By Chandrahas M. Halai
Figure 1
What is the ratio of the area of the red colured curve to the whole square?
Let us approach this problem in a step by step manner.
Figure 2
Let us first consider a unit square (p11, p12, p13, p14). Now, inscribe another square (p21, p22, p23, p24) within this square by joining it’s midpoints as shown in Figure 2. Now, shade the triangle formed by joining the points p21, p12 and p22. The area of this triangle will be 1/8.
Figure 3
Now, inscribe the square (p31, p32, p33, p34) within the square (p21, p22, p23, p24) by joining it’s midpoints as shown in Figure 3. The area of the square (p21, p22, p23, p24) will be half of the square (p11, p12, p13, p14). Hence, the area of the square (p21, p22, p23, p24) will be 1/2. Now, shade the triangle formed by joining the points p21, p31 and p34. The area of this triangle will be 1/8th of the area of the square (p21, p22, p23, p24). Thus the area of this triangle will be (1/2)*(1/8). Thus the area of the red curve at this stage will be 1/8 + ((1/8)*(1/2)).
Figure 4
Now, inscribe the square (p41, p42, p43, p44) within the square (p31, p32, p33, p34) by joining it’s midpoints as shown in Figure 4. The area of the square (p31, p32, p33, p34) will be half of the square (p21, p22, p23, p24). Hence, the area of the square (p31, p32, p33, p34) will be 1/4. Now, shade the triangle formed by joining the points p34, p41 and p44. The area of this triangle will be 1/8th of the area of the square (p31, p32, p33, p34). Thus the area of this triangle will be (1/4)*(1/8). Thus the area of the red curve at this stage will be 1/8 + ((1/8)*(1/2)) + ((1/8)*(1/4)).
Figure 5
Now, inscribe the square (p51, p52, p53, p54) within the square (p41, p42, p43, p44) by joining it’s midpoints as shown in Figure 5. The area of the square (p41, p42, p43, p44) will be half of the square (p31, p32, p33, p34). Hence, the area of the square (p41, p42, p43, p44) will be 1/8. Now, shade the triangle formed by joining the points p44, p53 and p54. The area of this triangle will be 1/8th of the area of the square (p41, p42, p43, p44). Thus the area of this triangle will be (1/8)*(1/8). Thus the area of the red curve at this stage will be 1/8 + ((1/8)*(1/2)) + ((1/8)*(1/4)) + ((1/8)*(1/8)).
Similarly, repeat the procedure as shown in figures 6, 7 and 8.
Figure 6
Figure 7
Figure 8
Now, the area of the red curve at this stage will be 1/8 + ((1/8)*(1/2)) + ((1/8)*(1/4)) + ((1/8)*(1/8)) + ((1/8)*(1/16)) + ((1/8)*(1/32)) + ((1/8)*(1/64)).
If this procedure is repeated ad infinitum, then the area of the red curve will be the infinite geometric series
A = 1/8 + ((1/8)*(1/2)) + ((1/8)*(1/4)) + ((1/8)*(1/8)) + ((1/8)*(1/16)) + ((1/8)*(1/32)) + ((1/8)*(1/64)) + …
Thus, we have
A = (1/8) * [1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + …] = (1/8) * 2 = 1/4
Thus the ratio of the area of the red colured curve to the whole square is 1 to 4.
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