The nice thing about this mathematical controversy is that everyone can understand it.

Τ (tau) equals Π times two. The circumference of a circle is 2Πr where r is radius of that circle, so it also equals Τr. Another way to express this is Πd where d is the diameter, or alternatively Τd/2 . These formulae are ratios.

Here’s what they’re arguing about. A circle is a real thing. The ratios mentioned above express, perhaps, something about the fundamental nature of reality. Putting the number ‘2’ in any of them is a bodge to get the result right. It would be better to have a formula that didn’t have a number in it. Two versions of such a formula are available: Πd and Τr.

This raises the question, which is the more fundamental measurement of a circle, the radius or the diameter? If a circle is defined as the set of points a common distance from a given point, the centre, then the fundamental properties of a circle are the position of the centre and the radius. If you take an abstract circle with no set position but just its size, the only property is the radius.

That’s what the Taoists are arguing: the radius is the measurement that should be used and so children should be taught to multiply it by Τ, instead of using Π.

A circle is defined by its centre and its radius. But which point of view does this support?

For circumferences and turns, T. The circumference is Tr, and there are T radians in a circle.

But for the area of a circle, Π. The area of a unit circle is Π.

The Tau manifesto lists some formulas which are simpler with T. But the Pi manifesto lists some formulas which are simpler with Π.

ESR’s solution is the best. Agree on a list of important formulas, and see whether the list is shorter using Π or T.

The history of the adoption of Π is interesting.

“which is the more fundamental measurement of a circle, the radius or the diameter?”

The radius, surely? If you chain a cow to a post, it will eat a circle of grass with a radius as long as the chain. That’s no different to using a pair of compasses to draw a circle.

If you have a ‘circle of friends’, you start with yourself in the middle, and then you have links in each direction (from work, from the neighbourhood, from school days, hobbies, whatever) and there’s a cut off point in each direction where somebody is no longer ‘friend’ but merely ‘colleague’ or ‘neighbour’. So the distance from the marginal ‘colleague’ to yourself is the same as the distance from the marginal ‘neighbour’ to yourself but in different directions.

You decide the radius first, the diameter is merely the end result, it just so happens that the diameter is twice the radius.

“And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and…a line of thirty cubits did compass it round about….And it was an hand breadth thick….” — Kings I, 7:23-26

So the measure from “one rim to the other” (the diameter) is primary. Also, pi=3.

Dom settles the argument.

“which is the more fundamental measurement of a circle, the radius or the diameter?”

I’ve mulled over this question for a bit, and would also ask:

“which is the more fundamental measurement of a circle, its surface area or its circumference?”

To my mind, it’s the surface area (conveniently given by pi r squared), making me a pi-ist, I’ve never felt the need to work out the circumference of a circle, but I have worked out the area. But if you think the circumference is more fundamental, then your a Tau-ist.

Then again, does a circle have area, or does that make it a disk?

“which is the more fundamental measurement of a circle, its surface area or its circumference?”

It all depends upon the number of dimensions you live in.

Is you were to live in a one dimensional world it would be the circumference but if you lived in a two dimensional world it would be the area. You cannot measure what you cannot envision. Basically if you were in the one dimensional world how could you specify area (okay Peano curves can but their pathological :-))

Assume you have a cirlce with radius r. Circumscribe a square about this circle. Now circumscribe a circle around the square. What is the area of this last circle?

Dom, it’s twice the area of the inner circle.

“‘which is the more fundamental measurement of a circle, its surface area or its circumference?’

It all depends upon the number of dimensions you live in.

Is you were to live in a one dimensional world it would be the circumference but if you lived in a two dimensional world it would be the area. You cannot measure what you cannot envision.”

No. A circle is a two-dimensional shape. Measuring its area or its circumference both requires two dimensions.

Perhaps what you’re getting at is that the formula for the circumference is a one-dimensional formula in that it only contains r once (2Πr), whereas the area contains r twice (Π(r^2)).

I’m tempted to say that the circumference is more “fundamental” than the area, and also the volume of a sphere. ((4/3)Π(r^3))

Therefore Tau wins.

You only have to ask: “what is the least amount of information needed to describe (a) a circle as a dimensionally-thin line centred on a fixed point, or (b) the maximum area that can be inscribed by rotating a line about a fixed point for the minimum distance?” You come up with

(i) The co-ordinates of the fixed point

(ii) The length of the line you project from that point.

So it’s 2 x Pi x R

And the area, beautifully drops out as the integral of 2piR(dR) which is piR^2

What to make of curiosities like 4/3 IIr^3, eh?