eJournal (Final entry 2) “The Complexity of Patterns”

“The true epicness and beauty of patterns is infinite, never ending.”

The egotistical author of this blogpost (trying to act smart again)

INTRODUCTION

This blogpost begins with a “hit the runway” activity (try to construct a line that is in line with the runway given to ensure the safety of all passengers, which involves the topic “straight line graphs” and “y=mx+c intercept form”); and the next two investigations purely revolves around fractals.

Part A: How to Land a Plane

How I saved the life of all the passengers

Step 1: Find the gradient of the line. I guess this is pretty self-explanatory.

Step 2: Find the y-intercept of the line, which is 3.1.

Step 3: Find the range of the line (how thick the runway is, from y1 to y2.) Range is shown in above screenshot.

Step 4: Find the step. (How much the graph shifts upwards or downwards.) In this case, the step is 0. (That is what I input).

Step 5: Plane lands on the red line you have just plotted. Lives are saved. The outcome of this mission could have been much worse. Congratulations!

Part B: Kosh’s Snowflake Legacy

Our tasks:

1. Calculate the perimeter of the snowflake at each iteration.
2. Calculate the area of the snowflake at each iteration.
3. Tabulate the results and explain the number patterns that could be observed.
4. Create a model that helps to generalise the perimeter and area at any iteration.

Topics involved: <DISCLAIMER> Topic about Kosh’s snowflake could not be found in textbook, but it purely revolves around fractals.

PERIMETER OF KOSH’S SNOWFLAKE:

Hence,

Perimeter of snowflake at first iteration (P sub 1) = 3 . 81 . (4/3)^1 = 324 cm.

Perimeter of snowflake at second iteration (P sub 1) = 3 . 81 . (4/3)^2 = 432 cm.

Perimeter of snowflake at third iteration (P sub 1) = 3 . 81 . (4/3)^3 = 576 cm.

AREA OF KOCH’S SNOWFLAKE:

The total area of the snowflake after n iterations is:

where s is the length of the original triangle.

Hence,

Area of snowflake = (2 . 81^2 . 3^0.5)/5 = 4546 cm^2 (to nearest whole number)

Iteration Number	Perimeter (cm)	Area (cm^2)
1	324	4546
2	432	4546
3	576	4546

From the table of results, we can conclude that the area of Koch’s snowflake is finite and will always stay the same, and its perimeter follows the pattern of 3 . 81 . (4/3)^n, while n = the number of iterations.

Therefore,

The total perimeter of the Koch’s snowflake after n iterations is:

where s is the length of the original triangle.

AND

The total area of the Koch’s snowflake after n iterations is ….. always constant.

where s is the length of the original triangle.

Part C: Sierpinski’s Triangle

Our Tasks:

1. Note the pattern observed in relation to the number of green triangles.
2. State the pattern in relation to the number of stages and the length and area of each green triangle.

The “family of 4” of the Sierpiński triangle.

Stage	0	1	2	3
Number of green triangles	1	3	9	27
Length of one side of one green triangle	1	1/2	1/4	1/8
Area of each green triangle	1	1/4	1/16	1/64

The pattern I observed in relation to the number of green triangles:

The pattern I observed in relation to the length of one side of one green triangle:

The pattern I observed in relation to the area of each green triangle:

where n is the number of stages (for all three patterns).

All these patterns are arithmetic sequences, which uses n as the index.

Stage	0	1	2	3	4	5	6
Number of green triangles	1	3	9	27	81	243	729
Length of one side of one green triangle	1	1/4	1/8	1/16	1/32	1/64	1/128
Area of each green triangle	1	1/4	1/16	1/64	1/256	1/1024	1/4096

As the number of stages increases, the length of one side of one green triangle decreases by a factor of 1/2, and the area of each green triangle decreases by a factor of 1/4.

REFLECTION

Fractals play an integral part in our everyday lives, unbeknownst to some of us. For example, fractals are used to predict or analyse various biological processes or phenomena such as the growth pattern of bacteria, the pattern of situations such as nerve dendrites, etc. Fractals also depict the beauty of nature, such as snowflakes, mountain ranges, and even the terrifying electrical powers of lightning is an amazing real life application of fractals. In the process of completing this blogpost urged me to think out of the box, be creative; be innovative; and be more cognitive of my surroundings and everyday life. I have thus learnt the IB learner profiles of being a thinker, as it is paramount to equip oneself with the skills of thinking out of the box and applying critical thinking; being open-minded as there is an infinite trove of knowledge only waiting to be discovered; being knowledgeable, keep exploring, keep striving, keep on achieving greater heights.

Unforgettable Moments

Even though I cannot find a picture to upload under this much-awaited section, but I cherish every moment in the 2 year IGCSE program in Class 4I dearly, all of which that neither words or pictures are able to depict.

CITATIONS:

1. Rebloggy.com. (2019). See more fractal eye candy ?. [online] Available at: http://rebloggy.com/post/popular-quote-tech-animation-math-fractal-fractals-x2-ibm-geekout/68189140477 [Accessed 20 Nov. 2019].Simplicity. (2019). Fun with the Koch snowflake. [online] Available at: https://dkwise.wordpress.com/2017/06/22/fun-with-the-koch-snowflake/ [Accessed 20 Nov. 2019]. (Simplicity, 2019)
2. En.wikipedia.org. (2019). Koch snowflake. [online] Available at: https://en.wikipedia.org/wiki/Koch_snowflake [Accessed 20 Nov. 2019]. (En.wikipedia.org, 2019)
3. Go Figure. (2019). Koch Snowflake. [online] Available at: http://gofiguremath.org/fractals/koch-snowflake/ [Accessed 20 Nov. 2019]. (Simplicity, 2019)
4. En.wikipedia.org. (2019). Koch snowflake. [online] Available at: https://en.wikipedia.org/wiki/Koch_snowflake [Accessed 20 Nov. 2019]. (En.wikipedia.org, 2019)
5. Go Figure. (2019). Koch Snowflake. [online] Available at: http://gofiguremath.org/fractals/koch-snowflake/ [Accessed 20 Nov. 2019].
6. GeeksforGeeks. (2019). Sierpinski triangle – GeeksforGeeks. [online] Available at: https://www.geeksforgeeks.org/sierpinski-triangle/ [Accessed 20 Nov. 2019]. (GeeksforGeeks, 2019)
7. En.wikipedia.org. (2019). Sierpiński triangle. [online] Available at: https://en.wikipedia.org/wiki/Sierpiński_triangle [Accessed 20 Nov. 2019]. (En.wikipedia.org, 2019)
8. En.wikipedia.org. (2019). Sierpiński triangle. [online] Available at: https://en.wikipedia.org/wiki/Sierpiński_triangle [Accessed 20 Nov. 2019]. (En.wikipedia.org, 2019)
9. En.wikipedia.org. (2019). Sierpiński triangle. [online] Available at: https://en.wikipedia.org/wiki/Sierpiński_triangle [Accessed 20 Nov. 2019]. (En.wikipedia.org, 2019)

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