Square spirals R. Ramanujam, The Institute of Mathematical Sciences, Chennai Look at Figure 1. What patterns do you see, what questions occur to you? Here are some questions worth thinking about. What shapes can you see in the diagram? Can you find the area of any of the shapes? Can you see any sequences? What fraction of the whole shape are the shaded regions? Well, we see many squares and triangles. Let us for the moment concentrate on the squares. Do you see a "square sequence", that is, a series of squares embedded inside each other? Then do you see a shaded "triangle sequence"? Squares are easier, so let us first try the area of a square sequence. Consider Fig 2. What fraction of the large square is each shaded square? Can we write the total shaded area as a sum of fractions? If the area of the large square is 1, we see that the shaded areas are fractions of the larger area. For instance, the largest shaded square is one fourth of the full square. The next largest shaded square is one fourth of this largest shaded square, that is, one fourth of one fourth or 1/16 of the total area. In fact, the fractions of each shaded square form a *geometric sequence*. Consider Figure 3. If we make the assumption that this pattern will be infinite, then the sum of the fractions will be: 1/4 + 1/16 + 1/64 + .... = 1/4 + 1/4^2 + 1/4^3 + .... We see that all terms are powers of the first term. This is the meaning of a geometric sequence. So the series goes on for ever. How can we find an "infinite sum" like this one? Notice that in Figure 2, the largest shaded region is in the bottom left of the figure, but there is an unshaded region above it and another to its right. But they are all identical shapes! We can thus collect three sets of identical shapes. Similarly, there are three identical copies of the second-largest shaded region, and so on. We have the same sequence appearing three times, and covering the whole of the square. Pull them out and you get Figure 4! Hence, 1/4 + 1/4^2 + 1/4^3 + .... = 1/3. Now we can try our hand at the "triangle sequence" we saw in Figure 1. To do this, we have to make precise some of our observations. We see that every square is inside yet another square and we see that this has a pattern. The inner square is given by four lines with opposite sides parallel. These lines are the diagonals of rectangles formed by connecting the midpoint of one of the sides to the opposite corner of the outer square. Figure 5 makes this clear. There are many ways we could find the areas of triangles and the relationships between them, but we will think about *similar triangles*. Let the shaded region be triangle ABC, consider the smaller triangle ABD to the "left" of ABC, formed by extending CB to BD completing the diagonal. Can you see that the two triangles are similar? See Figure 6. The larger triangle has sides that are double the length of the smaller triangle, and since they are similar, the area is four times as big. The two triangles together give 1/4th of the square. Can you see this from Fig 1? Let the bigger triangle have area A. Then the smaller one has A/4, and their sum is 5A/4, but this equals 1/4. Hence the bigger triangle has area A=4/5th of 1/4, that is, 1/5th of the whole square. Now look at Figure 7, which is like Figure 1, but shaded differently. There are 4 such triangles that are shaded, leaving a small square in the centre. Since the triangles together have an area = 4 times 1/5, or 4/5, the area of the small square in the centre is 1/5th of the area of the full square. To get Fig 1, we have to fill squares inside inside squares and triangles inside triangles in Fig 7. We have just seen that each square is 1/5th of the area of the previous square, and thus we have that each triangle is 1/5th of the area of the previous triangle. Thus, if the area of the large square is 1, the total shaded area in Fig 1 is: 1/5 + 1/5^2 + 1/5^3 + .... What is this sum? The shaded area is exactly one of four similar figures, which you could "pull out" (as we did for square sequences). Thus we get 1/5 + 1/5^2 + 1/5^3 + .... = 1/4. Writing it out, we have summed the infinite series, 1/5 + 1/25 + 1/125 + ... = 1/4! Amazing, isn't it? BOX Can you think of diagrams that represent the following sums? 1/2+1/4+1/8+1/16 .... 1/3+1/9+1/27+1/81 .... Do you see that the first sum is 1 and the second is 1/2? During the discussion we saw that: 1/4 + 1/4^2 + 1/4^3 + .... = 1/3. 1/5 + 1/5^2 + 1/5^3 + .... = 1/4. What pattern do you see here? Can you generalize it?