HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE 9 Frank Glaser THE UNIVERSALITY OF THE HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE Mathematics Continuing with the development of a theory of hyperpolar fractals, we present in this article a way of constructing hyperpolar universal sets in two and three dimensions which are mappings of two classical fractals: the Sierpinski carpet and the Menger sponge. This article is, therefore, a continuation of a previous one (Glaser, 1996) and its aim is to extend some of the results of the Polish mathematician Waclaw Sierpinski (1882-1969) as well as some of the work of his Austrian colleague, Karl Menger (born in 1902) into two and threedimensional hyperpolar spaces respectively. The Sierpinski Carpet The Sierpinski carpet is a fractal which is a generalization of the Cantor set into two dimensions. In order to construct this fractal we begin with a square in the plane, subdivide it into nine smaller congruent squares of which we drop the open central one, then subdivide the eight remaining squares into nine smaller congruent squares in each of which we drop the open central one. We carry out this process infinitely often obtaining a limiting configuration which can be seen as a generalization of the Cantor set. Figure 1 shows the first three steps in the constructions of this fractal. Figure 1. The first three steps in the construction of the Sierpinski carpet.
10 FRANK GLASER Fall 1997 The precise construction of the Cantor set is found on a line parallel to the base of the original square and which goes through its center. (Peitgen, Jürgens & Saupe, 1992). It will be shown that the complexities of the hyperpolar Sierpinski gasket (Glaser, 1995) and the hyperpolar Sierpinski carpet may at first look essentially the same, but there is a great topological difference between them. In order to obtain a hyperpolar image of the Sierpinski carpet we must first describe it analytically. Let Q 0 = (x,y) 0 x 1, 0 x 1 be the closed unit square. The lines x = 1 3, x = 2, 3 y = 1 3, and y = 2 3 divide Q 0 into nine squares from which we removethe open central square. Q 1 (0, 0) = (x,y) 1 3 < x < 2 3, 1 3 < y < 2 3. Next we use the lines: x = 3i + 1, x = 3i + 2, y = 3j + 2 3j + 2, and y = where i, j = 0,1,2, 3 2 3 2 2 3 3 2 in order to divide each of the remaining squares into nine smaller congruent ones, from each of which we remove the open central one from a total of 9 2 = 81 squares. Next we use the lines: x = 3i + 1, x = 3i + 2, y = 3j + 2 3j + 2, and y = where i, j = 0,1,2, 3 2 3 2 2 3 3 2 in order to divide each of the remaining squares into nine smaller congruent ones from each of which we remove the open central one from a total of 9 2 = 81 squares. The removed squares can be written as: Q 2 (i, j) = (x, y) 3i + 1 3 2, < x < 3i + 2 3 2, 3j + 1 3 2 < y < 3j + 2 3 2 where i, j = 0,1,2, = 3 2 We therefore remove the union of sets 2 j = 0 2 i = 0 Q 2 where the square Q2 (1,1) Q1(0,0) has already been removed in step 1 when Q 1 (0,0) was removed. At the third step, we remove the 81 squares. (i, j) Q 3 (i, j) = (x, y) 3i + 1, < x < 3i + 2, 3j + 1 < y < 3j + 2 3 3 3 3 3 3 3 3 where i, j = 0, 1, 2, 3, 4, 5, 6, 7, 8 = 3 3-1 - 1.
HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE 11 Observe that at the third step Q 3 (1,1) Q 2 (0,0),Q 3 (4,1) Q 2 (1,0), Q 3 (7,1) Q 2 (2,0),Q 3 (1,4) Q 2 (0,1), Q 3 (2,4) Q 2 (2,1),Q 3 (0,7) Q 2 (0,2), Q 3 (4,7) Q 2 (1,2),Q 3 (7,7) Q 2 (2,2), and furthermore, there are nine squares that are also contained in Q 1 (0,0): Q 3 (3,3) Q 1 (0,0),Q 3 (4,3) Q 1 (0,0), Q 3 (5,3) Q 1 (0,0),Q 3 (3,4) Q 1 (0,0), Q 3 (5,4) Q 1 (0,0),Q 3 (3,5) Q 1 (0,0), Q 3 (4,5) Q 1 (0,0),Q 3 (5,5) Q 1 (0,0), Q 3 (4,4) Q 2 (1,1) Q 1 (0,0). These 17 squares have already been removed at the second, and 9 of them at the first step; therefore, at step 3, we are removing 81-17 = 64 new open squares. Figure 2. The removal of the squares Q n (i,j) where i,j = 0,1,...,-1-1 for steps n = 1,2,3.
