Classic Iterated Function Systems

Right Triangle Divided by Nine Count

The triangles are numbered as follows:

Here is a Python program for counting the number of fractals, where two fractals that are mirror images across a vertical reflection are counted only once. The integers correspond to the triangles that are removed as indicated in the diagram above.

from itertools import combinations
a = [1,2,3,4,5,6,7,8,9]
m = 3                       # number of rectangles to remove
b = list(combinations(a,m)) # all 9 choose m combinations in lexicographical order
b2 = [list(x) for x in b]
c = []
dic = {3:1, 6:5, 7:4, 1:3, 5:6, 4:7}  # dictionary for symmetric locations
for k in range(len(b2)):
    d = [dic.get(n,n) for n in b2[k]] # replace triangles in kth item with symmetric locations
    d.sort()                          # put in lexographic order after swap
    if d not in b2[0:k]:              # check if already in list up to kth position
        c.append(b2[k])               # if not, append to the list
c2 = [tuple(x) for x in c]
print(c2)           
print(len(c2))

The output is shown in the following table:

Click to see all the fractals when 1, 2, or 3 triangles are removed, and the 8 symmetric fractals when 4 triangles are removed.