Snowflake Curve

“Snowflake Curve”

Part A: Find formulas for the number of sides (Sn), the length of a side (Ln), and the total length of the nth approximating curve (Pn).

Solution:

In the demonstration above, you can see that the equilateral triangle is divided into three equal parts where the middle part is then ‘raised’ or ‘pulled away’ from the initial triangle to create another equilateral triangle where the middle part is deleted. Therefore, the one side has now become four sides in just one step. Since there is initially 3 sides and we are breaking up 1 of these sides you would need to multiply your equation by 3 each and every time to account for each initial side of the equilateral triangle. With this said the formula for Sn is:

Sn= 3*4^n

Where n refers to the next repetition or step (i.e. the illustration above is going from step 0 to 1).

If one side of the initial equilateral triangle is divided into thirds, then the length of a side from the 1st repetition would be one third. The length of a side in the 2nd repetition would be a third of a third or (1/3)/3 and so on. Below is a table which shows this.

nth Power Length of One Side

0th repetition 1

1st repetition 1/3

2nd repetition (1/3)/3= 1/9

3rd repetition (1/3)/3/3= 1/27

4th repetition (1/3)/3/3/3= 1/81

From this table, we concluded that the formula which represents the length of a side is:

Ln= (1/3)^n

Where n refers to the next repetition or step.

nth Power Length of One Side Formula Proven: (1/3)^n

0th repetition 1 (1/3)^0= 1

1st repetition 1/3 (1/3)^1= 1/3

2nd repetition (1/3)/3= 1/9 (1/3)^2= 1/9

3rd repetition (1/3)/3/3= 1/27 (1/3)^3= 1/27

4th repetition (1/3)/3/3/3= 1/81 (1/3)^4= 1/81

Essentially to solve for the perimeter we would simply just combine the two formulas that we have come up with thus far for the number of sides and the length of these sides giving us…

Pn=(Sn)*(Ln)

Pn=[(3*4^n)*(1/3)^n]

Pn=3*(4/3)^n

Part B: Show that Pn ∞ as n ∞

Solution:

To