The Koch Snowflake
The snowflake model was created in 1904 by Helen von Koch. This snowflake appeared to be one of the earliest fractal curves. The fractal is built by starting with an equilateral triangle. One must remove the inner third of each side and replace it with another equilateral triangle. The process is repeated indefinitely. The length of each side is one which will help you determine the perimeter of each triangle. With having the perimeter of each triangle, the height can be determined so the area can be defined. The height must be determined because to use the formula A=12bh to find the area of the traingle, the height must be known. After repeating the process for the triangles, the graph below displays the number of sides (Nn) for each snowflake, the length of a single side (In), the length of the perimeter (Pn) and the area (An). In 0 1 2 3 1 1/3 1/9 1/27 Nn 3 12 48 192 Pn 3 4 16/3 64/9 An 3/4 3/4(1+13) 3/4(1+13+427) 3/4(1+13+427+16243)

The behavior of the graph above proves that each time a new snowflake is formed, the perimeter increases by 49. So, by simply multiplying 49to the area prior to, you will generate the area for the next sequence. When A4 occurs, these are the following results;

In 4 181

Nn 192

Pn 25627

An 3/4(1+13+427+16243+642187)

When A6 occurs, these are the following results; In 6 1729 Nn 3072 Pn 1024243 ) An 3/4(1+13+427+16243+642187+1024177147

The pattern that is emerged after experimenthing further into the snowflakes is An+1=An+(3*4n9n)A0 and n1. Theoretically the area gets too small to see so it is a limit is when N gets to 7. So, N could never reach with the area being noticable.

...Adrian Zwierzchowski
2 IB
Investigation – Von Koch’s snowflake curve
In this investigation I am going to consider a limit curve named after the Swedish mathematician Niels Fabian Helge von Koch. I will try to investigate the perimeter and area of Von Koch’s curve.
[pic]
The Koch’s curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage.
First of all I am going to suppose c1 has a perimeter of 3 units. I will try to find the perimeter of c2, c3, c4 and c5.
c1 s1 = 1 (s – side length)
c2 s2 = 1/3
c3 s3 = 1/9
c4 s4 = 1/27
c5 s5 = 1/81
If the original line segment had length s, then after the first step each line segment has a length s · ⅓. For the second step, each segment has a length s ·(⅓)2, and so on.
Assuming a unit length for the starting straight line segment, we obtain the following figures:
|iteration |segment |segment |curve |
|number |length |number |length |
|1 |1 |1 |1.00 |
|2 |⅓ |4 |1.33 |
|3 |1/9 |16 |1.77 |
|4 |1/27 |64...

...nature. Fractal or not, patterns give us something more to admire and wonder about.
Introduction
Fractals never fail to fascinate. If you aren't just gazing at their unearthly beauty, you ponder the mathematics behind them... and then you can't help but wonder how such prosaic, unsensational mathematical formulae can give rise to such intricacy. What is it that makes it possible for (to some) a short, ugly equation to generate the exuberant beauty of the Mandelbrot set? Or is it all just in the way our brains are wired?
Fractals are objects with infinite lengths that occupy finite volumes, resulting in a "fractional dimension" that is not 1-, 2-, or 3-D, but a combination of all three, depending on its spatial configuration.
The Kochsnowflake is the repetitive procedure of dividing the image into three equal parts and replacing the middle piece with two similar pieces.
Hypothesis
Fractals mimic nature. (true or false)
This is the basic belief of fractals, and a common concept among those who study fractals. In nature, symmetry is often remarked upon. To mimic is to be similar in to a certain object, and in this case, of a lesser proportion. Thus, we would like to propose that fractals may mimic nature.
Definitions
Fractals
1. A curve or geometric figure, each part of which has the same statistical character as the whole.
2. Any of various extremely irregular curves or shapes for which any suitably chosen part is similar in...

...Investigation:
VON KOCH’S SNOWFLAKE CURVE
Ha Yeon Lee 11B
Mathematics HL
• Introduction:
➢ History of Von Koch’s Snowflake Curve
The Kochsnowflake is a mathematical curve, which is believed to be one of the earliest fractal curves with description. In 1904, a Swedish mathematician, Helge von Koch introduced the construction of the Koch curve on his paper called, “On a continuous curve without tangents, constructible from elementary geometry”.
➢ In this mathematical task, I am going to investigate how the area and perimeter of a shape/curve changes and find out whether they increase by the same number every time,as the following process is repeated:
i. Start with an equilateral triangle.
ii. Divide each side of the triangle into three equal segments.
iii. On the middle part of each side, draw an equilateral triangle by connecting lines.
iv. Now remove the line segment that makes the base of the smaller triangle that was formed in step 3.
The above process (steps i~iv) can be repeated indefinitely. The shape that emerges is called “Von Koch’s Snowflake” for obvious reasons. An equilateral triangle, which is the shape used to start with to...

