Symmetry Of Fractal Objects

Question

I'm interested in knowing whether there has been any study of the symmetries of fractal objects. The symmetries acting upon an object usually, if not always, form a group to my knowledge, but I was wondering whether the same could be said in the case of fractal objects, which often have non-integer dimensions.

I'll try and describe a couple of possible fractal objects that might have some interesting properties. For example, let me make a construction using the sierpinski gasket. Label the corners $0$ (at the origin), $A$ (the point with the highest $y$ value), and $B$ (the point with the highest $x$ value). Say that $A$ has co-ordinates $(\frac{X}{2},Y)$, and hence $B$ has co-ordinates $(X,0)$. Then make a copy of the gasket and translate it by $(X,0)$, and then make another copy of the original gasket and translate it $(\frac{X}{2},Y)$. Then you have made a dilated copy of the gasket. Iterate this process ad infinitum. Then you have the first object I would like to look at. Now, the contraction mappings used to create the Sierpinski Gasket all fix this new object, as do their inverses. Furthermore, the reflection which maps $0$ to $0$ and $A$ to $B$ is fixes this new object too. Unlike the original gasket, however, as well as any triangle, the other elements of $Dih(3)$ don't fix this new object, because the two other reflections and the non-trivial rotations can't be defined. So for an initial problem, could anyone tell me what the symmetry group of this object is isomorphic to, and does anyone else have any experience with this area?

EDIT: I realised that only one of the three contractions on the Sierpinski Gasket is well-defined on this new object so the group of symmetries on it is just $\mathbb{Z}_2\times\mathbb{Z}$.

Are there more interesting examples maybe?

Answer 1

Of course, there's a lot that can said about the symmetries of fractal objects. The symmetry group of the Sierpinski gasket, for example, is $D_3$.

One interesting application of the symmetries of fractal objects is the construction of self-similar tilings of plane. For example, the Gosper snowflake:

This consists of seven copies of itself scaled by the factor $1/\sqrt{7}$. It's dimension is 2 but the dimension of its boundary is $2\log(3)/\log(7) \approx 1.129$. By repeatedly scaling up and shifting, we can generate a tiling of the plane:

Of course, then the symmetry group is necessarily one of the 17 wallpaper groups. For more information you might Google scholarly articles on Self-Affine Tiles.