Tromino divided by four

The scaling factor from one generation to the next is 1/2. This reptile has a single axis of mirror symmetry, allowing for choices in the orientation of the smaller units in cases where those units lack mirror symmetry. Half of the gasket at left below, but not the full gasket, can be formed from the square-divided-into-four-squares reptile. The gasket at right appears to generate the same fractal as the left one on the four-rectangles reptile page. The two examples below are carried through seven iterations. The ninth iteration of the left gasket is shown at a larger scale at the bottom of the page. The four options with this reptile that retain mirror symmetry (one is these is the gasket shown at left below) are described in the Wikipedia article on self-replicating tiles.

Two spatially-developing fractal-gasket polyhedra based on the left gasket above are shown below. In the left example, the layer thickness scales the same as the lateral features from one generation to the next. In the right example, the scaling of the layer thickness depends on which of the three positions the smaller copy occupies.

All images copyright Robert Fathauer

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