Explore Sierpinski's Triangle 1b

Fractal Gaskets A gasket is a flat sheet of some flexible material, in which one or more holes has been cut. They are used to seal the join between two pieces of a machine which may otherwise leak. Following this idea, a fractal gasket is a mathematical flat sheet in which holes are cut. The Sierpinski Triangle is a fractal gasket - it starts life as a triangle, in which triangular holes are cut. If we were to make one out of paper, we would progressively cut smaller and smaller triangles out of the paper. In fact, any tiny area of paper in the triangle that we could find would immediately have holes cut in it. This leads (at least conceptually) to the idea that the fractal gasket has no area. But this does not mean that the fractal isn't connected. Looking back at the steps we used to make the triangle, we see that when we cut a triangular hole out of any given triangle, we left the corners of the three mini triangles touching. This means the outside edge of the original triangle is never cut all the way through - the tiniest, thinnest piece of paper remains to connect the three corners of the original triangle. All fractal gaskets follow the same principle. Although they start life as a mathematical area, after each iteration there is less and less area left. In the end, the area has been reduced to zero, yet it remains a single shape. A real world example is a spider's web - a readily-identified object that is actually mostly empty space. The incredibly thin fibres that make it up hardly take any space at all, but they map out a construction that we readily recognize as a single "shape" which we think has area. A web might occupy a substantial space, but what is the actual area of a spider web? The spider web idea also helps demonstrate the concept of connectedness. Even though we cut away every area we can find from the Sierpinski Triangle, we leave the incredibly thin edges of those triangles behind. The ends of smaller ones are touching larger ones, so that it is possible to choose any two point on the triangle, and be able to find a path of fibres between them. To the right is another fractal gasket: the Sierpinski Carpet. This one starts with a square, from which smaller squares are progressively cut. It may seem more "solid" than the triangle, however it too has zero area, and yet is fully connected. Return to Explore Sierpinski's Triangle