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 f() f()