Subject:
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Re: Good read (Was: [faq FAQ How do I format an entry in the Lugnet FAQ?])
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Newsgroups:
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lugnet.off-topic.geek
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Date:
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Fri, 16 Jul 1999 02:30:56 GMT
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Viewed:
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3845 times
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In lugnet.faq, Sproaticus <jsproat@io.com> writes:
> Robert Munafo wrote:
> > My web pages (on http://www.mrob.com/ ) [...]
>
> Okay, I've been reading aout numbers for a half hour now. Very interesting
> reading, at least for a geek such as I. :-,
Me too! Great stuff, Robert! Say, here's a fun tidbit about the golden
ratio if you're interested --
If you plot multiples of phi on the unit interval [0,1), then the nth
multiple always falls into the largest "untouched" sub-interval. That is
to say, if you have the iterated map
f(n) = 0, n = 0
f(n) = frac(f(n-1) + phi), n > 0
where frac(x) is x - int(x), then as n counts upwards, the plot goes like
this:
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
|------|------|------|------|------|------|------|------|------|------|
0 x
1 x x
2 x x x
3 x x x x
4 x x x x x
5 x x x x x x
6 x x x x x x x
7 x x x x x x x x
8 x x x x x x x x x
9 x x x x x x x x x x
10 x x x x x x x x x x x
11 x x x x x x x x x x x x
:
etc.
(Or visualize it as points on a circle, since eventually the interval
connecting .94 back to 0 through 1 gets chosen. :-)
IIRC, any irrational can be used to fill the unit interval (via its
multiples) this way, but only the golden ratio conveniently chooses the
largest untouched region at each iteration. I don't remember the proof (my
dad showed it to me many years ago) but it's got some interesting
applications, one of the most interesting of which is the subject of
_phyllotaxis_ in plant left arrangements to help maximize light absorption
for certain leaf-shapes. :)
Another interesting application (my personal favorite) is choosing a
least-visited area of the screen when doing progressive image refinement on
an unknown image (such as a 2D fractal) -- although integer bit-reversal
works just as well at partitioning the space and doesn't have the same
floating point errors to worry about.
--Todd
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