(That was quick!  I thought I'd have to wait weeks for my cue.)

> Anton Sherwood wrote:
> > ... tau is irrational (in a sense the most irrational number) ...

JRG wrote:
> Forgive my ignorance, but in what sense?

In the sense that the rational approximations to it
converge most slowly.

Every real number can be expressed as a continued fraction:

	x = a0 + 1/(a1 + 1/(a2 + 1/(a3 + 1/(a4 + 1/( ... )))))

where each a is a positive integer (except that a0 may be zero or
negative).  If x is rational, the sequence {a} is finite; conversely,
if you truncate the sequence, you get a rational number.

Now, it's obvious that the residue 1/(ai + ...) is somewhere
between 1/ai and 1/(ai+1), right?  Which implies that if ai is
high, the error in cutting off the sequence before ai is small.

Example.  The sequence for pi begins 3,7,15,1,292 ...
So the rational approximations to pi are:

	  (3)	     3/1	3.0
	  (7)	    22/7	3.143
	 (15)	   333/106	3.14151
	  (1)	   355/113	3.1415929 : pretty good!
	(292)	103993/33102	3.141592653
		     ...

(Which is why the knot you just saw is a union of 355 spheres.)

To make the error at each step as big as it can be, you pick the
smallest possible integers for {a}, i.e. all 1.  What does that give?

	x = 1 + 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ ...))))
	x = 1 + 1/x
	x^2 = x + 1

This quadratic has one positive root: (1+sqrt(5))/2, called tau,
the golden ratio.  (Also called phi, but I use phi for angles.)

-- 
Anton Sherwood  --  http://www.ogre.nu/

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