06-28 is TauDay, a day devoted to the 1 true CircleConstant (the 1st 3 digits of the decimal representation of Tau (τ) is 6.28).

I have to finish my series about electoral methods, but I work much overtime (this week, I worked 6 days). I shall tie this to the series:

An HalfTauist tried to argue that Tau (τ) is not as good as Half Tau with me. I mopped the floor with him. The only argument which held water is that τ collides with other uses of τ (unfortunately, collisions are unavoidable). Still, it might be better to choose a symbol for c/r with fewer collisions. This is how I would do it:

1. Nominate alternative symbols, which are already part of the UnicodeStandard.
2. Have a score election with values from -9 to +9, skipping over 0, thus forcing Mathematicians to come down 1 way or another.
3. Sum the votes.
4. Remove all symbols with negative totals
5. Have a ScoreVote runoff of the symbols with positive scores
6. Have the top-2 candidates have a plurality runoff
7. Have the winner go up against τ in a plurality runoff.
8. If the new symbol wins, we adopt the new symbol.

6000252
Why should we switch to Tau? Convince me: Present in what way it is significantly better than Pi as the periodic constant (I maintain that ‘circle constant’ rather undersells the fundamental importance of the number, regardless of which is chosen).

Edit: In addition, a challenge for your symbol election: If the majority of mathematicians reject your proposal of Tau altogether and refuse to participate in your election at all, how can you claim the results to be legitimate?

6000397

• The radius defines the circle.
• One does not have extraneous factors of 2 in circle-measurement in trigonometry.
• Tau comes up in all of the basic equations.*
• e^iτ=1

* An HalfTauIst said "¿What about the area of circles?". This is just ignorance:

The interior of an n-sphere is:

(d-1)/dτr^d=interior

2D:
(τ/2)r^2-area

3D:
⅔τr^3-volume

4D:
¾τr^4=interior

τ comes up all of the time because the radius is in the definition of the circle:

"A circle is all points equidistant from an origin on a 2D Euclidean Plane."

That is why the CircleConstant is "c/r=τ".

For more information, I recommend that you read the primer at TauDay.Com:

TauDay.Com

6000996

The radius defines the circle.

No more so than the diameter.

One does not have extraneous factors of 2 in circle-measurement in trigonometry.

Yes you do, they're just in a different place. It simplifies some expressions, but complicates others. I see no reason to believe it simplifies more than it complicates. And in fact, where it complicates, it adds a factor of two to the denominator, which is more bad than removing a factor of two from the numerator is good, because division is more complicated than integer multiplication.

Tau comes up in all of the basic equations.

Again, no more so than Pi does. That's literally how mathematical substitution works.

e^iτ=1

1e^(πi) = -1
My equation has four special numbers, while yours only has three.

Then you follow up with a single example of where an equation is simplified, but fail to provide an argument for why that would generalize to all equations, formulas, and functions that utilize the periodic constant. In fact, I'm not entirely sure your given equation is correct: According to Wikipedia (which tends to have very good math articles), the function for the n-dimensional interior measure of an n-dimensional ball is:
Vₙ(r) = ((π^(n/2))/Γ((n/2}+1))r^n
(where Γ(x) is the gamma function)
This only matches your formula up to n = 3.

τ comes up all of the time because the radius is in the definition of the circle:

That doesn't follow; Pi shows up in many places where circles are not clearly present, so it shouldn't matter what it's use is in defining a circle specifically.

I've read and watched videos about this before; it's never convinced me. And frankly, they've put up better arguments.

I asked for a way in which Tau is significantly better than Pi. You failed to provide a single clear example of it being better at all, let alone significantly, which would require you presenting an argument as to why it would have a real impact beyond minor aesthetics.

6001148

> > "'The radius defines the circle.'"

> "No more so than the diameter."

A circle is all points equidistant from the origin. That is how one uses a compass to draw a circle.

> > "'One does not have extraneous factors of 2 in circle-measurement in trigonometry.'"

> "Yes you do, they're just in a different place. It simplifies some expressions, but complicates others. I see no reason to believe it simplifies more than it complicates. And in fact, where it complicates, it adds a factor of two to the denominator, which is more bad than removing a factor of two from the numerator is good, because division is more complicated than integer multiplication."

I do not see the extra factors of 2 you mention. half a circle is half instead of 1. A third is a third instead of 2 thirds. It all makes sense.

> > "'e^iτ=1'"

> "1e^(πi) = -1"

> "My equation has four special numbers, while yours only has three."

That is numerology.

> "This only matches your formula up to n = 3."

Given that I live in an universe with 3 spatial dimensions, I shall take it.

6001493

A circle is all points equidistant from the origin. That is how one uses a compass to draw a circle.

