Now substituting by their conjectures, we get,
Therefore we come to the conclusion that the formula involves the calculation of a geometric series and that difference between successive terms, say .
- Now we graph the table values.
(here)X-axis represents ‘n’ and Y-axis represents.
(here)X-axis represents ‘n’ and Y-axis represents .
(here)X-axis represents ‘n’ and Y-axis represents .(values in reference to table)
(here)X-axis represents ‘n’ and Y-axis represents .
3. Generalizations
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 deduction 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 ( 3 onwards) gets multiplied by 4 units and the graph slopes steeply.”
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.
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.
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 deduction 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 ( 3 onwards) gets multiplied by units and the graph slopes gradually.”
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.
So far we have discovered,
.
Considering the terms found, now we attempt at deriving a simplified formula for ,using the above formula,
“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 can be obtained by substituting the n value in the above conjecture and the graph slopes gently.”
This statement generalizes the behavior of the graph.
Verification:
Therefore, with reference to the graph, we are now convinced that the generalization applies consistently to the table’s values for.
- Investigation at n=4.
When n=4, the fractal at the third stage undergoes the iteration process and we obtain a new set of values for
STAGE 0
STAGE 1
STAGE 2
STAGE 3
STAGE4
VERIFICATION:
With Reference to the diagram;
It is evident that the iteration process, as described in the given description, has taken place for each side of the triangle.
As we observe the changes that have taken place from stage three to stage four, it is evident that each side of the third stage gets transformed to form 4 new complete sides but of a smaller length.
Therefore the predictions that are true and
Indeed is true.
After each iteration the mid third part of a side is removed and an equilateral triangle without a base, the length of whose side is equal to the length of the side removed, that is, one third. .This process continues indefinitely.
Therefore the predictions that is true and
Indeed is true.
Perimeter
Let us consider the sub-parts of the fractals of the first two successive stages.
In the first case Perimeter
In the second case Perimeter
Therefore the prediction that is true and
Indeed is true.
The difference in the area between two successive triangles is represented by the dark triangles in the figure below.
Stage 1 Stage 2
This difference is demonstrated in step 1 as.
For Stage 4,
It is also demonstrated in step 3 that the as ‘n’ increases it corresponds to a converging increase in the value of the geometric series of the conjecture. It is evident that in the fourth stage although there is an increase in area the increase itself is lesser than the any former increase.
Therefore the predictions that and are true and, Indeed
Hence Verified.
-
Using the relationship between successive terms in Step 1 we discovered values for stages 0 to 4. Using theses values we discovered, in the third step, the conjecture for.
The Conjecture can be further simplified in terms of decimal places as.
We graph this new conjecture in mathematical software called ‘Advanced Grapher’. Note:
The software allows us to generate values of for n=0 to 5000.
Investigating for which value of ‘n’ , the software deduces the following:
On observing the tabular data we can see that when,
It is evident that considering the values in terms of 6 decimal places both equal to 0.692200.
Therefore we can conclude with the result that when n=17,
.
- Now we investigate what happens to the Perimeter and Area as ‘n’ gets very large.
Recalling that,
Now as ‘n’ increases, the is raised to this increased ‘n’ so the ‘3’ in the right side of the expression gets multiplied by larger numbers at every increase in ‘n’ leading to the ultimate increase in the perimeter.
Since which is >0,is bound to increase with increasing ‘n’. In other words its graph diverges.
(The tables below are screenshots of the table values from ‘Advanced Grapher of the equations mentioned below).
The first table shows the value of for increasing ‘n’ and the second table shows the value of for increasing ‘n’.
Thus it is evident that as
Now we compute what happens to the area as ‘n’ gets larger.
Note that as ‘n’ increases decreases since it is the ratio is less than 1. The series in is just a simple geometric series with the base 4/9, and we know that since this number is less than 1, the series converges. (The convergence is evident in the graph and value table of step 5).
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)