The Circle Constant

A mathematical constant that we all know well is π, this concept is describing the “the circle constant”. This constant is defined as the ratio of a circle’s circumference to its diameter

\pi \equiv \frac{C}{D} = 3.14159265...

Why did we choose the diameter though? Why not define this constant based on the radius?! Unfortunately, there are many shapes with a constant diameter – yet there is only one shape with a constant radius, the circle!!

To illustrate the different choice we could make, try unrolling the circle below using the drag handle. The first figure measures distance with the diameter, the second with the radius!

Unit Diameter

Unit Radius

As you see above, we could have just as easily picked the radius to define the circle constant. This constant, when defined using the radius is called “tau” or τ.

\tau \equiv \frac{C}{r} = 6.283185307...

This choice is important! A lot of concepts are based on a unit-radius, rather than a unit-diameter! Radians, the unit circle, trigonometry, imaginary numbers, and many more!

Special Angles

We use π a lot is when referring to angles, specifically for special angles in radians. However, as we saw above π is defined with the unit length of the diameter, not the radius! So when we compare the “turn” - where one rotation is one turn - we see that our definition of π is always “off” by a factor of two!! Is it “wrong”, no, but it sure is confusing!

Turn	Radians (π)	Radians (τ)
0, 1	0, 2\pi	0, \tau
\frac{1}{12}	\frac{\pi}{6}	\frac{\tau}{12}
\frac{1}{8}	\frac{\pi}{4}	\frac{\tau}{8}
\frac{1}{6}	\frac{\pi}{3}	\frac{\tau}{6}
\frac{1}{4}	\frac{\pi}{2}	\frac{\tau}{4}
\frac{1}{3}	\frac{2\pi}{3}	\frac{\tau}{3}
\frac{1}{2}	\pi	\frac{\tau}{2}
\frac{3}{4}	\frac{3\pi}{2}	\frac{3\tau}{4}

Evolving Ideas

If you are interested in the consequences of choosing “tau” instead of “pi” then you should go and read the Tau Manifesto. Michael Hartl does an amazing job of explaining why we probably picked the “wrong” unit dimension from the standpoint of clarity of the idea - we should have used the radius, not the diameter. As a consequence of this early blunder, there are factors of two that have propagated throughout the world, because π is only equal to half a turn!

This causes unnecessary pain in learning and using the idea, even small pain of converting from degrees to radians, and remembering to divide/multiply by two!

What if we could from __future__ import tau and use that in our writing of equations, and let the document system compile to whichever is more appropriate for the reader?

Circular Area: A = \frac{1}{2}\tau\pi r^2
Euler's identity: e^{i\tau}=1e^{i\pi}=-1
Fourier transform: f(x) = \int_{-\infty}^\infty F(k)\, e^{\tau ikx}\,dkf(x) = \int_{-\infty}^\infty F(k)\, e^{2\pi ikx}\,dk
Polar coord integrals: \int_0^{\tau}\int_0^\infty f(r, \theta)\, r\, dr\, d\theta\int_0^{2\pi}\int_0^\infty f(r, \theta)\, r\, dr\, d\theta

Beyond dividing by two and simple substitutions, what if we could move from a static representation of an idea to a computational representation?

What if we could bring some of the things that are working well in programming over to the world of communication? How do we bridge that gap?

What would an import statement in writing look like?

How would you release a new version of an idea? How do we connect fragments of ideas across our projects?

iooxa

These are some of the questions and ideas that we are thinking about at iooxa.com. We are working on an editor for interactive content like this! If you would like to learn more please sign up for our mailing list or checkout some of our open source projects.

Written with love by iooxa