Premium Essay

Essay On Snowflakes

Submitted By
Words 1005
Pages 5
1. Introduction
As winter approaches, I get fascinated by the idea of snow. I can still recall the first time I ever saw snow and snowflakes. I especially found snowflakes’ unique shapes and structures absolutely magnificent and beautiful. Therefore, I found myself naturally drawn to explore more about snowflakes, when I had to chose a topic for my mathematical exploration.
The aim of the investigation is to find the patterns of a snowflake and try to make a mathematical sense out of it. In addition, I wanted to explore how to easily calculate the area of a complicated shape as snowflake.
As I began to research how exactly I can approach the question, I encountered fractal geometry, which seemed to explain and address my question. Hence, by using a fractal, Von Koch Island, I have tried to make a “mathematical sense” out of snowflakes by exploring its …show more content…
Dimensions
Moreover, while Euclidean geometry belong to an integer dimension of either 1, 2, or 3, fractal geometry belongs to so-called fractal dimension. In Euclidean geometry, the number of dimension can be defined as “a set, region, object, or space possesses indicates how many mutually perpendicular directions of movement are possible”; Euclidean geometry obeys the following law:
, where D is dimension, L is linear scaling and S is the resulting increase in size. This is what we are used to.

0-Dimension
1-Dimension
2-Dimension
3-Dimension
Size
Point (0)
Length
(length)2 = Area
(length)3 = Volume

However, fractal geometry does not obey this law. I will talk more about fractal dimensions later in the exploration.
4. Von Koch Island
Von Koch curve is one of the most famous examples of fractal geometry. It was created by a Swedish mathematician, Niels von Koch, in 1906. It looks as following and evidently, the same image is reproduced repeatedly when zoomed in.



So how does one form such a shape?

Firstly, have a line. This very beginning can be called the initiator.

0th

Similar Documents

Premium Essay

Snowflakes Argumentative Essay

...Arthur Caplan says, “...embryo adoptions are a genuine boon, but rather a cover for an about the status of embryos.” Meaning embryo adoptions are a complete waste of the governments’ money and are false hope for infertile couples. The Snowflakes are a group funded by the Nightlight Christian Adoption agency. The whole goal of the Snowflakes is to take care of the embryos that are produced as extras after a couple produces their baby. They are then asked if they wanted to donate their extra embryos to benefit other couples who need help producing a baby. These embryos are produced in a lab after removing from the mom's eggs from the ovaries. Her eggs are fertilized in a lab dish and put back into the mother. When a couple is seeking help to produce a baby, it is best to not use frozen embryos because the longer an embryo is frozen the more ineffective it is. This article is very one-sided because it is so negative towards embryos when there could be positive outcomes, but the positive cases are ignored in this article. The goal...

Words: 1505 - Pages: 7

Premium Essay

Descriptive Essay

...Descriptive Essay: Describe a Place During a Holiday Celebration Holiday celebrations are often a very fun and festive time in many cultures. One such holiday is the Christmas time of year. Many families have a fun day filled with love, gift sharing, eating, and fun times and laughs. In my family the entire family gathers at my grandparent’s home. Imagine several rooms filled with family of all age ranges. My grandfather is the eldest of 85 years old, and I have a second cousin who is only 3 months old. Not only are the human family members present, everyone brings along their fury friends who are an extension of family. Upon arriving at my grandparent’s home I am amazed at the time they have spent ensuring the décor is spectacular. On Christmas day they keep the decorative lights on all day, so when we drive up to the house all we see is pretty, bright Christmas lights in the windows and the bushes in front of the house. Beaming through the front bay window we are able to see the tall, beautifully snow-flocked Christmas tree. There are also lit up Christmas figurines that line the horse-shoe driveway. This past year my grandparents added a twist to the décor by having a door wreath that was lit up and playing Christmas music and fake snowflakes on the front lawn. They wanted to ensure we got a wintery holiday experience, and they accomplished just that with ease. Once we get past the outside décor and make our way inside, the beautiful decorations do not stop....

Words: 777 - Pages: 4

Free Essay

Michel de Montaigne "On Education Children"

...just that, identical. So it would come to no surprise to Montaigne that out of a class of 30 individuals, only one or two students would have properly understood the material that was “taught” to the whole class. Montaigne states, “Spewing up food exactly as you have swallowed it is evidence of a failure to digest and assimilate it; the stomach has not done its job if, during concoction, it fails to change the substance and the form of what it is given.” (Screech, 169). Our education system has been out-of-date for quite some time now. Teaching groups of individuals, each with their own mindset, a one set strategy is failure within itself. No one mind thinks exactly alike, so it would be like trying to teach snowflakes the proper way to crystallize. Individual snowflakes follow...

