2 Pi or Not 2 Pi?

June 28, 2015 — Giorgia Fortuna, Consultant, Advanced Research Group

Three months ago the world (or at least the geek world) celebrated Pi Day of the Century (3/14/15…). Today (6/28) is another math day: 2π-day, or Tau Day (2π = 6.28319…).

Some say that Tau Day is really the day to celebrate, and that τ(=2π) should be the most prominent constant, not π. It all started in 2001 with the famous opening line of a watershed essay by Bob Palais, a mathematician at the University of Utah:

“I know it will be called blasphemy by some, but I believe that π is wrong.”

Which has given rise in some circles to the celebration of Tau Day—or, as many people say, the one day on which you are allowed to eat two pies.

But is it true that τ is the better constant? In today’s world, it’s quite easy to test, and the Wolfram Language makes this task much simpler. (Indeed, Michael Trott’s recent blog post on dates in pi—itself inspired by Stephen Wolfram’s Pi Day of the Century post—made much use of the Wolfram Language.) I started by looking at 320,000 preprints from arXiv.org to see in practice how many formulas involve 2π rather than π alone, or other multiples of π.

Here is a WordCloud of some formulas containing 2π:

I found that only 18% of formulas considered involve 2π, suggesting that τ, after all, would not be a better choice.

But then why do τ supporters believe that we should switch to this new symbol? One reason is that using τ would make geometry and trigonometry easier to understand and learn. After all, when we learn trigonometry, we don’t measure angles in degrees, but in radians, and there are 2π radians in a circle. This means that 1/4 of a circle corresponds to 1/2 π radians, or π/2, and not a quarter of something! This counterintuitive madness would be resolved by the symbol τ, because every ratio of the circle would have a matching ratio of τ. For example, 1/4 would have an angle of τ/4.

I personally do not have strong feelings against π, and to be honest, I don’t think students would learn trigonometry faster if they were to use τ. Think about the two most important trigonometric functions, sine and cosine. What’s most helpful to remember about them is that sin= cos(2 π) = 1, and sin = cos(π) = –1. I have not only always preferred cosine simply because it’s easier to remember (there are no fractions in π and 2 π), I’ve also always recognized that sine and cosine are different because one is nonzero on integer multiples of π and the other is nonzero on some fractions of it. By using τ instead, this symmetry would be lost, and we would be left with the equalities sin = cos(τ) = 1 and sin = cos = –1.

Given these observations, it seems like choosing τ or π is a personal choice. That’s fair, but it’s not a rigorous approach for determining which constant is more useful.

Even the approach I had at the beginning could lead to the wrong conclusion. The Tau Manifesto, by Michael Hartl, gives some examples of places where 2π is most commonly used:

And indeed, all these formulas would be easier if we used τ. However, those are just six of the vast number of formulas that scientists use regularly, and as I mentioned before, not many mathematical expressions involve 2π. Nevertheless, it could happen that formulas not involving 2π would still be simpler if written in τ. For example, the expression 4 π² would simply become (τ²).

For this reason I looked back at the scientific articles to see whether using τ instead of 2π (and τ/2 instead of π) would make their formulas simpler. For instance, these are some that would be simpler in τ:

And these are some that would not:

Let me now try to explain what I mean by simpler by looking at an example: if I take the term containing π in the bottom-left formula of the Tau Manifesto equation table:

I can replace π with τ/2 using ReplaceAll, and I get:

Just by looking at these two expressions, you can see that the second one is simpler. It’s not just your intuition that tells you that; it’s clear that there are fewer symbols and constants in the replaced expression. We can look at their corresponding TreeForms to demonstrate it explicitly:

To get a numeric difference, we can look at the leaf counts (number of leaves on the trees), which correspond to the number of symbols and constants in the original formulas:

To see whether τ had an overall simplifying impact, I computed the complexity of each formula (defined as their leaf counts, as computed above) involving π that appeared in the articles when using π and τ. To be more precise, I first deleted all the formulas that were either equal to π or 2 π. I felt it would have been unfair to consider those as well because very often, if they appear by themselves, they do not stand for formulas. I then compared the number of times the τ formulas were better with the number of times they were not, and only 43% of the formulas whose complexity changed at all were actually better, meaning that using τ would make more than half of them look more complex. In other words, based on this comparison, we should keep using π. However, this is not the end of the story.