Yes, you use half the diameter to draw a circle with a compass.

I do not see the extra factors of 2 you mention. half a circle is half instead of 1. A third is a third instead of 2 thirds. It all makes sense.

Your mistake is to think that Pi only shows up regarding circles. I already explained that that is not the case.

That is numerology.

You're the one who presented the first equation as if it proved a point. I was just following your lead.

Given that I live in an universe with 3 spatial dimensions, I shall take it.

Seriously? I just pointed out your formula could be mathematically wrong, and you don't care? Even creationists don't usually deny math. If you don't care if your math is accurate, why do you care whether it's Pi or Tau?
And yes, I'm taking this one a little personally; I'm a math geek.

Here is the reasoning that convinced me that τ is superior:
1. The diameter of a convex shape is, from Wikipedia, "defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary". By this definition, there are an infinite number of 2D planar shapes with constant width equal to the diameter; one such shape is the Reuleaux Triangle. However, the circle is the only shape with a constant radius; by extension, n-spheres are also the only shapes in their respective dimensions with this property. As a result, the radius is more fundamental to the circle than the diameter.
2. The circle constant relates the circumference to a straight component of a circle. Since the radius is fundamental to the circle, it follows that the circle constant ought to be defined by the ratio between the circumference and the radius. Therefore τ is also more fundamental than π.

6001712

> > "'A circle is all points equidistant from the origin. That is how one uses a compass to draw a circle.'"

> "Yes, you use half the diameter to draw a circle with a compass."

No, one uses the radius for drawing the circle. The circle does not have a diameter until after it exists:

One must choose the radius and origin. Then one draws the circumference. Finally, after the circle is finished, the diameter exists.

> > > "'You're the one who presented the first equation as if it proved a point. I was just following your lead.'"

> > "'Given that I live in an universe with 3 spatial dimensions, I shall take it.'"

> "I just pointed out your formula could be mathematically wrong, and you don't care?"

The formula (d-1)/dτr^d=interior is a great mnemonic:

Many students learn that area is (τ/2)r^2=area. A common error is assuming that the volume of a sphere is (τ/2)r^3-volume. When they learn that the formula is ((4/3)(τ/2))r^3=volume, they often accidently use ((¾)/(τ/2)r^3=area. If they learn that for the 2nd and 3rd dimensions, the formula is (d-1)/dτr^d=interior, they will never err.

"Even creationists don't usually deny math."

No they do not:

The creationists believe that τ=6.

6003428

¡Excellent point!

6003477

No, one uses the radius for drawing the circle.

Everyone uses a half-diameter to draw a circle.

The circle does not have a diameter until after it exists:

By that logic, a circle doesn't have a radius until after it exists, either. Unless you're saying that a radius exists in concept because you've decided it's length, in which case you've also necessarily decided the length of the diameter, in which case the diameter also exists before the circle does.

The formula (d-1)/dτr^d=interior is a great mnemonic:

I don't care if it's easy to remember if it's wrong. And ‘never err’? You just erred when you tried to apply it to a four dimensional shape; but apparently that error doesn't matter to you. This is a serious thing; you're essentially saying that you think a formula is right because it's useful for your argument; backwards reasoning if I ever saw it.

No they do not:
The creationists believe that τ=6.

I've had actual conversations with creationists. If you think they believe that, you apparently haven't.
“But it's in the Bible!”
I know it's in the Bible, but that doesn't mean the average creationist believes it. They don't tend to read the Bible through, you see.

6003428

2. The circle constant relates the circumference to a straight component of a circle. Since the radius is fundamental to the circle, it follows that the circle constant ought to be defined by the ratio between the circumference and the radius.

1. You are oversimplifying things by assuming the periodic constant is only relevant to geometry. It's about more than just circles.
2. If you want the constant to be the ratio of fundamental components of a circle, then you'll also have to find something to replace the circumference, because there are plenty of shapes that have circumferences.
3. The constant is defined by the ratio between the circumference and the radius. That it is then divided by a factor of two does not change that.
4. Why should it matter if the constant is mildly more aesthetically pleasing, anyway? That's all you're really saying here.

6003490

> "'The circle does not have a diameter until after it exists:'"

> "By that logic, a circle doesn't have a radius until after it exists, either. Unless you're saying that a radius exists in concept because you've decided it's length, in which case you've also necessarily decided the length of the diameter, in which case the diameter also exists before the circle does.

One must set the radius of the compass before drawing the circle. At that point the radius exist. The circumference comes into being as one draws the circle. After one draws the circle, one can draw the diameter-line. The radius creates the circle.

> "I don't care if it's easy to remember if it's wrong."

It is right for the 2nd and 3rd dimensions. I have never had to figure out the interior in more than 3 spatial dimensions. ¿Have you?

