Pi may be wrong, but so is Tau!

[@ravtastic9802]

missed a real gem on euler's identity's true form: e^inη = i^n

[@sbaumei]

Eta really only works better than Tau for contexts where 90° angles have special meaning. Cancelling out 1/4 factors therefore only serves to conceal this context, similarly to Pi cancelling out 1/2 factors in contexts where 180° angles have special meaning. Using the same logic you could demonstrate cases where a circle constant representing tau/8 works so much better in contexts where 45° angles are the focus. However, choosing contexts where factors cancel out in your favorite field of science is not the fundamental target when choosing a circle constant. I really think Pi was a good choice for Archimedes and classical mathematics, and so would have been Eta or Tau. Because it was all about geometry. In modern mathematics, everything is about calculus as the mathematics of change. And there, Tau really shines like no other, no other choice can reveal so much hidden symmetry when it comes to integration, differentiation, and limits.Thats why these geeks (like me) really love Tau, calculus is the bread and butter of the modern scientific world. And that's why Euler as the most important founding father really should have left Pi in the dark ages where it belongs. At least he never mentioned Eta. And about revealing fundamental things: e^(i*tau) + 0 = 1 tells you above all else that if you turn once, you are back where you started. It reveals the fundamental property of periodicy in the complex plane, something you must never forget when dealing with complex functions, and too often forget because neither Pi nor Eta reminded you! Similarly, the most important choice in music scales is not how many half tones you chose, or how they are organized - that's just taste and practicallity. It is the choice of the octave distance that governs the repetition pattern, i.e. the periodicy of the scales. Beat that. :)

[@valeweinmann9907]

Sascha Baumeister wow,can I have your email or something, I would like to get in touch with you :)

[@bretthawkes5957]

The perfection achieved when using the true exact Pi ratio would keep the geeks entertained for Millenia. The geometry involves 3 ratios, Pi phi and the Royal Ancient Egyptian Cubit

[@guardiandogoargentinos1385]

I wish i knew wtf y'all were talkin about

[@isavenewspapers8890]

@@Imperio_Otomano_the_realest A symbol that varies in value representing the length of arc on a unit circle enclosed by an angle? Believe it or not, we still have that: it's called θ. But we also need *some* fixed value if we were going to talk about circles effectively, so we settled on π. And I don't know what you have against fractions, but you can't just pretend they don't exist. For instance, remember the time Euler used π to represent 1/2 the circumference of the unit circle? Key term: 1/2. You still have a fraction in there. Only now, you're proposing to hide it within the notation itself, which will just serve to cause unnecessary confusion. You'll have to define the symbol every single time you use it, and if you actually do any math with it, there's a good chance you'll just end up dealing with fractions anyway. I can't think of a practical reason anyone would do this unless they're lazy.

[@sistajoseph]

This is fantastic. Hope you are well, nine years ago. I have been looking for this for at least 3 years, I didn't know. You are unto something but you don't know what it is. Very good work. The half circle is symmetrical to the two points and there is also a right angle, symmetrical to both of them. This is a significant development.

[@BillShillito]

I see it like this. τ is the fundamental surface constant of 2-dimensional Euclidean space. But η is a fundamental constant of any n-dimensional Euclidean space from which can be built any number of constants for surfaces and volumes. τ is one of those constants.

[@PianoMastR64]

Also, π is the constant that is useful in engineering because the diameter it the only thing about a circle that is directly measurable.

[@chrisvolk4730]

PianoMastR64 So, take your measurements, divide by two, and the REST (almost) of your calculations will be simpler and more-intuitive.Tau on!

[@PianoMastR64]

Chris Volk Thanks for your response. Looking back at what I wrote, I said something that I could have said in a better way. The circumference is directly measurable. It's just that the diameter is very quick and easy to measure precisely. I'm nitpicking though.

[@chrisvolk4730]

PianoMastR64 Not at all. It seems obvious that is the reason that C/d was chosen in the first place (measurement). Now, while beneficial, I see that as having largely passed. Just in working with my pre-teen kids, it's amazing how much simpler and more-intuitive the concept of tau*r is (anecdotal I know). Also, it is fun to see the movement progress. Who'd thunk a number as seemingly as unassailable as pi is be under assault (and rightfully so!)? Best of luck!

