&pi,

*the ratio of the circumference to the diameter of a circle*,
is one of the most important constants in Math. However, &pi could
just as easily have been defined as

*the ratio of the circumference to the radius of a circle*.
This would not change math in any serious way, but
it would make some formulas simpler.
Think about how often `2*&pi' comes up in formulas.

This theme was explored by

Bob Palais in

this article.
He makes a good case. I look at two examples
not in the article, one of which supports his case, and the other
is a matter of taste.
During this blog I will denote the ratio of Circumference to Radius by PII.

EXAMPLE ONE:Consider the volume and surface area of an n-dim sphere.
There is no closed form formula (that I know of)
but there is a recursive formula.
See

this.The following table shows, for each n, the volume of an n-dim
sphere divided by R

^{n}.

n | Trad Vol/R^{n} | New Vol/^{n} |

1 | 2 | 2 |

2 | &pi^{2} | (1/4)*PII^{2} |

3 | (4/3)*&pi^{3} | (1/6)*PII^{3} |

4 | (1/2)*&pi^{4} | (1/32)*PII^{4} |

5 | (8/15)*&pi^{5} | (1/60)*PII^{5} |

6 | (1/6)*&pi^{6} | (1/382)*PII^{6} |

7 | (16/105)*&pi^{7} | (1/1640)*PII^{7} |

Is the New Volume easier or harder?
A little easier in that all of the numerators are 1.
But no real pattern.
Similar is true for surface area.
Are these formulas better? That is a matter of taste.

EXAMPLE TWO:The Zeta Function is

&zeta(n) = &sum r

^{-n}(The sum is from r=1 to infinity.)

It is known that

&zeta(2n) = (-1)

^{n-1} ((2*&pi)

^{2n}/2(2n)!)B

_{2n}
where B

_{n} is the nth Bernoulli Number.If we use PII instead we get the simpler

&zeta(2n) = (-1)

^{n-1} ((PII)

^{2n}/2(2n)!)B

_{2n}
This is BETTER!

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Posted By GASARCH to

Computational Complexity at 8/07/2007 03:06:00 PM