Back to Fractals

Figure 2: (top) 2-simplex for {1/a} Sierpinski Gasket, and bottom: after collapsing a2-a3 to a single point Figure 1: Solution for space {3/a} Sierpinski Gasket

We start with the solution to the problem posed at the end of the last blog entry. The sketch of the correct {3/a} space is shown in Fig. 1. Note the vertices of the holes are all self-consistent which is an important part of the solution. Thus, the three holes are presented so there is no overlap of their vertices.

As before the fractal mass density is: rho(f) = {N(s) - N(h)/ N(s)}

so: rho (f) = {9 - 3/ 9} = {6/9} = 2/3

The fractal dimension D_f = 1/ rho(f) = 1/ (2/3) = 3/2

Now, let's look at the {1/a} base space but in a new guise, this time as an oriented 2-simplex(see Fig. 2- top). For this oriented 2-simplex we can write: a1a2a3 = a2a3a1 = a3a1a2 = -a1a3a2 = - a3a2a1 = -a2a1a3, so that provided the arrows can also be applied to the Sierpinski Gasket {1/a} base space, the two are homeomorphic. Consider now the boundary of a similar 2-simplex such that:

@2(a1a2a3) = a2a3 - a1a3 + a1a2

By definition, the group C_n(x) of oriented n-chains (e.g. a2 a3 is a 1-chain) of x is the free-abelian group generated by oirentied simplexes of x. Thus, every element of C_n(x) is a finite sum of form SIGMA_i s(i)m(i) where the s(i) are the n-simplexes of x and m(i) ( Z.

Problem:

Take the 2-simplex at the top of Fig. 2 and let a2, a3 collapse to a point as shown. Call the lower branch of the loop 'Y' - a sub-complex of the simplicial complex X.

If the generators for C1(x) are: a1a2, a2a3, a3a1, find:

a) the generators of C1(Y)

b) the value of Z1(X,Y) and H_k(X,Y) = kth relative homology group arising frmo sub-complex (C(Y)) of chain complex C(X)

Hints: The factor quotient group H_n(x) = Z_n(x)/B_n(x); for all chains of Y set = 0

We will look at this solution next time.