[@luisuribe104]

nice argument. i'll have to rethink my preference of using tau and reset my favorite identities using eta. thanks for giving me a bit more grist for the mill. the fact that there are coherent arguments for the use of pi, tau and eta just confirms in my mind the elegance of mathematics. changing the geometric/trigonometric paradigm from the line/turn to the right angle allows for new perspective. this will keep me thinking for a few days...

[@rohanjoshi3630]

I think we can agree that tau is the circle constant, pi is the semicircle constant, and eta is the quarter circle constant (or right-angle constant).

[@aeon00000000]

The only problem is that ½ τ r² is clearly better than 2 eta r² and the simple fact that now you'll have 4 eta popping up EVERYWHERE. I honestly feel that nature itself depends very clearly on τ, however we should also use eta for many of the reasons you proposed above. Also, we can just forget about π.

[@BlackEyedGhost0]

The best way to figure out the FUNDAMENTAL circle constant is to take any equation in which it would appear and generalize it to work with any given angle. The equation for circumference is πD, τr, or 4ηr. Generalizing this to work with any angle gives the equation for arc length: θr. The equation for area of a circle is πr², τr²/2, or 2ηr². Generalized, you get the equation for the area of a section contained by the arc: θr²/2. In any area where π, τ, or η would come up, generalizing the equation to dis-include them gives you the same form as τ. This makes it the fundamental circle constant more than anything else. I certainly think that η is useful in many many cases as is π (although less so), but it's not the fundamental circle constant, τ is.

[@austingarcia6060]

My thoughts

[@kevinmorais]

Why can't we use Tau and Eta and Pi when they are simplest in using in a formula for what ever makes it easier for the Reader to understand...Can't we all just get along ;) hehehe You are BRILLIANT by the way with your Eta, I really, really enjoyed your video it blew me away :)

[@DavidButlerAdelaide]

Happy Eta day everyone (the 11th of July being 11/7). Interestingly 11/7 is closer to eta than 22/7 is to pi and also has the pleasing property of being the ratio of two consecutive primes ;)

[@IntergalacticPotato]

Interesting! Though I'd expect 11/7 to be closer to Eta, as the gap decreases with the values

[@DavidButlerAdelaide]

@SzlampStudios No. If you inscribe a triangle in a circle, then the ratio of Circle Area to Triangle Area is 4 pi / (3 sqrt(3)) = 2 tau / (3 sqrt(3)) = 8 eta / (3 sqrt 3). The pentagon is worse!

[@SzlampStudios]

area of a circle: 1/2eta D^2 volume of a ball: 1/3eta D^3

[@NuclearDuckie02]

2π (τ): Good for formulae involving circular arcs (e.g. orbital motion in physics) π: Makes finding roots of sinθ & tanθ easier; one might also argue a straight angle is fundamental to geometry (in terms of lines) π/2 (η): Good for complementary trig functions and perpendicular lines/vectors π/4 = arctan(1): Good for fast calculation via infinite series You're going to run into ugly coefficients somewhere no matter what, so the important thing is to realise there is no single best choice.

[@TokrokK]

This is pretty interesting. I don't think eta prevails in all of your arguments, but I do like what eta says about trigonometry. Tau, I think, is still better when defining a circle, its area, circumference, etc. At this point it seems to me that in most contexts tau is best, eta in some, and pi in very few. Perhaps this whole issue is an iteration of the natural tendency of humans to try to categorize things when the universe doesn't really care.

[@rasputozen]

That's what I'm saying. The fact that we use 90 degrees as our "fundamental angle" ie. our known angle, is completely arbitrary. We like our coordinate plane and it happens that it's based on 90 degrees instead of 135 or 49 - again, arbitrary. But to us humans, 4 is a nice number. Now I think most of us will agree that the coordinate plan is pretty much how we want it, wouldn't it be nice if it lined up without fractions with our constant? Patterns would certainly be seen more easily.

[@aeon00000000]

2013-07-28 #3 And the problem with η still stands: If you use η, you're going to have 4η everywhere in Physics and Chemistry instead of just τ. 2π comes as a unit so damn often, it's as if nature is telling us that we should just give it a name itself. However, η has it's uses. If we used τ, π, and η when appropriate, we would see τ the most by far, then η would be second, and finally π.

[@gouskin]

Better than pi, but not as good as tau

[@ayushsharma9270]

gouskin way better than tau

[@dirac17]

The arclength for a right angle is more fundamental than the arclength for a circle. A sine or cosine function is just a repetition of one quarter turn around a circle (ie. a right angle). Which means eta is the true arclength constant.

