HL TYPE 1

By Abhishek Chhabria

                                                                                    (Math HL)  

International Baccalaureate

B.D.Somani International School

INTRODUCTION

The Koch curve was named after Helge von Koch in 1904. The generation of this fractal is simple. We begin with a straight line of unit length and divide it into three equally sized parts. The middle section is replaced with and equilateral triangle and its base is removed. After one iterations, the length is increased by four-thirds. As this process is repeated, the length of the figure tends to infinity as the length of the side of each new triangle goes to zero. Assuming this could be iterated an infinite number of times, the result would be a figure which is infinitely wiggly, having no straight lines whatsoever. The construction is very simple but still looks really beautiful.

Throughout, let us consider-

1. Using an initial side length equal to 1, we deduce a table which shows the values of the above mentioned variables with respect to changes in ‘n’.

     Stage 0                   Stage 1                  Stage 2                  Stage 3

Now we study the relationship between successive terms for each of the above geometric deductions of Koch’s snowflake.

Let us consider the term in the stage to be respectively, according to the general term being studied, and ‘r’ to be the ratio between successive terms.

• For ,

This statement generalizes the behavior of the graph.

To verify the generalization we derive the conjecture from the table’s values.

Therefore, with reference to the graph, we are now convinced that the generalization applies consistently to the table’s values for.

• For , (second graph)

Again, we attempt at deriving a conjecture from our deductions in step1.

We enter values of the graph’s points in the statistics list of a Graphic Display Calculator (Texas Instrument).

Then, we undertake Exponential Regression

and the deductions from the calculator are as follows:

This implies that the conjecture for  is:

“The conjecture suggests that as we move right along the x-axis (0 onwards), i.e. as the value of ‘n’ increases by 1 unit, the corresponding value on the y-axis (1 onwards) gets multiplied by  units and the graph slopes gently.”

This statement generalizes the behavior of the graph.

To verify the generalization we derive the conjecture from the table’s values.

Therefore, with reference to the graph, we are now convinced that the generalization applies consistently to the table’s values for.

• For , (third graph),

We attempt at deriving a conjecture from our deductions in step1.

We enter values of the graph’s points in the statistics list of a Graphic Display Calculator (Texas Instrument).

Below are value tables of.

 It is evident that the increase in is a converging one which means as ‘n’ increases the difference between successive terms decreases.

Now as ‘n’ approaches infinity, that is , we compute:

Therefore we can say that (recalling that the area of the original triangle)

As

Verification:

Imagine drawing a circle around the original figure. No matter how large the perimeter gets, the area of the figure remains inside the circle.

Percentage of original Area enclosed by curve of infinite perimeter is:

Comments:

Clearly the perimeter will increase in the further stages and become infinite, but the area of the figure will be less than the area of the

circumcircle of the original equilateral triangle. This figure has

an infinite perimeter but a finite area!  The area enclosed by the closed curve of infinite length is actually only 60 percent more than that of the original area of the equilateral triangle we started in Stage 0. A remarkable property indeed. It is the significant property of a fractal shape that has self-similarity to an infinite depth. That is, you can enlarge a portion of the boundary to any extent and find shapes similar to the original figure.

7. In step 1 we found that .

The general expression found in step 3 is

Proof: (By Principle of Mathematical Induction)