One observation I made is that if an expression gets either more or less complex, it’s likely to have a leaf count that is less than 40. In fact, if you look at the percentage of formulas that are better when using π or τ and that have a number of leaves that is less than a fixed number, you get this picture:

where the x axis represents the upper bound on the number of leaves. This suggests that almost all formulas that become simpler have complexities less than 50, regardless of the symbol we choose.

A more relevant observation is that the situation changes drastically as the complexity of the formulas increases. Already by only considering formulas that have complexities greater than 3, like from earlier, only 48% are simpler in π against 52% that are simpler in τ. The graph below shows how the percentage of formulas that are better in either π or τ changes as a function of the complexity:

As you can see, as the number of leaves exceeds 48, the situation becomes chaotic. This is because only 0.4% of formulas have complexities greater than 50. There are not enough of these for us to deduce anything stable and reasonable about them, and the previous observation tells us that we should not really worry much about them anyway.

What this graph tells me is that in everyday life, and for anything more complex than fairly easy expressions like , we should indeed use τ for simplicity. But there is still something else I have not considered. What about different subjects?

It might be that formulas in physics look simpler in τ, but formulas in other subjects do not. The initial search I made included articles from different subjects; however, I didn’t initially check whether the majority of π-containing formulas were from a limited subset of those subjects, or whether the ones that became simpler with τ were mostly from a limited subset. In fact, if I just restrict analysis to articles in mathematics, the situation becomes the following:

Basically, only 23% of formulas benefit from using τ, and those benefits come only when the complexity is fairly high. For instance, something of this sort:

would be an expression that would be simpler in τ, and you probably have not seen many of this type of expression. This suggests that either scientists in different subjects should use different conventions depending on their field-specific formulas, or that all scientific disciplines should switch to τ even though it does not really make sense for some of them to do so. After all, in a democracy, the majority wins, and it is impossible to accommodate everyone.

However, the above formula shows something else that I want to point out. With τ, it becomes this:

And that is not much of an improvement: even though an expression could be easier in τ, the improvement might be so small that it is irrelevant. Consider for instance these two expressions together with their leaf counts:

And the corresponding expressions in τ:

The first formula is simpler in τ, but the leaf count is only 1/13 smaller than the original complexity, whereas the second expression is simpler in π and the replaced expression is 1/6 higher than the original complexity. In other words, the first case’s improvement was 1/13 and the second’s was -1/6 (the minus sign indicates negative improvement, as the expression in τ was worse). The mean of the vector is –0.044, a negative number, which means that using τ in these two expressions makes the whole vector 0.044 worse, although π and τ each improved one formula.

This vector approach is different from the one-count-per-equation one that I used earlier. It considers quantity of improvement instead of just an either/or binary, and it completely reverses the previous conclusions. I have computed these vectors for formulas having complexities bounded from below in the same way I did in the previous example. What I’ve seen is that the overall improvement in going from π to τ, computed as the mean of these vectors, looks like this as the complexity increases:

where the least worsening, -0.04, is achieved at a complexity of 5. As you can see, the improvement stays below 0 the whole time, meaning that while more formulas may be shorter with τ (depending on the field), on average those length decreases are outweighed by the length increases in the formulas that are getting longer.

To make my point, at the end of this scientific investigation: I think we should be happy with our old friend π and not switch to τ.

I have two final observations. The first is that if we had already lived in a τ world, the conclusion would have been different, and we would have chosen to stick with τ. If our expressions were already in τ and we were investigating whether switching to π would make them simpler, our vector-based graph would look like this:

That difference in behavior is because the vectors used to construct the graphs depend on the original complexities, and so change when the original changes.

This shows that for formulas that have a complexity greater than 2 (most of them do) and for which the complexity is not always greater than 18, the improvement in switching from τ to π would be negative again, suggesting that we should not accept the switch. Unfortunately for supporters of τ, we do not live in a τ world.

The second observation, which was brought to me by Michael Trott, is that 2/3 of the formulas shown in The Tau Manifesto (the green table at the beginning) don’t just have 2π in them, but the complex number 2πi. This suggests that maybe the question I was trying to answer is not the correct one. A better one could be this: would it make sense to have a new symbol τ for the complex number 2πi?