> > "'No they do not:'"

> > "'The creationists believe that τ=6.'"

> "They don't tend to read the Bible through, you see.";

⸘Creationists do not tend to read the Bible‽ ¡Say it ain't so!

6003497

One must set the radius of the compass before drawing the circle. At that point the radius exist.

How many people still use compasses to draw circles? A raise of hands? Nobody? You're saying you all use digital drawing programs instead? Wow; it's almost as if an argument that assumes all circles, real or hypothetical, necessarily start with a compass being set is faulty.

It is right for the 2nd and 3rd dimensions. I have never had to figure out the interior in more than 3 spatial dimensions.

You literally did exactly that a few comments ago:

4D:
¾τr^4=interior

This is wrong. When an honest person is wrong, they admit it. They don't make increasingly frivolous objections defending their mistakes. I repeat: This is a serious problem; you can't claim to care about math enough to weigh in on Pi versus Tau and then shrug your shoulders when you use a false equation, and you can't present yourself as a rational person while defending a false position with essentially a shrug and an ‘I don't care’. Yet you've done both, here.
I don't honestly care if it's Pi or Tau. I do, however, care about truth; and you appear to be rejecting truth because it's inconvenient for your argument.

6003492

1. You are oversimplifying things by assuming the periodic constant is only relevant to geometry. It's about more than just circles.

That's not what I'm assuming. What I listed is "the reasoning that convinced me that τ is superior". Of course, there are many ways of defining the circle constant. One way, and very likely the way people first came across it, is the ratio between the circumference and a linear length of a circle. Other ways involve the period of sin(x), cos(x), or e^iθ (all of their periods are τ, by the way), or through integrals, and most if not all of them are connected to circles. Sure, the circle constant is found in lots of (unexpected) places, but those are rarely used to define it. Granted, I am no expert at university+ level math, so there may be something important I missed.

2. If you want the constant to be the ratio of fundamental components of a circle, then you'll also have to find something to replace the circumference, because there are plenty of shapes that have circumferences.

The circumference is an essential component of a circle, though. Obviously 2D shapes will have a border because that is what makes them a shape; without a border, the shape wouldn't exist.

3. The constant is defined by the ratio between the circumference and the radius. That it is then divided by a factor of two does not change that.

By that logic, τ/k or τ*k for any positive integer k would be equally good. However, with τ, there is no need to multiply or divide; C/r is enough. That extra factor of 2 (or any other k) is what one can avoid by using τ.

4. Why should it matter if the constant is mildly more aesthetically pleasing, anyway? That's all you're really saying here.

Why make it less aesthetically pleasing when one can make it more aesthetically pleasing without a loss of function? Anywhere where π is used, it can be replaced by τ/2. Plus, many expressions involving π come with a factor of 2, so are simplified by using τ.

Now that we have made arguments as to why τ is superior to π, maybe you should tell us some of your arguments, backed up by evidence, as to why π is better, as I am not convinced. So far you have only said why τ isn't significantly better, but not why it is worse.

6003508

> > "'One must set the radius of the compass before drawing the circle. At that point the radius exist.'"

> "How many people still use compasses to draw circles? A raise of hands? Nobody? You're saying you all use digital drawing programs instead? Wow; it's almost as if an argument that assumes all circles, real or hypothetical, necessarily start with a compass being set is faulty."

Computers draw circles by defining the origin and radius.

> > "'It is right for the 2nd and 3rd dimensions. I have never had to figure out the interior in more than 3 spatial dimensions'".

> "You literally did exactly that a few comments ago:"

4D:
¾τr^4=interior

I admit that I am wrong about d>3. It still does not change it the mnemonic value. That is 1 of the 1st things to jump out at me:

I always forgot the formula for a sphere. When I saw it in τ, it jumped out at me:

Area=t/2r^2, volume=⅔τr^3; so therefore, one can generalize to interior=(d-1)/dτr^d. I was wrong, but the fact remains that we can teach children that interior=(d-1)/dτr^d holds for the 2nd and 3rd dimensions. That probably will cover >99% of cases that they will encounter. Rather than learn 2 separate equations for interiors in 2 and 3 dimensions, ¿is it not easier to learn 1 generalized equation, with the caveat that it breaks down in higher dimensions?

6003522

The circumference is an essential component of a circle, though. Obviously 2D shapes will have a border because that is what makes them a shape; without a border, the shape wouldn't exist.

By that reasoning, the diameter is also an essential component; a circle without a diameter doesn't exist, either.

By that logic, τ/k or τ*k for any positive integer k would be equally good.

Yup.

That extra factor of 2 (or any other k) is what one can avoid by using τ.

