[ExI] symmetrical 11-Venn discovered

[Anders Sandberg]

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

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.extropy.org/pipermail/extropy-chat/attachments/20120814/4a31d533/attachment.html>