate, as 6.28... is basically as important as the Sun (not to mention all the orbit references invoked by that)."
Loved your e^(i\tau) = 1 + 0
You have forgotten the most important use case of Tau (1 Turn). It's the Simple Continued Fractions (SCF) representation of an angle in terms of the circumference. It's perfectly expressed under this topic -> ua-cam.com/video/NboydwaYaaQ/v-deo.html
The discussion is superfluous.
noice
10:15 It's as if you learnt a new language and know it even better than your mother tongue. Or you switched to metric or seximal.
good luck distinguishing tau and proper time of the Sun in GTR
(just a joke. no problem at all.)
even if the whole agree to use tau the americans will insists on using pi