This is what I was trying (in a horribly-worded way, I admit) to get across in my first point: Unless you're measuring an actual radius of an actual circle, you'll generally come across the constant multiplied by some number or another; whether that number is a factor of two larger or smaller doesn't really matter accept in the very specific case of exactly 1 Tau or 2 Pi.

Why make it less aesthetically pleasing when one can make it more aesthetically pleasing without a loss of function?

The order of importance should be:
1. Functionality
2. Practicality
3. Aesthetics
If you admit that it's only a matter of aesthetics, then you have to justify how it would not be burdensome to shift one of the most basic numbers in mathematics, physics, and engineering by a factor of 2; otherwise, the practicality of leaving it as Pi overrides the aesthetics of changing it to Tau.

Plus, many expressions involving π come with a factor of 2, so are simplified by using τ.

And many expressions would also be complicated by dividing by a factor of 2; for instance, the formula that was discussed earlier in this thread:
“The function for the n-dimensional interior measure of an n-dimensional ball is:
Vₙ(r) = ((π^(n/2))/Γ((n/2}+1))r^n ”
The term π^(n/2) would turn into the much uglier (τ/2)^(n/2) , which as I pointed out in an earlier comment not directed at you, would be more burdensome than an extra factor of two added by converting Tau to 2 Pi, because division is more burdensome than multiplication.

Now that we have made arguments as to why τ is superior to π, maybe you should tell us some of your arguments, backed up by evidence, as to why π is better,

I never claimed that Pi was better. It is the people who are making the claim that Tau is better who have the burden of proof, here. I'm of the position that they are equivalent, and as you pointed out, so would be any other constant multiple.

6003538

Computers draw circles by defining the origin and radius.

But not by using a compass. And again we come back to the radius just being half the diameter.

I admit that I am wrong about d>3.

Finally.

¿is it not easier to learn 1 generalized equation, with the caveat that it breaks down in higher dimensions?

Easier? Yes. Good teaching method? No. Easy isn't always good, and learning isn't always easy. It's better to teach the correct methods from the beginning than it is to give people flawed information and hope that they never end up in the situations where it doesn't work. An engineering professor would say that 0.1% failure is still 0.1% of bridges collapsing. Cutting corners is never a good idea.

6003657

> > "'¿is it not easier to learn 1 generalized equation, with the caveat that it breaks down in higher dimensions?'"

> "Easier? Yes. Good teaching method? No. Easy isn't always good, and learning isn't always easy. It's better to teach the correct methods from the beginning than it is to give people flawed information and hope that they never end up in the situations where it doesn't work."

I wrote to give them the caveat that it does not work in dimensions higher than 3.

> "An engineering professor would say that 0.1% failure is still 0.1% of bridges collapsing. Cutting corners is never a good idea."

Most students will not have to figure out the volume of a 6-dimensional tank. Besides, the correct equations are probably too advanced for grammar school:

Interior=((τ^(n/2))r^n)/n!! if n is even
interior=(((2τ)^((n-1)/2))r^n)/n!! if n is odd

I got that from the TauManifesto. I see why people just memorize the formulæ for each dimension. Speaking about memorizing formulæ, rather than memorizing 1 formula for 2 dimensions and another formula for 3 dimensions, ¿why not learn just 1 formula for both dimensions?, with the warning that the formula does not hold for other dimensions.

6003693

rather than memorizing 1 formula for 2 dimensions and another formula for 3 dimensions, ¿why not learn just 1 formula for both dimensions?,

And how is thinking of it as one formula that only applies in two cases better than thinking of it as two formulas that look similar?

6004529

It prevents the mistake of using ¾ by mistake.

6004532
Frankly, I'm not sure someone who would do that is capable of remembering a generalized formula anyway.

6003652
I've shown in my first post a reason why tau is better, at least aesthetically. True, there are some formulas that become more complicated when you use tau, but if I remember correctly, someone did an analysis and found that the number of formulas getting simplified is about the same as those becoming more complicated when one switches from one to another. However, at least in the formula for the area of a circle, 1/2*tau*r^2, using tau reveals how it can be derived through integration, or even by finding the area of a triangle. Using pi in conjunction with r in an equation is a bit weird, since pi = C/d, so both d and r are in the equation.
Tau also makes more sense in trigonometry,
since when learning, people would be less likely to be confused, as 1 turn = 1 tau but = 2 pi.
Therefore tau does have better functionality, though for engineers, pi may be more practical, but outside of that, tau is superior. If you think pi and tau are equally good, I hope the reasons I have shown do convince you that tau is indeed at least somewhat better a choice than pi. Personally, I use tau now and it is, for me, easier to use.
I'm not saying that everyone should change to using tau, but simply that it is better.