[@TheoryXI]

i will begin to use tau and eta interchangeably depending on the situation

[@johannesh7610]

I love this constant, but I also love τ! I denote it with a three legged pi (I call tri, uncreatively), which continues in a logical way that adding a leg halves the circle constant, so τ = 2π = 4ш (reflected). Really, τ is everywhere where full circles are involved, actually everywhere in physics and much of mathematics.

[@Waltonruler5]

I was actually pretty convinced but I'm still convinced by Tau's simplification of radian measure and how the area formula better reflects that area is the integral of circumference and mirrors equations from physics like a free-falling object, kinetic energy, elastic energy of a spring, etc. The equations that matched the derivatives of the sin function nearly had me crying though. That was beautiful.

[@alexroodman4318]

While eta is important as a right angle, the definition of eta is unfundamental, for the reason that to get that circular path requires a radius and a center, and why would it be fundamental to have the radius be half of the path with the center being the midpoint? The reason a right angle is important is that the lowest number of dimensions it takes to work with minimally complex geometry is the second, letting us build off of that in future dimensions, and eta is the angle between two axes. Essentially eta appears when you cut a full circle in to four separate pieces, each yielding eta. The only reason the axes split the circle into four pieces is that each axis goes into the origin and comes out the other side; eta is important because 2 x 2 = 4.

[@Justinian42]

Really, each definition is great for different tasks. I think that the education should teach each instead of limiting everything to π. If children learn about all these (π,τ,η) from a young age, they would be able to use the "right tool for the right job." Who says only one is the best?

[@shadyparadox]

As bad as this is, it's a much better case than I've ever seen for pi.

[@samisiddiqi5411]

I was skeptical but once you mentioned the right angle, I finally understood. Thanks for the vid!.

[@loudej]

True. Thinking of it as the rotational distance to any orthogonal axis - the fact it's also quarter circumference of unit circle is really a secondary property.

[@DavidButlerAdelaide]

@reinux Well it IS absurd -- which was part of the reason for my video in the first place. But on the other hand it's not SO absurd -- when new mathematics is made, there are often very heated discussions about the best way to go about it and represent it. What you're seeing here is a version of that -- albeit about maths that's been around for a while that we can't really change...

[@okuno54]

As a person who has done a fair bit of physics, I can confidently assert that our civilization, and beyond that the world at large, cares much more about frequency and period. That whole path/area difference ratio thing (which I expect to hold in higher dimensions, but you didn't go into it) simply does not appear when we try to understand the universe. Relatedly, when you describe the "beauty" of those 4-step repeating patterns, what I see is the destruction of the natural, continuous SO(2, R) symmetry in favor of this weird, discrete D2 group that's artificially imposed on something that so obviously is not discrete. In particular, there is nothing natural about scaling an element in the unit interval [0,1) to a parameter of SO(2, R) in [0,\tau) by a factor of 4*\eta, especially when mapping by a factor of \tau has no need for an arbitrary integer constant to be agreed on.

[@NuclearDuckie02]

I'd have thought pi would turn up more in trig, since it is the sum of internal angles of any Euclidean triangle. The only real significance tau has (in that field) is as the period of sine and cosine, which isn't really their most important property.

[@vgcellomusic]

Making all of trigonometry simple and elegant is way more important than simplifying these couple odd circumstances where you compare a straight line and a perfectly circular curve.

It seems to me that π is definitely more wrong than τ. I think it would be a good idea if we used both η and τ, depending on the use case. I think, τ has still too many advantages to be completely replaced by η. But I'm not sure if there is any use for π. If we use both η and τ, we might get rid of it completely, because at least I don't see any argument going for π.

[@AdderSIG]

The thing is, a circle is defined as the locus of all points a set distance (r) from a central point. It is not defined by a diameter. The fundamental distance of a circle is the radius. The circumference distance (which is where we get pi from) is tau units for a unit circle.

[@boptillyouflop]

TBH what I really wish is that the sin and cos functions would take their angle in turns instead of in radians or degrees. I've lost count of the number of times I've typed in something like wave[x] = sinf(x * 2 * 3.14159265f / 256.f); Yes, using radians simplifies calculus (and I guess the Taylor expansion). But other than that, there's no way you'd ever want to measure things in radians.

[@jesusthroughmary]

The thing that sold me was seeing radian measures in terms of tau. So intuitive and easy to explain (and remember).

