Sierpinkski Gasket

To create a Sierpinski gasket, start with a triangle and cut out the inscribed middle triangle. This results in three smaller triangles. In each of the smaller triangles, again cut out the middle inscribed triangle in the same way. Repeat this process indefinitely.

The gasket is perfectly self similar, an attribute of many fractal images. Any triangular portion is an exact replica of the whole gasket. The dimension of the gasket is log 3 / log 2 = 1.5849, ie: it lies dimensionally between a line and a plane.

Of particular interest is the area of the holes and the circumference of the solid pieces. If the area of the original triangle is 1 then the first iteration removes 1/4 of the area. The second iteration removes a further 3/16 and the third a further 9/64, etc.

If the circumference of the original triangle is 1 then after the first iteration the circumference increases by 1/2. After the second iteration it increases by 3/4, etc.

The above shown gasket has no area but the boundary is of infinite length.  See the Matlab code used to generated the picture.