Archimedes revisited

My proposal to use the name “Archimedes” for Θ = 6.283… got a reply from Peter Harremoës from Denmark. Peter argues that engineers and artisans in Archimedes’ time found it more efficient to measure circles by their diameter d and not with the radius r = d / 2, so that Archimedes calculated π = Θ r / d = Θ / 2 = 3.141…. Hence the latter number is called Archimedes’ number, historically. Peter discovered that the Persian mathematician Jamshid al-Kashi in 1424 apparently was the first to use 6.283… as a separate number. Hence Peter suggests to use Al-Kashi’s constant τ, where he also adopts the symbol tau as do Robert Palais, Michael Hartl and Vi Hart as shown on my proposal page.

Bear with me. I have been aware of Archimedes’ historical position, see the proposal text indeed. The point is that there is only one mathematical constant. The values 6.283… and 3.141… are mere transforms of the same constant. Thus we should select only one name. Moreover, Θ / 2 would be vocalized as “one half Archimedes” such that Θ is a unit of account and not just a number discovered by some person.

It may be fun to say that Isaac Newton discovered one Newton and Alessandro Volta discovered one Volt while Archimedes discovered only one half Archimedes, but that would stretch what we mean by a mathematical constant. Archimedes really was the first to determine the mathematically correct way to catch that mathematical constant. So, there is no conflict between using the Archimedes as the unit of account and accepting that 3.141… was historically seen as Archimedes’ number.

Subsequently, Archimedes’ reasoning was didactical, since he adopted the common usage in his day of the diameter. We have switched to the radius so let us switch consistently. Perhaps Al Kashi was instrumental in that switch but he was aware of Archimedes’ important discovery and I like to think that he would agree that Archimedes receives all honour.

I have really thought deeply about tau. I really don’t mind what is actually chosen as long as it works best in education. I considered tau independently from the others but rejected it because it looks too much like r. The capital theta looks nicely like a cirle. The little mark in the center is not a slash like for the diameter or crosssection Ø. My proposal is that we research what works empirically best in education.

It might be a nice idea to put the choices up for an opinion poll. The true vote would be to use either current π or one of the alternatives for 2 π. But this vote would be biased when there is a difference in opinion about what that alternative will be. A vote now cannot be decisive since it is a matter of empirical research. However, voters can have an opinion about what should be tested in that research, or have a forecast about what would work best, at least for themselves. Thus, an opinion poll can be somewhat informative.

See this page for the vote.

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