[@jesusthroughmary]

The radians argument is more convincing on a practical level. I said to myself, "Yeah, it makes more sense, but pi is what everybody uses, so why change it now?" Then I saw how much easier it becomes for students to actually understand formulas using tau.

[@aeon00000000]

To quote from the TauDay website, Hartl writes: (I liken the difference between τ and η to the difference between the electron charge e and the charge on a down quark q_d=e/3: the latter is the true quantum of charge, but using q_d in place of e would introduce inconvenient factors of 3 throughout physics and chemistry.)

[@Walabio]

You bring up an interesting point: The way we derived the formula for the area of a circle is by dividing a circle into an infinite number of triangles and then adding up their area. The formula for the area of a circle is height times width divided by 2: a = (h*w)/2 The formula: a=((r^)τ)/2 tells us something about how we got the formula, as in we had to figure out the area of an infinite number of triangles. I am almost out of characters, but I shall reply to myself with shortcuts.

[@DavidButlerAdelaide]

Totally agree about the area thing. When I look at a circle I don't see its circumfrence either, but its area.

[@herbie_the_hillbillie_goat]

1. A circle is all points on a plane equidistant from a central point. Radius is that distance. The radius is therefore fundamental to a circle as opposed to some arbitrary cord. 2. A line can only intersect a circle in one of two ways. It will either intersect at one point (tangent) or two points (secant). The trigonometric ratios, secant and tangent were historically defined as: _Secant_ is the ratio of a secant line through a circle to the circles radius. _Tangent_ is the ratio of a tangent line through a circle to the circles radius. Point 1 justifies the circle constant being the ratio of the circumference of a circle to its radius. Point 2 shows precedence. It's not really an argument about pragmatism or utility, but one of mathematical definition. π and η have practical use. No doubt. However, τ is as fundamental to a circle as circumference and radius.

[@JayCee235]

I'll agree that 'eta' would be a much more versatile constant, but I don't think it's the more fundamental of the 3.

[@rasputozen]

As a convention we use 90 degrees as the "known" angle in most geo, trig and calc functions. We could have alternatively set up a system that uses 89 or 86.5 degrees as the "known" angle but that would needlessly complicate things because we'd have to bring in fractions more frequently. So since 90 degrees is a convention we use, pi and tau don't make the best matches for it. Eta makes the 90 degrees 1, the simplest you could possibly make it, contrary to tau's 1/4 and pi's pi/2.

[@rasputozen]

I was saying that it's arbitrary that we use 90 degrees as the "known" angle in trig - this is true. This 90 degree "known angle" convention is much more ingrained than the circle constant because it's related to our preference of quarters (4 cuts of 90 degrees) that we use for our coordinate plane. So even though it's arbitrary, this "Eta" is more suited to it than either pi or tau. Pi and tau would work better for other coordinate plane conventions though, but I don't see us changing that.

[@dirac17]

With all this Tau talk going about, I thought pi/2 actually made more sense than 2pi. I'm glad more people agree. I like your justifications, well done sir.

[@DavidButlerAdelaide]

Glad you liked the needle. Yes it doesn't on the surface have anything to do with circles, but that's one of the cool things about pi/tau/eta (whichever of them you like) -- they appear in more places than just circles.

[@HighKingTurgon]

1) τ is fundamental because it is the constant ratio of the two criteria for the definition of a circle: the distance from a hypothetical centre, and the distance compassed by all the points equidistant from them. 2) Trigonometry is not predicated on right angles. Never has been. Far too many angles are very awkwardly termed in terms of η--2η/3 is far less cogent than π/3, and certainly less geometrically apparent than τ/6. 3) η fails to convey periodicity in sinusoidal functions. Saying that the function repeats its shape every η completely ignores the information the graph is giving you. Ultimately, expressing angles as a fraction of a turn is much more useful than saying 'one-quarter-turn is fundamental and there are four of them in a circle'. Every advantage you list as having multiples of η is as well expressed as fractions of τ. Given the myriad other benefits of τ--and most fundamentally, its relationship to a circle's most important measure, the radius--there is no reason to adopt η (that is, the ratio of Circumference to twice the Diameter) as a circle constant.

[@rx327prime]

The circle and the sphere are kind of fundamental shape in the universe, from soap bubble to planets. I choose TAU. Taurus, Thora, Thor , Tao, ... Bi-bowl or Bi-bull (Bible) which makes a TAU See truthyracy and david lapoint.

