[ExI] symmetrical 11-Venn discovered
Anders Sandberg
anders at aleph.se
Tue Aug 14 08:10:34 UTC 2012
On 14/08/2012 02:51, Mike Dougherty wrote:
> Is there a relationship to fractal imagery? Why are the numbers prime?
There is a theorem by Henderson that states that a necessary condition
for the existence of a symmetric n-Venn diagram is that n is a prime number.
D. W. Henderson,/Venn diagrams for more than four classes/, American
Mathematical Monthly, 70 (1963) 424-426. http://www.jstor.org/stable/2311865
This later paper (which is freely accessible) states the basic argument
(which is very clean and simple), and then goes on patching a serious
hole in it:
http://www.math.umn.edu/~webb/Publications/monthly645-648-wagon.pdf
<http://www.math.umn.edu/%7Ewebb/Publications/monthly645-648-wagon.pdf>
Fractals: I think this is mainly because of how the human mind works. We
are bad at maintaining large amounts of information in working memory,
so we like to work with modular or recursive objects: while the object
is complicated, the method of making it is simple. So the successful
methods of making multidomain Venn diagrams (see the examples on
Wikipedia, http://en.wikipedia.org/wiki/Venn_diagram ) all tend to
produce patterns that are fractal-like.
In general, I expect that fractals often emerge because the universe
also tends to follow simple rather than complex programs. Even when
these programs make organisms it is easier to grow and evolve recursive
structures than non-recursive ones. So the important and deep question
is what forces affect what programs get run, and what kind of simplicity
is optimized for.
BTW, here is a stained glass Venn diagram from Gonville and Caius
college in Cambridge:
http://www.flickr.com/photos/arenamontanus/3212655985/
I was eating breakfast in the hall when I noticed a familiar diagram in
the stained glass windows. Then I noticed the other windows:
http://www.flickr.com/photos/arenamontanus/3213502484/in/photostream/
http://www.flickr.com/photos/arenamontanus/3213502248/in/photostream/
Venn, like the others, was a fellow there.
--
Anders Sandberg,
Future of Humanity Institute
Philosophy Faculty of Oxford University
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