12 FRANK GLASER Fall 1997 At the end of step 3, we have removed the union of squares Q 1 (0,0) j = 0 2 2 i = 0 Q 2 (i, j) 8 j = 0 8 i = 0 Q 3 (i, j) At the nth step, we remove the 3 2(n-1) squares defined by Q n (i, j) = (x, y) 3i + 1 < x < 3i + 2, 3j + 1 < y < 3j + 2 where t, j = 0,1,..., -1-1. These are removed from a total of 9 n squares of which a certain number has already been removed in previous steps. To find this number, we construct the following table: Step 1 2 3 4... n Total number of squares 9 9 2 9 3 9 4... 9 n Number of squares not removed 8 8 2 8 3 8 4... 8 n Number of squares removed 1 17 217 2465... 9 n - 8 n Thus at the nth step, we remove 9 n - 8 n squares. As n tends to infinity, we are left with the set of points Q 0 n = 1 j = 0 i = 0 Q n (i,j) which is the Sierpinski carpet. The Siepinski carpet contains all possible one-dimensional sets in a topological sense; i.e., not as such sets appear independently but rather as one of their topologically equivalent mutants. The main result was obtained by Sierpinski in 1916, and we restate it as the following theorem: The Sierpinski carpet is universal for all compact one-dimensional objects in the plane. Compactness for a set in the plane or in space means that the set is bounded, i.e., it lies entirely within some sufficiently large disk in the plane or a sphere in space, and that every convergent sequence of points from the set converges to a point within the set.
HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE 13 Compactness can be assumed for any drawing on a sheet of paper. Suppose we draw a curve on such a sheet which fits on it. Then we have drawn a one-dimensional object and, since it fits on the sheet, it is compact. No matter how complicated this curve is and how many self-intersections it has, we always will be able to find within the Sierpinski carpet a subset which is topologically equivalent to the curve we have drawn. In this sense, the Sierpinski carpet is said to be universal, and it can be described as a super object or even as a house which contains all possible one-dimentional objects (Peitgen, Jürgens & Saupe, 1992). The Menger Sponge In order to maintain the topological character of one-dimensional objects, we may have to go into space. This is already necessary in the famous problem of supplying three houses, A, B, and C, with water, gas and electricity from W, G, and E, as shown in Figure 3, so that the supply lines do not cross if drawn in a plane. The only way to avoid crossing the supply lines is to go into space and run them at different levels. A B C W G E Figure 3. The problem of supplying three houses with water, gas and electricity without crossing the supply lines. The question then arises, is there also a superobject in space that holds all compact, onedimensional objects? In 1926, the Austrian mathematician Karl Menger generalized the work of Sierpinski by constructing such a superobject: the Menger sponge. Let C 0 be the closed unit cube; i.e., C 0 = {(x, y, z) 0 x 1, 0 y 1, 0 z 1}. The planes x = 1 3, x = 2 3, y = 1 3, y = 2 3, z = 1 3, and z = 2 3, divide C 0 into 27 homothetic, mutually congruentsubcubeswith side length 1, and we removethe innermostof these cubes, that is, 3 the subcubewhose outer surfacedoes not meetthe outer surfaceof C 0 We also remove the six subcubes that have a face in common with the innermost cube. The remaining 20 subcubes must retain their entire boundaries; i.e., we have removed 7 open cubes. In each of the 20 remaining subcubes, repeat the process, dividing them into 27 smaller cubes. Continuing with the process, in the nth step, one obtains 20 n cubes of side length 1/ each; these are called cubes of the nth step of the generation of the Menger sponge, and their union is denoted by M n. The set, which is one-dimentional and continuously parameterizable, is the universal one-dimensional set called the Menger sponge (Edgar, 1993).
14 FRANK GLASER Fall 1997 We can also write at each step the regions of the unit cube that are removed at that step. At the first step we remove the region where R 1 = R 1x (0,0,0) R 1y (0,0,0) R 1z (0,0,0) R 1x (0,0,0) = (x,y,z) 0 < x < 1, 1 3 < y < 2 3, 1 3 < z < 2 3 R 1y (0,0,0) = (x,y,z) 1 3 < x < 2 3, 0 < y < 1, 1 3 < z < 2 3 R 1z (0,0,0) = (x,y,z) 1 3 < x < 2 3, 1 3 < y < 2 3, 0 < z < 1 Figure 4. The first two steps in the generation of the Menger sponge. At the nth step, we define (i,j,k) R 1x = (x,y,z) 0 < x < 1, 3j + 1 < y < 3j + 2, 3k + 1 < z < 3k + 2 (i,j,k) R 1y = (x,y,z) 3i + 1 < x < 3i + 2 3 n, 0 < y < 1, 3k + 1 < z < 3k + 2 (i,j,k) R 1z = (x,y,z) 3i + 1 < x < 3i + 2, 3j + 1 < y < 3j + 2, 0 < z < 1 where I, j, k = 0,1,2,...,-1-1.
HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE 15 We then remove the region R n = R nx (i,j,k) R ny (i,j,k) R nz (i,j,k) The Menger sponge is the subset of the unit cube given by C 0 R n. n = 1 By DeMorgan's law, this is equal to n = 1 C 0 R n cubes of the nth step of the generationof the Menger sponge. The Hyperpolar Sierpinski Carpet The hyperpolar image Q 0 of the closed unit square is the region where C 0 R n = M n is the union of the Q 0 = (u,v) 0 tan 1 v u 1, 1 u 2 + v 2 e. At the first step, we remove the region Q 0 (0,0) = (u,v) 1 3 < tan 1 v u 2 3, e1 1 3 < u 2 + v 2 e 2 3 At the second step, remove the regions Q 0 (i, j) = (uv) 3i + 1 < tan 1 v u < 3i + 2, exp 3j + 1 < u 2 + v 2 < exp 3j +2 where i,j = 0,1,2 = 3 2-1 - 1.
16 FRANK GLASER Fall 1997 Figure 5. The first three steps in the generation of the hyperpolar Sierpinski carpet. At the nth step, we remove the regions Q n (i, j) = (uv) 3i + 1 < tan 1 u v < 3i + 2 exp 3j + 1 < u 2 + v 2 < exp 3j +2 where i,j = 0,1,2,..., -1-1. As n tends to infinity, we are left with the set of points Q n n = 1 j = 0 i = 0 which is the hyperpolar image of the Sierpinski carpet. Q n (i,j)
HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE 17 The Cylindrical Hyperpolar Menger Sponge In cylindrical hyperpolar space, the closed Cartesian unit cube C 0 maps into the region C 0 = (u, v, z) 0 < tan 1v u < 1, 1 < u2 + v 2 < e, 0 < z < 1 Figure 6. The first two steps in the generation of the cylindrical hyperpolar Menger sponge. At the nth step, we define 0 < tan 1 v u < 1 R nx (i,j,k) = (u,v,z) exp 3j + 1 < u 2 + v 2 < exp 3j + 2 3k + 1 < z < 3k + 2 R ny (i,j,k) = (u,v,z) 3i + 1 < tan 1 u v < 3i + 2 1 < u 2 + v 2 < e 3k + 1 < z < 3k + 2
18 FRANK GLASER Fall 1997 R nz (i,j,k) = (u,v,z) 3i + 1 < tan 1 u v < 3i + 2 exp 3j + 1 < u 2 + v 2 < exp 3j + 2 0 < z < 1 We then remove the region R n = R nx (i,j,k) R ny (i,j,k) R nz (i,j,k) The cylindrical hyperpolar Menger sponge is the subset of the unit cube given by C 0 R n n = 1 or C 0 R n n = 1 where C 0 R n = M n is the union of the cylindrical hyperpolar image of the cubes of the nth step in the generation of the Menger sponge. Universality of the Hyperpolar Sierpinski Carpet and of the Cylindrical Hyperpolar Menger Sponge The hyperpolar Sierpinski carpet contains all possible one-dimensional sets in the plane because they are topologically equivalent to those contained in the ordinary Sierpinski carpet since in the domain Q 0 = [0, 1] x [0, 1], the hyperpolar transformation is one-to-one. Similarly the cylindrical hyperpolar Menger sponge contains all one-dimensional sets in space because they are topologically equivalent to those contained in the ordinary Menger sponge and the transformation to cylindrical hyperpolar coordinates is one-to-one in the domain Q 0 = [0, 1] x [0, 1] x [0, 1]. Therefore, both the hyperpolar Sierpinski carpet and the cylindrical hyperpolar Menger sponge are universal one-dimensional sets. Conclusion In this article, we have introduced two new fractals, the hyperpolar Sierpinski carpet and the cylindrical hyperpolar Menger sponge, as well as a new notation that shows how to describe regions taken out of the unit square or out of the unit cube in the steps performed to generate these fractals. Furthermore, we have established that the hyperpolar Sierpinski carpet is a universal set for one-dimentional sets in the plane and that the cylindrical hyperpolar Menger sponge is a universal set for all one-dimensional sets in space. References Edgar, C. A., (1993). Classics on fractals. Addison-Wesley Publishing Company Glaser, F., (1996). Journal of Interdisciplinary Studies. Vol. 8. Peitgen, H. O., Jürgens, H. & Saupe, D., (1992). Chaos and fractals, new frontiers of science. Springer-Verlag.