[@knightriderultimate]

However the earliest references of the circle constant is as follows: "If you travel 7 parts in a linear path to a point, it would take 11 in a semicircle" So the most natural way of defining a circle constant would be to divide the two, which gives 'eta'. PS: the approximation pi=(22/7) comes from this definition.

[@DavidButlerAdelaide]

@XFi6 Oh, that's easy -- just make the string into a loop and put one end around a pin at the centre and the other end around your pencil. Indeed, if you use TWO pins for the centre rather than one, you'll be able to make any ellipse this way.

[@anticorncob6]

I recognize you from many videos. Can I ask what your opinion on the whole pi/tau (and maybe eta) issue is?

[@anticorncob6]

Area of a circle, volume/surface area of a sphere, nth roots of unity, fourier series and integration over polar coordinates, circumference of the earth at a certain point latitude? Reduced planck constant and pendulum time formula? Can you explain how eta fits neatly into all those formulas?

[@dirac17]

I've always thought pi/2 should be the true circle constant.

[@MartianSpore]

In terms of Tau, the area of a circle is 1/2tau*r^2. When differentiated, this  gives tau*r, or the circumference of a circle. This relationship is much easier to see with Tau than with pi, or even Eta.

[@dirac17]

My thinking is aligned with yours. Right angle is the important angle. Eta for the win.

[@sacul804]

very interesting. I like the approach, especially b/c 4η = τ. I think that, for teaching, it is a better since most people know the four quarters is a whole concept, and having the fundamental unit being a quarter is a very good idea. Only thing I would say is we should keep the τ notation for 4-tuples of η since it's cleaner in formulas. I agree that η feels more fundamental, but I still see an argument for τ, though I may just be having a hard time letting go... I'll try it out.

[@DavidButlerAdelaide]

@gmam96 All depends on location and teacher -- different people use different letters -- especially if they're using different languages! Technically you can choose to write an unknown in a formula as whatever letter you like.

[@zmanroc]

One 'turn' corresponds to 180 degrees in common English language" No, if you were to tell someone to turn, they would simply rotate on an axis. If a minute hand on a clock makes one full turn, did a minute pass?

[@dracoprimus]

I agree. The first assumption for eta is that each circular path has a ratio to the straight path, and the ratio for a circular path is eta, but how do you differentiate a circular path from other curved paths? it's the path with a ratio of eta. Circular reasoning. At least with tau, the argument is that since the definition of a circle involves the radius, not the diameter, the radius should be used in the defining ratio, not the diameter.

[@rrr00bb1]

What you are proposing is a major pet peeve of mine when dealing with computer code that people write (where people create things can return right answers but create a bunch of special cases that are exceptions to the pattern.) "areaOfLinearlyIncreasing(x) = 1/2 * x^2" + c means something, and when you plugin x, you can give it units (inches, feet, rotations, etc). If you use a stupid unit like "half rotations", then you can't automatically MATCH the data structure; doubled your API size.

[@aeon00000000]

2013-07-28 #2 My dad is in his 60s and recently became interested in Calculus. The Area-Circumference relationship of the circle was completely missed by him and this is entirely due to the fact that we use π, which obscures the integration performed to get Area. This problem would still exist if we used η. Lastly, take the equation in context of similarly derived equations: A = ½ τ r² (Area) s = ½ a t² (Position) E = ½ m v² (Kinetic Energy) F = ½ k x² (Potential Energy in Spring) etc.

[@christydavidpallanivel1708]

But A = eta(d^2)/2

[@Walabio]

To figure out the area of a circle bound bu a square, just figure out the area of the square and multiply it by .785398163397448. If one wishes to find the perimeter of the circle, just multiply the perimeter of the square by the same number. The volume of a sphere is .523598775598299 of the volume of a cube encompassing the sphere. The surface-area of a sphere is also that number times surface-area of a cube encompassing the sphere. These heuristics allow one to do the mathematics faster.

[@TheTrumanZoo]

this divides the sin motion in 4 parts right? positive up, positive down, negative down, negative up ? i kinda like this.

[@DavidButlerAdelaide]

Right angles were good enough for Euclid -- so important that he went to the effort of stating that they were all the same. Interestingly he defined a right angle pretty much as half a straight-angle, so it would seem Euclid thinks that 180 degrees (ie pi) is a more fundamental angle.

