### Reasons#

The elliptical area formula πab is easy to remember by analogy with the circle, but surprisingly there is no elliptical circumference formula!

After investigation, it was confirmed that there is no precise elementary function expression, and it is necessary to use infinite series expansion, and there is even a mathematical branch of elliptic integrals.

### Thoughts#

The circle is a special case of an ellipse, and π is a simple and crude "artificial" number.

Only when the eccentricity of the ellipse is equal to 0, there is a fixed ratio of circumference to diameter - just like many empirical coefficients in formulas are obtained by assuming a direct proportion within a small range, it cannot be generalized to more general cases. Just like the formula for a pendulum T=2π√(L/g), it is accurate only within a small range.

Moreover, considering that such an important constant, apart from being used in calculations related to perfect circles and testing computer computing power, has no other use. Therefore, it should start from the ellipse to derive the formula for the curve's circumference, so that it is possible to discover more important, more in line with general rules, and more widely applicable "natural" numbers.

To clarify, it is not denying the significance and accuracy of π as a number, it is also objective, "artificial" is in quotation marks. Many empirical coefficients in engineering formulas are "correct" and valuable. It's just that I feel that π, among these "artificial" and "natural" numbers, is fundamentally closer to "artificial" rather than "natural". On the other hand, a constant like e, it cannot be denied that it is a more "natural" constant than π, just as its name suggests.

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Lofter old post