Sierpiski's Triangle, Carpet, and Sponge
The Sierpiski triangle (also know as the Sierpiski gasket or sieve) is a fractal first described by Sierpiski in 1915. To construct it, draw the outline of an equilateral triangle on white paper. In its middle draw a black upside down triangle with side length 1/3 the side of the original. Now there are 3 small white right side up triangles inside the large one. In the middle of each of these triangles draw a black upside down triangle. Repeat this process for the 9 small white triangle that are left. Continue this process forever to get the Sierpiski triangle.
Here is a way to obtain something that is a lot like Sierpiski's triangle. Start with any object whatsoever. Make three copies of it and place them one of them symmetrically above the other two. Now shrink all three object by half and make three copies of their union. Place the three copies so that one of them is symmetrically above the other two. Continue the process of shrinking, making three copies and placing indefinitely. If we had started with a triangle then we would end up with Sierpiski's triangle but regardless of what object we choose the result "looks like" Sierpiski's triangle. Probably for this reason, the Sierpiski triangle is commonly used to illustrate fractal properties. One such property is that the white part of the triangle has no area yet it has infinite perimeter. Another property is self-similarity, that is if we look through a microscope at any triangle inside Sierpiski's triangle then what we see looks a lot the whole triangle. One other property defines a fractal: a geometric object whose topological dimension is not equal to its Hausdorff dimension (i.e., fractal dimension; Felix Hausdorff was a German mathematician who helped develop the field of topology). The topological dimension of an object is defined as follows. Let the object be covered by balls (a "ball" is the set of all point y that are at a distance at most r from some central point c for some fixed number r and point c). A refinement of the cover is a collection balls so that each ball fits inside one of the balls of the cover and the union of all balls of the collection covers the object. The order of the refinement is the largest number n such that there is a point in the object that is contained in n balls of the refinement. The topological dimension is equal to minus one plus the smallest number n such that for any cover of the object there is a refinement of order n. It can be shown that the topological dimension of n-dimensional real space is n as expected. The Hausdorff dimension is defined as the minus the limit as the radius r goes to zero of the quantity log(N)/log( r) where N is the smallest number of balls of radius r required to cover the object. The Hausdorff dimension of the Sierpiski triangle is log(3)/log(2) = 1.58...
To obtain the Sierpiski carpet, draw an outline of a square on white paper. In its middle draw a black square whose side length is a third as long as that of the first square. The white space left over is made up of eight squares each equal in size to the middle square. Treat each of these white squares like the first square: draw a black square in the middle of each whose side length is a third that of the white square. Now the remaining space is made up of 8^2 = 64 small white squares. Treat each of these like the first white square. Continue this process forever. Although it is very similar to the Sierpiski triangle, the Sierpiski carpet has one property that the triangle does not have: every compact one-dimensional curve that can be drawn in the plane has a topological equivalent inside the Sierpiski carpet. This means the following: suppose we draw a curve (or overlapping curves) on a piece of paper. Now suppose that the curves drawn are made of rubber and so can be stretched, compressed or bent at will. Then they can be deformed so that they fit inside the white part of the Sierpiski carpet.
What is called the Sierpiski sponge should really be called the Menger sponge (or Menger curve) after Karl Menger who first described it in 1926. It a three dimensional analogue of the Sierpiski carpet. To construct it, start with a wooden cube. Draw a three by three grid on each its faces. Drill a square hole that starts with the middle-square of some face and continues to the other side. Do this for each face. The object is now like twenty small cubes that have been glued together. For each of these smaller cubes do the same thing that you did with the original cube. Continue this process forever. The result is the Menger sponge. Unlike the Sierpiski carpet, every compact one-dimensional curve has a topological equivalent inside the Menger sponge, not just those that can be drawn on paper.
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