This new convention would require changing from πi to τ/2 as well, but that doesn’t affect the complexity of πi. In general, formulas having a πi term inside would either become simpler or preserve their complexity. To give you an idea, here’s a word cloud of formulas that would become simpler:

Which, after substituting τ= 2πi, become these:

You could argue that the percentage of improved formulas may not be high enough, and changing from 2πi to τ is not worth the effort. What evidence shows, however, is the opposite: of all formulas having a πi term, 75% would be simpler, and the remaining 25% would keep their original complexity—none would get worse. This is a strong point to make, and I am not in the position to do it, but I think the equality τ = 2πi looks more promising (and less historically disruptive) than τ = 2π.

Whatever your opinion on τ, I hope you have a lovely Tau Day. Please enjoy two pi(e)s today—imaginary or otherwise.

Posted in: Mathematics

 How did you process the formulas to get from arXiv into Mm? If general that would be a very useful capability! Posted by Mike Burns    June 28, 2015 at 9:50 am
 I used articles that have been preprocessed and converted to .xhtml. I got them from this website: https://mir.fi.muni.cz/MREC/ After you import them into Mathematica, you can simply search for math formulae using StringCases, and process them a bit in order to use ToExpression[ , MathMLForm]. The Maths Information Retrieval research group, that created the dataset of articles, did a great job. Posted by Giorgia Fortuna    July 6, 2015 at 1:31 pm
 The use of tau instead of pi would ruin the simplicity of Euler’s equation. That for me eliminates it from consideration as a replacement. Posted by Bruce Crawford    June 28, 2015 at 2:41 pm
 No, it wouldn’t. e ^ (tau i) = 1. That’s just as beautiful if not more beautiful than the original. Posted by Alexi G    August 7, 2016 at 11:36 pm
 The total leaf count for formulas in pi was 8 891 003, and 8 991 052 for formulas in Tau. The ratio of these two numbers is approximately 0.989, meaning that the total number of leaves stays almost constant. Posted by Giorgia Fortuna    June 28, 2015 at 2:52 pm
 Giorgia, I enjoyed reading your article. To improve my use of Mathematica, would you please add the code that you used to generate your graphics and plots. Thank you for your consideration. Posted by Anonymous 2    June 29, 2015 at 1:24 pm
 Hi, I am glad you enjoyed reading the article! The nice pictures were made using WordCloud, the rest was simply made with ListPlot. The very first picture was made in this way: WordCloud[MapAt[ToString[#, TraditionalForm] &, formulasContaining2pi, {All, 1}], Disk[], MaxItems -> 100, WordSpacings -> 5] where formulasContaining2pi is an array of formulas containing 2pi that I extracted from scientific articles. The red and black pictures were made using WordCloud as well, by playing with different options. The first was creating using WordCloud[ToString[#, TraditionalForm] & /@ formulasBetterinPi, Rasterize[Style[\[Pi], 80, FontFamily -> “Helvetica”], “Image”], MaxItems -> 60, ColorFunction -> Red, Background -> Black] where formulasBetterinPi is an array of formulas containing that are simpler in Pi. If this does not answer your question (or if you want to know more), let me know. Posted by Giorgia Fortuna    July 6, 2015 at 1:52 pm
 It looks like, with the two Wordclouds the most common formulas are in terms of tau. This paper in general suggest to me that both should be taught. That way, we can benefit knowing when ether are used as to reduce complexity the complexity is reduced. This suggest a combined third Wordclould with both, after all 2pi is changed to tau. Posted by John    November 16, 2015 at 9:43 pm
 It does look like the most common formulas are in terms of tau indeed, however by looking at the formulas people deal with when doing science it turns out that using pi is the best choice. I agree that using both Pi and Tau would make everything look simpler, but this argument would apply everywhere… If we could use one new symbol for every composed expression we could have extremely simple formulas but we would have to learn and remember lots of symbols. I believe the purpose of this post was to say that if we have to choose only one symbol, than Pi is the one we should pick. Posted by Giorgia    November 18, 2015 at 10:40 am
 I am not sure I understand your objection. If by “constant multiples of pi which could easily hide arbitrary factors” you mean things of the form “c Pi”, where c is a parameter, then no matter what c is, the pure formula “c Pi” is simpler than “c Tau/2″. Posted by Giorgia    February 12, 2016 at 5:49 am
