Counting the Number of Triangle Fractals

Each triangle fractal can be described by a sequence of three digits corresponding to the transformations applied to each of the three triangles in the pattern for that fractal. Because there are 6 possible transformations, there are 6x6x6 = 216 possible sequences.

The pattern for a symmetric triangle fractal can be described by showing the location of the upper triangle that is removed from each of the three triangles in the pattern, which we color as black. For the fractal to be symmetric with respect to a horizontal reflection, these three black triangles must be symmetrically located with respect to a vertical line through the middle of the original triangle. For the middle triangle, there is only one possibility for where the black triangle can be located.

In the left triangle, the black triangle can be placed in any of the three corners. Once that choice is made, however, the location of the black triangle in the right triangle must be chosen in a way to maintain the symmetry with respect to the vertical line. There are therefore only 3 possible choices for locating the black triangles among the three triangles in the pattern. These are shown below. Click on any of the images to toggle between the pattern and the corresponding triangle fractal.

     

 
Now as long as another transformation that is applied to a triangle does not change the location of the black triangle, the attractor will not change. For each pattern above, there are two allowable transformations for each of the three triangles that will preserve the symmetry. One of these transformations is the identity. The other is either 4, 5 or 6, the reflections across the altitudes. Which reflection is allowed for a triangle depends on the location of the black triangle within that triangle. For example, in the pattern

  

the possible transformations are {1,5} for the left triangle, {1,4} for the middle triangle and {1,6} for the right triangle. There are therefore 8 different 3-digit sequences that will leave the black triangles in this pattern in the same location and thus produce the same fractal. These are shown below.

The same is true for each of the other 2 symmetric patterns. This accounts for 24 of the 216 possible 3-digit sequences.

The remaining 192 possible 3-digit sequences correspond to triangle fractals that do not exhibit any symmetry. They do, however, come in pairs in which each member of the pair is symmetric to the other with respect to a horizontal reflection. If one member of the pair is given by the sequence (a,b,c), then the other member is given by the sequence (4•c•4, 4•b•4, 4•a•4), where 4•w•4 is the product (read right to left) in the dihedral group for the symmetries of the triangle.

To get a sense of why this works, first note that in reflecting the pattern horizontally, the left and right triangles will trade places. This is why the order of the sequence is reversed. Second, if F represents the iterated function system for the pattern (a,b,c) and if A is the attractor for that IFS, then T₄(A) would represent the horizontal reflection of A, where T₄ is the transformation corresponding to the horizontal reflection. Since F(A) = A and T₄T₄ is the identity transformation, we would have

T₄FT₄(T₄(A)) = T₄F(A) = T₄(A).

This shows that T₄(A) is the attractor (fixed point) for the IFS given by T₄FT₄.

According to the dihedral group multiplication table, we have

4•1•4 = 1
4•2•4 = 3
4•3•4 = 2
4•4•4 = 4
4•5•4 = 6
4•6•4 = 5

So for example, if we start with the sequence 251, its symmetric counterpart would be 163. These patterns and corresponding fractals are shown below.

251     

163     

 
Since the non-symmetric fractals can be grouped into pairs, there are essentially 96 different non-symmetric triangle fractals.

See a Gallery of all 99 triangle fractals.