I'd read a while back about Tau Day and the idea that *τ=6.28...* is a better mathematical constant than *π=3.14...*, for a variety of reasons. (Go read the Tau Manifesto and learn several of them if you haven't already.)

One of them was the idea that far from being a strength of π, the area formula *A=πr ^{2}* is actually a weakness, because it camouflages the fact that there should naturally be a ½ in there, deriving from its integral relationship with the circumference formula. By contrast,

So anyway. I was thinking about the volumes of spheres, and I recalled that the formula was *V=⁴/₃πr ^{2}*; of course I knew that because I'd memorised it many years ago, not that it had any reason behind it:

A=πr^{2} |
V=⁴/₃πr^{3} |

A=½τr^{2} |
V=⅔τr^{3} |

V_{2} (area) |
=πr^{2} |
=(1/2)τr^{2} |

V_{3} (volume) |
=(4/3)πr^{3} |
=(2/3)τr^{3} |

V_{4} |
=(1/2!)π^{2}r^{4} |
=(1/2!∙4)τ^{2}r^{4} |

V_{5} |
=(8/5∙3)π^{2}r^{5} |
=(2/5∙3)τ^{2}r^{5} |

V_{6} |
=(1/3!)π^{3}r^{6} |
=(1/3!∙8)τ^{3}r^{6} |

V_{7} |
=(16/7∙5∙3)π^{3}r^{7} |
=(2/7∙5∙3)τ^{3}r^{7} |

V_{8} |
=(1/4!)π^{4}r^{8} |
=(1/4!∙16)τ^{4}r^{8} |

But wait! What if we take that awkward extra power of 2 in the even-dimension formulas and distribute it over the factorial?

V_{2} (area) |
=(1/2)τr^{2} |

V_{3} (volume) |
=(2/3∙1)τr^{3} |

V_{4} |
=(1/4∙2)τ^{2}r^{4} |

V_{5} |
=(2/5∙3∙1)τ^{2}r^{5} |

V_{6} |
=(1/6∙4∙2)τ^{3}r^{6} |

V_{7} |
=(2/7∙5∙3∙1)τ^{3}r^{7} |

V_{8} |
=(1/8∙6∙4∙2)τ^{4}r^{8} |

Check it out! Even if we don't have a deep understanding of what a double factorial is or how to compute the *Γ* function, we can clearly see the recurrence relation among the various dimensions, and the relationship between the even-numbered dimensions and the odd-numbered dimensions, and that they're much more closely related than might first appear from reading the Wikipedia article on *n*-spheres that I linked above.

So, chalk up one more success for the τists!

"When I go to get a new driver's license... or deal with the city inspector... or walk into a post office... I find public employees to be cheerful and competent and highly professional, and when I go for blood draws at Quest Diagnostics, a national for-profit chain of medical labs, I find myself in tiny, dingy offices run by low-wage immigrant health workers who speak incomprehensible English and are rud to customers and take forever to do a routine procedure." --Garrison Keillor

Posted by blahedo at 10:26pm on 22 Dec 2010Comments