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. What's wrong with the "airline marginal cost pricing" argument? What I add to the numerator depends on whats in the denominator. wraps angles in lambda, in radians, to the interval [0, [0, 2*pi]. Thank you. To improve my use of Mathematica, would you please add the code that you used to generate your graphics and plots. 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. 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.

This new convention would require changing from πi to τ/2 as well, but that doesn’t affect the complexity of πi. Accelerating the pace of engineering and science. However, the above formula shows something else that I want to point out. 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. The nice pictures were made using WordCloud, the rest was simply made with ListPlot. Seems like it’s between a rock and a hard place, unfortunately. But then why do τ supporters believe that we should switch to this new symbol? Use MathJax to format equations. 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. I got them from this website: 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]. That way, we can benefit knowing when ether are used as to reduce complexity the complexity is reduced. June 28, 2015 — Giorgia Fortuna, Advanced Research Group ... 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. MathJax reference. 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. However, this is not the end of the story. Today (6/28) is another math day: 2π-day, or Tau Day (2π = 6.28319…). So what was the final total leaf count for all considered formulas (including duplicates) in pi form vs tau form? This is because only 0.4% of formulas have complexities greater than 50. 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. Should I speak up for her?

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. The red and black pictures were made using WordCloud as well, by playing with different options.

rev 2020.11.5.37957, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. What’s most helpful to remember about them is that sin= cos(2 π) = 1, and sin = cos(π) = –1.

That for me eliminates it from consideration as a replacement. The notation becomes easy to grasp. where formulasBetterinPi is an array of formulas containing that are simpler in Pi. 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. For instance, if you have $\dfrac{\pi}{5}$, and you add $2\pi$ to it, then you should get $\dfrac{11\pi}{5}$. 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. 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. June 28, 2015 — Giorgia Fortuna, Advanced Research Group. Please enter your comment (at least 5 characters). Choose a web site to get translated content where available and see local events and offers. Tau is 1 turn or 1 cycle or 1 rotation. I get it now.

I also found a third, important bias: a few formulas involving the character π do not refer to Archimedes’ constant! You’ve already given the example of the tau unit circle. 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. 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. Posted by Mike Burns June 28, 2015 at 9:50 am, Posted by Anonymous June 28, 2015 at 2:26 pm, Posted by Bruce Crawford June 28, 2015 at 2:41 pm, Posted by Alexi G August 7, 2016 at 11:36 pm, Posted by Anonymous 2 June 29, 2015 at 1:24 pm, Posted by John November 16, 2015 at 9:43 pm, Posted by Giorgia November 18, 2015 at 10:40 am, Posted by Anonymous February 12, 2016 at 2:06 am, Posted by Giorgia February 12, 2016 at 5:49 am, Posted by Gavin March 14, 2018 at 5:46 pm, Posted by Richard January 26, 2017 at 5:12 pm, Posted by Rémi February 5, 2018 at 7:35 pm, Posted by Jesse February 9, 2018 at 4:58 pm, Posted by Dave September 30, 2019 at 8:04 am, New Publications Using Wolfram Technologies, New Wolfram Language Books on Wolfram|Alpha, Calculus, Applied Engineering and System Modeler.

Thanks for contributing an answer to Mathematics Stack Exchange! This means that 1/4 of a circle corresponds to 1/2 π radians, or π/2, and not a quarter of something!

For instance, these are some that would be simpler in τ: 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. This suggest a combined third Wordclould with both, after all 2pi is changed to tau. I have an angle with a given radian measurement and need to express it differently by adding integer multiples of $\pi$. After learning τ, π becomes an elegant way to express one half rotation and its meaning becomes clear. For example, the expression 4 π² would simply become (τ²). It might be that formulas in physics look simpler in τ, but formulas in other subjects do not.

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