[@Quintinohthree]

Radius is defined coincidentally with circle. The one does not define the other. I'll admit, that does undermine my argument that circles are defined by their radii entirely, and so I must concede as much. This does not take away from the fact of the matter that the radius is more fundamental than the diameter, or any circle constant one might propose, or the circumference even.

[@rrr00bb1]

for example. would you represent 2 as: i*i*i*i + i*i*i*i ?? If i is the fundamental unit, then the answer is "yes". think about a cube root of -1 "m". 1 is still special because it's where we repeat, and "m" versus "i" ... 1 shows up either way. e^i*2pi is a rotation unit. e^i*pi/2 is a right angle. anyways, the point isn't to cancel stuff. the point is to match patterns so that you can say that one thing is isomorphic to another thing, so that you can give it a name.

[@darkcynite]

Dimensions exist at right angles right.

[@someman7]

How about this: Teach a short trig course using eta, then get someone to do it with tau. Then we can decide. Ask perhaps if coursera or udacity want to host it (maybe even khan academy, i don't know, there are lots of them now). Be sure to let me know if you decide to do this! Thanks.

[@DragonNoirification]

I still think that tau is overall the most comprehensible and functional, but I can definitely understand the reasoning behind eta. Also, the needle argument was really eye-opening (even if didn't really have anything to do with circles).

[@Robinsonero]

Actually the needle thing is all about circles. If you imagine the length of the needle is the diameter of a circle, than it's circumference shows us the probability that it will cross one of the parallel lines.

[@Xeroxias]

I don't know... I prefer to think of a right angle as a quarter circle, rather than a circle as four right angles. I think we can all agree it's silly to use Pi, and I can see the appeal of Eta to some extent, but seems more natural for me to think of a quarter turn.

[@joshbuckler]

next question: what is the Fundamental Integer? In other words, what is the most integery integer?

[@vanavm]

π/4 is more fundamental constant than π. The simplest expansion of pi (Gregory-Leibniz series) is a 4 divided by series. Many generalized expansions of π are actually 4 divided by series. Ponder why? Trigonometric properties of π (/2 or /4) may be a just a side effect.

[@unownmew]

Eta may be the key to spiral mathematics, which are fundamental in nature.

[@aeon00000000]

2013-07-28 #1 Oh, sorry I didn't explain why ½ τ r² is better. It makes it transparent that the Area of a circle is derived from *integrating* the Circumference. That is a very important point that is missed by a lot of students-one of those kinds of insights where things suddenly "click" when people encounter Calculus. I know it was like that for me, but I discovered it the hard way. It would be obvious with τ.

[@Moinsdeuxcat]

10:18 : The thing that seems the most obvious to me at this point is that it matches VERY WELL the powers of i. Therefore the link between piêta is pretty clear. (Your explanation is equivalent, since that's related to differentiation of e^(i*x) too). In a certain way, if real numbers are "cos" and pure imaginary numbers "sin", we find a coherent system, which immediately leads to : 1/ The geometric interpretation of complex numbers ; 2/ De Moivre's formula ; 3/PROFIT?!?. To me, the power of your system is that the analytic "i" is the base of complex numbers in the same way as the geometric "êta" is the base of trigonometry. They're both linked and make an immediate bridge between Geometry and complex numbers. i^2=-1 is exactly the same thing as "two right angles, and you're going backwards !", that is "2*êta=pi". ONE MORE ARGUMENT : arctan(1)="sum of inverse odd integers"=Pi/4. That's one of the most popular formulas for calculating Pi (veeeeery slooooowly).... Oops, did I say pi ? Cause what's calculated is êta/2 :p By the way, the definition set of x->tan(x) is WAY easier to describe : that's R minus "êta"N Your "êta" is very "analytical", whereas "pi" is very "practical" (for engineers, diameters can be measured etc...) and "tau" is very "intuitive".

[@iustinianconstantinescu5498]

Béranger Seguin That's TRUE!

[@ano0maly]

Hi DavidButlerAdelaide, Is there any source advocating eta or tau/4? I couldn't find any. It would be nice if this idea started gaining traction like tau did. While it sounds promising in some areas relating to right angles, I still think tau better encapsulates the circle as a whole.

[@RMarsupial]

It's not just that Euler's identity still works, it's that it's better. I just think it makes way more sense. I mean, it's in line with all the other equations similar to it now. I realise it's not that big of a dead, but it just seems way nicer if pi = 2pi.

