Square divided by nine

The scaling factor from one generation to the next is 1/3. This reptile has four-fold rotational symmetry and mirror symmetry, allowing for choices in the orientation of the smaller units in cases where those units have less symmetry than a square. If the center square is removed, there are no additional choices for subsequent iterations, the lone fractal being the Sierpinski Carpet (at left below). There are many other possibilities for removing one or more squares. One example is shown at middle below, through five iterations.

At far right is a path I found that essentially traces out a Sierpinski carpet. This path can be iterated such that it will trace out the carpet at any stage in non-crossing fashion and remain unicursal (a single strand).

All images copyright Robert Fathauer

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