mathsgiup

This is a pure combinatorial question:

 

To take it into the real world let's discuss Buckminsterfullerene:

 

http://en.wikipedia.org/wiki/Buckminsterfullerene

 

 

 

http://en.wikipedia.org/wiki/Truncated_icosahedron

 

See that:

 

Buckminsterfullerene has 60 atoms and 90 edges, edges are equivalent to connections, yes?

 

Now add 1 atom in the middle, there is a line or 'connection from each of the 60 atoms to this new atom:

 

Now we have, 61 atoms and 150 connections, see?

 

Combinatorially, brain connections is a massively huge number, easily more than atoms in the universe.

 

Let's say the maximum number of connections n atoms can have is X, as a function, we can represent this as f(n)=X

 

Now add 1 more atom:

 

This new atom can connect to maximally n other atoms so:

 

f(n+1) = X + n

 

but

 

f(n)=X therefore

 

f(n+1)= f(n) + n

 

but then, f(n) = f(n-1) + (n-1)

 

so we have

 

f(n+1)= f(n)+n = [ f(n-1) + n-1 ] + n = f(n-1) + 2n -1

 

now

 

f(1)=0 - dot

f(2)=1 - line

f(3)=3 - triangle

f(4)=6 - cross hatched square or tetrahedron

 

Dimensionality also has an effect.

 

In 2d you will see that f(n)= Summation (Binomial n,k) for k=1 to k=n-1.

 

binomial tree:

1

11

121

1331

14641

 

f(4) = 6 =

f(5)=1+3+3+1=8 - cross hatched pentagon

 

however 5 points in 3 dimensions, what happens,

 

.......work it out, it will take you a few hours

 

OR

 

type in

 

maximum number of edges and vertices in 3 dimensions

 

into google

 

Thanks,

D

 

 

 

 

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