...This piece of literature first appeared in the New Republic. It was written by former New York City mayor, Edward Koch. This essay was written primarily to justify his position in favor of capital punishment. He explains to readers that although there have been attacks on his opinion, he has closely examined the opposing arguments and still supports the death penalty. Edward Koch does an excellent job in his efforts to persuade those who are morally opposed to capital punishment to change their opinions, or at least to examine the arguments more closely.
In his essay, the author includes seven main arguments opposing capital punishment and refutes them. People may find that the death penalty is a barbaric act and Koch argues this point by suggesting that the method of lethal injection is actually quite humane and literally painless. He also argues that although no other democratic country imposes the death penalty as a form of punishment, no other country boasts a murder rate as high as the United States. The author contends with those who believe capital punishment diminishes life’s value by suggesting the contrary. He has found those who are sentenced to death have been judged fairly and with a great deal of examination. Koch then refutes the argument of capital punishment as a state-sanctioned murder by acknowledging that the state holds much different rights and responsibilities than the individual.
Edward...

...A Report on Snowflakes Dessert Café, SS15, Subang Jaya
INTRODUCTION :
You might be wondering what is it about Snowflakes that makes a lot of people would line up just to enjoy a bowl of shaved ice. Well, it’s not just an ordinary bowl of dessert. Jimmy Tsai, owner of Snowflakes Dessert Café, came up with the idea of this business because he missed the Taiwanese desserts that his mother and grandmother used to make back in his hometown, Taichung district in Taiwan. Started with a small shop in SS15 area in Subang Jaya, Jimmy opened and run the café with his Malaysian wife. Slowly, more and more people came to Snowflakes because of the affordable price and the very big portion of the desserts offered, which then makes the business grow bigger and makes Jimmy open 3 branches, in Pavilion KL, Kota Damansara and in Kuchai Lama.
RESEARCH METHOD :
In order to retrieve data for the research on Snowflakes, we use 2 kinds of method. The first one is to give questionnaire to Snowflake’s customer to know their satisfaction level towards the Taiwanese dessert café. And the second method is to pick up information and reviews about Snowflakes from food bloggers in Malaysia.
SELECTION CRITERIA :
When we go into Snowflakes, we know that it is a successful shop straight away. If you go to Snowflakes around 9pm to 11pm in the weekends, you will be faced with a super...

...* He was born in Germany on December 11, 1843.
* Koch decided to change his area of study to medicine from natural science, as he aspired to be a physician.
* In July of 1867, following his graduation from medical school, Koch married Emma Adolfine Josephine Fraatz, and the two had a daughter, Gertrude, in 1868
* After his graduation in 1866, he worked as a surgeon in, and following his service, worked as a physician in what today is known as Wolsztyn, Poland the Franco-Prussian War
* He is a german
* Koch served as an administrator and professor at Berlin University
* Koch’s marriage with Emma Fraatz ended in 1893, and later that same year, he married actress Hedwig Freiberg from 1880 to 1890
* Koch suffered a heart attack on April 9, 1910 and never made a complete recovery
* On May 27, only three days after giving a lecture on his tuberculosis research at the berlin academy of sciences
* Robert Koch died at baeden baeden at the age of 67
His contributions are as follows:
* Anthrax
* Koch’s four postulates
* Isolating pure culture on solid media
* Cholera
* Tuberculosis
ANTHRAX:
Koch is widely known for his work on this disease. He discovered the causative agent for this disease as Bacillus anthracis.
Koch discovered spore-formation in the anthrax bacteria, which could remain dormant under specific conditions....

... ¬¬¬¬¬
ENGINEERING STABLE PROT¬¬¬EINS
Stable and soluble HIV spike proteins
Appendix
Abstract……………………………….................................................1
Introduction………………………………..........................................1
Structure and function of gp120 and gp41.…………………………2
Mutation 1……………………………….............................................5
Mutation 2.……………………………................................................6
Mutation 3.……………………………................................................7
Conclusions…………………………………………………………...9
References……………………………….............................................9
Abstract
The functional unit of the human immunodeficiency virus type 1 (HIV-1) envelope glycoproteins consist of compact trimers of noncovalently associated gp120 exterior glycoproteins and gp41 transmembrane glycoproteins.The glycoproteins so called spikes have the role in virus-cell attachment and is of particular importance and is the major target for vaccine design. The lability of intersubunit interactions has hindered the production and characterization of soluble and stable forms of envelope glycoprotein trimers. Here I describe the generation of soluble and stabilized HIV-1 spike proteins by protein engineering.
Introduction
The HIV infection in humans is considered pandemic by WHO and an...

...rogue physicist travels back in time to kill the apostle Paul." (This is the summary for my first novel, Transgression.) The sentence will serve you forever as a ten-second selling tool. This is the big picture, the analog of that big starting triangle in the snowflake picture.
When you later write your book proposal, this sentence should appear very early in the proposal. It's the hook that will sell your book to your editor, to your committee, to the sales force, to bookstore owners, and ultimately to readers. So make the best one you can!
Some hints on what makes a good sentence:
• Shorter is better. Try for fewer than 15 words.
• No character names, please! Better to say "a handicapped trapeze artist" than "Jane Doe".
• Tie together the big picture and the personal picture. Which character has the most to lose in this story? Now tell me what he or she wants to win.
• Read the one-line blurbs on the New York Times Bestseller list to learn how to do this. Writing a one-sentence description is an art form.
Step 2) Take another hour and expand that sentence to a full paragraph describing the story setup, major disasters, and ending of the novel. This is the analog of the second stage of the snowflake. I like to structure a story as "three disasters plus an ending". Each of the disasters takes a quarter of the book to develop and the ending takes the final quarter. I don't know if this is the ideal structure, it's just my...