My book Conquest of the Plane (COTP) uses Θ = 2 π = 6.2831853…. My proposal in supplement to COTP is to use the name “Archimedes” for this particular symbol (“capital theta” with such assigned value). It will be a new mathematical constant.
One Archimedes thus is the circumference of a circle with radius 1. Another relevant format is 1 / Θ = 0.15915494… When you take a circle with a radius of about 16 cm then the circumference will be about 1 meter. A circle with radius r has circumference C = r Θ and surface S = r 2 Θ / 2.
In wikipedia (today 2012-02-17) we can read that π is already called “Archimedes’ constant”. However, we commonly speak about “pi” and not about “Archimedes”. Thus the name is free to use as the name of Θ.
There is some momentum in the USA to use tau, thus τ = 2 π. Bob Palais (2001) originated the idea but used an own new symbol (pi with three legs like m), Peter Harremoes and Michael Hartl convinced him to use tau, and Vi Hart has a presentation on YouTube. One argument is that tau refers to “turn” or Greek “tornos”.
However, turns are counted along the unit circumference cirkel C = 1 and not along the unit (radius) circle r = 1. Thus this association of tau would be confusing. Also, there is not much difference in writing r or τ. This can create a lot of confusion in handwriting, doing homework or checking exams.
Independently from Bob Palais I also came up with the idea that 2 π is the proper unit of account. Looking at the various symbols available on the keyboard I rejected tau because of the similarity to r, and settled for Θ since it neatly looks like a circle. I wasn’t quite happy with its uninformative name Theta but we had that also with pi or “meter”. Vi Hart pointed out that lower case theta is often used for angles which causes the problem of “theta Theta”. This disappears when we use “Archimedes”.
The proposal is to take the plane itself as the unit of account for angles. We know how to cut up a pie in those pointy bits radiating from the center, and we can do the same with the whole plane, getting a half plane, a quarter plane, etcetera. All those pointy bits add up to 1 plane. When we make circles we can find one with a circumference of 1 by which we can measure the angles. Comparing circles, the Archimedes unit shows up as a proportionality factor.
We need empirical tests whether this indeed works out better for students.
|Unit circumference circle =
|Unit radius circle =
|C = 1
r = 1 / Θ
angles α, β
functions Xur and Yur
|C = Θ
r = 1
arcs φ = α Θ and ψ = β Θ
functions Cos and Sin
See COTP page 41. Here Xur[α] = Cos[α Θ]. Angles can be measured by arcs or possibly be identified by them. It helps to separate the notions somewhat by putting emphasis on angles on the angular circle and arcs on the unit circle.