Sierpiński's triangle or gasket

sierpinski triangle

Five stages in the construction of Sierpiński's triangle.

At each stage you remove the centres of each of the remaining triangles.
It is easy to see that the area remaining converges to zero.

In fact the limit is a fractal set with Hausdorff dimension log2(3)1.5850\log_{2}(3) \approx 1.5850

Picture due to H K Strick