[@NuclearDuckie02]

Ah, the rational trigonometry guy? I remember my teacher mentioning this.

[@whytauisrightandpiiswrong3296]

Because it makes more sense with tau. Take a look at the sin wave. It has period tau. Each period has two intersections, one with derivative 1, and one with derivative -1. The x-intercepts with derivative one are tau*Z (as Z is an integer) and the intercepts with derivative -1 are tau*Z + tau/2. That makes sense.

[@whytauisrightandpiiswrong3296]

Michael Hartl did. It doesn't make equations look "prettier", it makes it easier to understand how mathematics works. I've heard several people say that they would've aced trigonometry if they would have been taught radians in tau.

[@airman122469]

@TheGilmer I'm not sure. There are valid arguments for both sides. So, I don't think it's a complete joke.

[@whytauisrightandpiiswrong3296]

That's not my trigonometry class, we use radians all the time. And trust me, a factor of 2 (which doesn't exist at all until you fudge it with bad choice for circle constant), it gets confusing and annoying.

[@rasputozen]

Actually there are no special properties to right triangles. The formula for their area is quite nice to look at but similar formulas could be derived for any other angle and then it could be used as the "known" angle. All principles of mathematics are arbitrary.

[@jonathanmorris1248]

This is not the first time eta has been proposed as a circle constant. In 1958, an english mathematitian "Albert Eagle" proposed replacing pi with pi/2 to simplify formulas, although he represented it with the tau symbol since it is half of pi. Overall, I personally think that tau should be the fundamental circle constant because it represents a circle not a quarter of it but that eta would definately be the next best choice due to its properties.

[@cya9973]

No1. It is our day to day activity to simplify things by finding constants. The only available tool when trying to find how things that relate to one another is the ratio. In life, aslo in math a constant gives you comfort and safety because you have eliminated the unpredictable

[@DavidButlerAdelaide]

@drmarvin613 Well, you could always have Eta Approximation Day on 11th of July, since 11/7 is the best fractional approximation to Eta with denominator less than or equal to 7 (like 22/7 is for pi)

[@Anonymous-df8it]

Four days later, you can celebrate 15/7 as well!

[@BattistaLonardi]

Some months ago I wrote a quite angry post about this topic on my blog. I'm sorry for the rude words I used (and then removed), as today Michael Hartl has released an update to his Tau Manifesto, where he explains that η may be convenient when dealing with a hypersphere's surface area and volume.

[@anticorncob6]

Yes. I can. With tau. nth roots of unity, fourier serious and integration over polar coordinates and pendulum formulas all contain tau in them, or 2pi, or 4eta. Area/circumference/surfacearea/volume is more elegant in terms of tau with integral calculus. Circumference of a circle is 4*eta*r? How does this make sense? Where does the 4 come from?

[@DavidButlerAdelaide]

@anticorncob6 Of course I cherry-picked the evidence! That's what people do when they want to be persuasive ;)

[@DavidButlerAdelaide]

Alternatively you could use the 11th of July being 11/7, which is one of the continued fraction approximations to eta. (Or in American date notation, the 7th of November)

[@Quintinohthree]

Well then what is the definition of a circle? As far as I'm aware, it's a curve formed by the set of points a constant distance, that constant being the radius, from a specific point, that point being the center.

[@cya9973]

No2. Now your curiosity challenges you and you start digging dipper and dipper feeling confident with new constant in hand. The moment you start breaking that diameter in two, you loose your point of reference(the C/D, meaning of pi) and thus your comfort. Now you have to ask yourself, is pi a letter, a ratio or a number you don't understand. It's a matter of simplicity or comfort, you choose your path.

[@tylerborgard8805]

