Applicability of Maths to the Universe

[polymath257]

polymath, can you tell me if there is any issue with this answer?

Why is this number equal to 3.14159....? Why is it not some other (ir)rational number?

Answer is that with the usual Euclidean metric that is the number that one gets, the value of π is dependent on the geometry that is being used, so on a sphere the π used to obtain the area will be different.

Another question to ask is that if S=π(sub1)r^2 is the area of the circle and C=π(sub2)2r is the circumference why π(sub1)=π(sub2)=π ? What geometries or metrics will result in π(sub1)≠π(sub2) ?

From:
https://math.stackexchange.com/questions...to-3-14159


If no issues, then this is generally the answer that satisfies my question. Even if pi was definitely a normal number, I still would've asked this question.

On a sphere, the ratio between radius and area (or circumference) of a circle is not a constant at all. For example, if you are at the north pole, the equator is a 'circle' whose circumference is 4 times the radius (not 2pi). But, the limit as the radius goes to 0 will give you pi back. Similarly for the area.

We choose circles because we have a distance function making R^2 into an inner product space, leading to the Pythagorean metric. If you use a 'taxicab' metric, the ratio would be 4sqrt(2) for the circumference and 2 for the area.

Or, you could live on a cone. At the vertex, you would have a constant ratio of radius to circumference (and for that matter between area and the square of the radius) and the ratio would depend on the opening angle of the cone, but at other points it would be the usual value of pi (if you measure distance in the cone).