 No, it becomes c’.tau, where c’ = c/2 Posted by Gavin    March 14, 2018 at 5:46 pm
 I appreciate this exercise, which begins to answer the question would tau really simplify the steps in working with formulas. Whether or not an advanced math practitioner has 46 or 50 leaves in solving an equation is minor compared with the insight that tau brings to equations, especially for fledgling math practitioners. Tau is a better way to teach math, and to students who learn math with tau, pi gains greater meaning. You’ve already given the example of the tau unit circle. This extends also to radians. There are τ radians in a circle, τ radians = 360 degree; 1 radian = 360/τ; 1 degree = τ/360. Tau is 1 turn or 1 cycle or 1 rotation. In its wave form, in your example, sin (τ/4) = cos(τ) = 1. In this form, it becomes obvious that the value of the cosine at one cycle is 1 and that the sine is ¼ turn or 90 degrees out of phase with the cosine. Likewise, with sin (¾τ) = cos (τ/2) = –1, the sine wave passes through -1 at ¾ of its cycle and the cosine wave passes through -1 at half its cycle. The notation becomes easy to grasp. Further, with an like equality z = π, the meaning is not immediately clear and neither is z = τ/2, however if we simplify to 2z = τ, it becomes clear that we need 2 of whatever z is to compete 1 rotation. I am not a proponent of abolishing π in all cases. After learning τ, π becomes an elegant way to express one half rotation and its meaning becomes clear. Posted by Richard    January 26, 2017 at 5:12 pm
 Quite an interesting analysis! :-) Though it is almost impossible to get rid of many biases… For instance, the fact that π is today the standard constant may have lead to using formulas somehow optimized for π: for instance, is is well known that π² ≈ 10 (and you can find this formula in many places), but the fact that √τ ≈ 5/2 is seldom noticed… Another possible bias was observed by a previous commenter: if for some reason you write « let x = rπ for r a rational number », then clearly the “correct” translation in the τ convention would be to write « let x = rτ », not « x = rτ/2 »… I also found a third, important bias: a few formulas involving the character π do not refer to Archimedes’ constant! For instance, I think that the formula « πN » which appears in one figure actually refers to the field of probability or statistics, where π is just an arbitrary letter to denote a probability (πN then being the expected number of occurrences when you carry out N experiments). But, conversely, some biases may let τ look better than it really is! Typically, I think that multiplication is a simpler concept than division, so that the gain when you go from « 2π » to « τ » is smaller than the loss when you go from « π » to « τ/2 »… Also, formulas like « π²/6 » and « τ²/24 » do technically have the same tree complexity, but the former is obviously (slightly) simpler, because 6 is simpler than 24…! So, what for a conclusion…? The author’s analysis is much worthy, and undoubtedly needed to be performed; however, at this point the only viable conclusion would be that the differences found are too small to be sure that they overcome the effect of biases. Personally I am rather a proponent of τ; but the case is tough, and in some fields, τ/2, or also τ/4 or √(τ/2) may appear as better primitive constants… As a final word, I would say that, contrary to many people, I definitely think that it is a worthy investigation to try and understand which is the most fundamental constant, because is help unfolding the very deep structure of mathematics…! Posted by Rémi    February 5, 2018 at 7:35 pm
 These are good observations! I agree with all of them except the very first one: I don’t believe it’s a bias to think that usage of Pi as the more common constant should continue if the stepping stones with which we’ve built theorems on have used it. In fact, I think it’s a valid argument for the insistence of Pi as the standard constant (and Giorgia sort of addressed this in another response earlier, where she suggested it would be an added unseen cost to completely relearn already established theorems just to gain an intuition that is not guaranteed when you use Tau in place of Pi). Past theorems may have been optimized for Pi, but it is not the job of researchers, and certainly not students today, to have to relearn theorems in terms of Tau when there is very little incentive to do so. Seems like it’s between a rock and a hard place, unfortunately. Just my \$0.02. Posted by Jesse    February 9, 2018 at 4:58 pm