You guys are all wrong! The true circle constant is eta/2=pi/4=tau/8. I will call this number upsilon, because no one ever seems to use that Greek letter for anything. Why is upsilon superior to pi, tau, and eta? Here are my arguments: Argument 1: If you inscribe a circle inside a square, the area of the circle divided by the area of the square is exactly upsilon! Even better, upsilon is also the ratio of the corresponding perimeters! This is very important, because the perimeter of the square is also the circumference of the circle when using the taxicab metric. Argument 2: The formula for the area of a circle is upsilon*d^2, where d is the diameter of the circle. This is much simpler than the alternatives: eta/2*d^2, pi/4*d^2, and tau/8*d^2 Argument 3: Upsilon = 1-1/3+1/5-1/7+... Argument 4: In an isosceles right triangle, the two non-right angles are both upsilon radians, since tan(upsilon)=1. This shows that the angle between the graph of the identity function and the x-axis is equal to upsilon. It's also the angle between the identity function and the y-axis! Isn't that symmetry just beautiful? Clearly upsilon is the fundamental angle, not a right angle or a turn. Argument 5: |sin(theta)|=|cos(theta)| precisely when theta is an odd multiple of upsilon. Therefore, marking the x-axis at multiples of upsilon is clearly the most sensible approach when dealing with trigonometric functions. Argument 6: While e^(pi*i)=-1, e^(tau*i)=1, and e^(eta*i)=i, e^(upsilon*i)=sqrt(2)/2+sqrt(2)/2*i. You might argue that this is uglier, but putting upsilon in the exponent clearly shows that complex exponentials don't only result in real numbers and pure imaginary numbers. They also result in true complex numbers, where the real and imaginary components are both non-zero. You just don't get that kind of information from using the other circle constants.

[@PianoPlayerT]

I was π expecting this τ video to be a η-hour, contrived troll argument … until you got to Euler's identity. I'll never understand why e, of all things, behaves that way, but e^(iη)=i is beautifully concise, and it really sparks the imagination.

[@w00tklumpWn]

+PianoManT This. As a optical learner i never understood what e really sad or how i have to imagine it. I eventually gave up and just accepted it, making me strugle at every calculation involved e. What? It's positive? Ok, next is negative. Fine an alternating pattern! No, in the next power it's negative again. I always had problems with pi beeing very unintuitive, mixing fractions of pi in eulers just ruined the wonderfulness of math for me. Tau saved me the trigonometry, and eta the imaginery numbers. (but still tau > eta :P)

[@RobertAdoniasCostaGomes]

I see the eta you propose as relevant to basic math but only to it. I mean, my first worry about changing pi wasn't calculating the angle of triangle or trigonometric functions or even the needle experiment (which I'd never heard of), but about calculus of volumes, periodic waves and this kind of stuff where it's wiser to think of a circle. I don't know but, as an engineering student, I find it easier to relate to an actual thing (circle) than a consequence (angle). I'm still a tau-believer

[@plukerpluck]

About pi winning for circular area. Are you referring to r*pi^2 = area? If so I'd like to point out that tau is actually quite intuitive here, especially if you do calculus. Obviously the circumference of the circle would be tau*r. Now integrating that gives the area which is (tau*r^2)/(2!). The reason this makes sense is it's how a lot of quadratic terms look. Integrate again (because we can for fun) and you get (tau*r^3)/(3!). So tau matches the quadratic pattern. So pi & tau are equal here.

[@NoActuallyGo-KCUF-Yourself]

Good point, and the 4 in the denominator of the exponent clearly means fourth-root.

[@jimzamerski]

Let me put it this way. I think people should know what each one is and use whatever one is appropriate for non-fractionally. See proof. (Pi, Tau, Eta)=(Understanding 1 value)/(3 ways to use it) (Pi, Tau, Eta)=1/3(understanding value/ways to use it) 3*(Pi, Tau, Eta)=understanding value/ways to use it 3*ways to use it(Pi, Tau, Eta)=Understanding value 3*ways to use it/Understanding value (Pi, Tau, Eta)= 1 All that non-sense equals one, right?

[@carnright]

What I like is that pi is being challenged.

[@andremeIIo]

The key thing from my point of view is not how easy it makes things or how much it simplifies them, it is, instead, how well it correlates with their nature. If we defined a circle in terms of its deviation from a straight line, then I would support eta, but we define a circle from the idea of a circunference - "all the points equally distant from a point". In an ideal world, I would support the adoption of both, but having to decide when to apply one or another would complicate things.

[@RMoribayashi]

I never got to maths levels where I had to use radians, just degrees. I encountered them when I studied for the higher levels of the Amateur Radio exam (in the days when you had to LEARN electronic theory, not do a multiple choice test). I thought they made no sense. Why would the circumference of a circle be 2 anything? 1 or 4 seem more logical, and seeing that most of the calculations are done on a 4 quadrant Cartesian plane, Eta makes even more sense.

[@whytauisrightandpiiswrong3296]

You first stated that using the right angle for the circle constant is elegant and beautiful and makes things so simple, yet you just stated that it's arbitrary and that there are no special properties to right angles. Or triangles, you stated. Pythagorean theorem?