## Happy Tau Day!

Let’s celebrate by having 2 pies.

I mean, 2 pi!

Posted in science. Tags: . 4 Comments »

### 4 Responses to “Happy Tau Day!”

1. Calculus Says:

After watching this, I can’t help but wonder why we use pi=3.14159…. If I had to guess (which I like to do) I would guess that in ancient times when pi was first discovered, it was easier in general to deal with the diameter than it was to deal with the radius. The tools that you have available do impact what can easily accomplish.

Consider two of the biggest circles that would be seen most days and nights–the sun and the moon. It would be easier to identify the diameter of these celestial objects than the radius. The diameter would be the widest the sun or the moon (or any other circular or spherical object) gets. To get the radius, you can take the extra step of dividing the diameter by two, or you can try to find the centre of the circle and then measure from there. I’ll posit that it is easier to determine where a circle is widest than it is to accurately find its centre.

Of course, the above could be completely off (this is afterall a sort of reverse-guess, or backwards justification if you will).

Xan says:

I think you are quite correct!

2. CodeSlinger Says:

Calculus:

You are thinking analytically – which is generally a good thing, but it does make for a bit of a disconnect with how the ancients thought about geometry.

The analytic tradition in geometry is only a few hundred years old. For most of human history, geometry was synthetic – it was concerned with constructions that could be carried out with a straight edge and a compass.

And when you set out to construct a circle with a compass, you see right away that its primary attributes are its centre and radius. Everything else follows from these.

Like the ratio of the area of a circle to the square of its radius, which is the same for every circle, and which we denote by the symbol pi.

3. Calculus Says:

If I had to take another guess as to why we use ~3.14 for pi, it would be related to the ease and accuracy of numerical representation. Though the concept of pi was known in ancient days, there was always the problem of how to simply express it.

Pi is approximately 3.141592654.
Tau (as described in video) is twice that so it is about 6.28315307.

When it was first discovered, the numbering systems used at the time were extremely clunky for expressing fractions (let alone irrational numbers). Fractions (if memory serves from a high school history class so many years ago) were often expressed as sums of unit fractions (numerator = 1, denominator = another whole number). Using this scheme, both of these values can be approximated …

Pi is approximately 3 + 1/7 (or 3.142857)
Tau is approximately 6 + 1/4 (or 6.25)

The accuracy of (3 + 1/7) is far better than (6 + 1/4). The former is off by about 0.04%. The latter is off by about 0.53%.

4. CodeSlinger Says:

Calculus:

It is confusing to think of pi as a number. Think of it instead as a relationship; a proportion.

Its numerical representation in terms of fractions or decimal digits, or its lexical representation as a particular Greek letter are entirely secondary. These are just ways of writing its name, and focusing too much on its name obscures its meaning.

In my previous comment, I described pi as a ratio, which is an invariant property of all circles. But there is another ratio that leads us to pi, and that is definition of angle. An angle is the ratio of arc length swept out to the radius of the arc.

This leads to a natural unit of measure for angles, namely that angle for which the arc length is equal to the radius. This angle is called one radian. A rotation that sweeps out a half circle, for which the ratio of arc length to radius is pi, is thus called a rotation of pi radians.

Why is it natural to single out the half circle, and not the full circle? Because such a rotation is equivalent to a reflection, and all rotations are built out of pairs of reflections.

To see this, take any two nonparallel lines and call their intersection point the origin. Now pick any point in the plane of the lines but not on either line, reflect it across the first line and then reflect the result across the second line. You have just rotated the point about the origin by an angle equal to twice the acute angle between the two lines.

There is a limit to this procedure, of course: as the lines approach perpendicularity, the generated rotation approaches a reflection.

So all rotations are built out of a pair of reflections, but only one rotation is equivalent to a single reflection and that is a rotation by pi.

Thus, when we look at the natural properties of circles and rotations, we always come back to this relationship which the Greeks denoted by the letter pi. It is only after we realize the importance of this proportion that we have any reason to think about how to write its name.