This rep-7 fractal is sometimes known as the Sierpinski Snowflake. I believe that this is because of perceived similarities between its construction and that of the Sierpinski Triangle. This fractal has a dimension of less that 2.
The central element can be removed to to create the fractal to the right, which I term the 0-cyclohexamer. (This is one of two cyclohexamers with dihedral symmetry.) This also has a dimension of less that 2, and also less than that of the Sierpinski Snowflake.
If the interior and exterior boundaries of this fractal are examined it will be seen that they have the same shape. This means that a transform can be added to the IFS to fill the gap. (The 6 elements of the 0-cyclohexamer are reduced 3-fold with respect to the original figure; the element added to the central is reduced 3½-fold, and rotated 30° with respect to the original figure.
This is the Koch Snowflake, which is a fractal with a similarity dimension of 2, which tiles the plane, and with an appropriate weighting of the transforms (the central element being weighted at thrice the others) a uniform measure.
There is a teardrop fractal, derived from the Sierpinski Snowflake, as shown on the left.
It can be seen, that like the 0-cyclohexamer, this contains interior voids whose boundaries have the same shape as the exterior boundary. This also can be filled in to create a new plane tiling fractal discovered by R. William Gosper.