Words: 1463 - Pages: 6

Premium Essay

The First Time of Taking Toefl Exam

...The First Time of Taking TOEFL Test Outside the window, only a few houses still had their light on. The street lamps turned the glow color from white to yellow, a warm color that could pacify the anxiety of the passerby. I looked up at the sky. The dark blue sky extended to the boundless distant place, to the other side of the earth—America—the country where I wanted to go to school. To achieve this goal, I had to take two exams, the TOEFL and SAT. I took a deep breath, thinking about my TOEFL exam. Test Of English as Foreign Language, the first barricade I need to go through, now bothered me a lot. I was forced to practice it again and again, to make sure my score was good enough to apply the American universities. But my reading, listening, speaking and writing skills could not assist me to get an excellent grade. I knew it well. So I could not stay outside for a long period of time. I needed to come inside and begin my practice once more. “It’s time”. Dad sipped his hot tea, reminding me to continue my work. “Go do your work.” His gentle voice sounded like a ringing clock, low and deep but powerful, pushing me to obey his order. “Yes”. I responded, and turned on the computer to listen to the online TOEFL course. I attempted to note down all the key point the teacher said. She then started a listening practice. As the lecture began, I worked on taking note of the key words for the questions that would come out after the lecture. Hours later, after finishing...

Words: 1519 - Pages: 7

Premium Essay

Operation Iraqi Freedom Case Study

...operational environment (OE) and incorrect problem definition resulted in an impractical operational approach to the Iraq war until reframing occurred with the 2006 surge. This paper describes General Franks’, Casey’s and Petraeus’ use of operational design. This essay discusses civilian policy shortfalls...

Words: 514 - Pages: 3

Premium Essay

Essay Robert Frost Snowy Evening

...Poetry Essay Snowy Woods: a Peaceful Moment or Crucial Crossroads? An essay on Robert Frost's "Stopping by the Woods on a Snowy Evening". "Stopping by the Woods on a Snowy Evening", when read casually, presents the reader with a quiet moment, a respite, a short break from a journey. Peaceful and sleepy it almost comforts the reader with its imagery and sense of calm. Yet under that calm is a lot of emotional undercurrent, similar to the surface of a frozen river. Its what's under the ice that is more interesting. Our main character, tells us in the first stanza that he has paused a moment (line 3). He has paused to take in a peaceful scene. The visual imagery of the woods filling "up with snow" (line 4) is rich. The snow is falling fast enough that it seems to be filling as he watches. How long does he pause? That depends on how fast the snow is falling. Long enough to see the "woods fill up" is what he says. The second stanza tells us that he has paused long enough that he's contemplating his horse's thoughts wondering what he thinks of standing there between the woods he is describing and a frozen lake (line 7) on an very dark evening. The darkest. (line 8) The third stanza makes us listen and asks the question, "Are we really supposed to be here?" (Line 10) We hear the horse shaking or tossing his head and correspondingly the harness bells jingle (line 9) and fade into the sound of a light wind and tiny pats of snowflakes landing. (Line 12) The fourth stanza tells...

Words: 902 - Pages: 4

Free Essay

Relaxation

...Brooke Bohinski January 24, 2012 English 101 Process Analysis Essay Sit Down and Relax Do you ever get the chance to just sit down and take a deep breath? Well many people say they never get that chance between school and work, but guess what? You finally have the weekend off and you can do whatever you want. Take the time to relax. The most relaxing weekend is the one that you just sit down and do what you like, such as spending time to yourself, outdoors or with people you love. People go to great lengths to relax, however relaxing can be as simple as an enjoyable weekend with no responsibility. First thing in the morning take a long, hot shower or bath to wake you up. You should get out and put on your most relaxed clothing, such as sweats or whatever you prefer to be comfy in. Almost all people wake up hungry, so after getting cleaned you need to fix something hot for breakfast, whether it is pancakes, waffles or just a fried egg. Next comes everyone’s favorite, sit down and watch either your favorite movie or the three episodes of your favorite TV show that you haven’t watched in over a month due to studying all night. After that get Facebook and check up on the latest gossip. By then anyone would be hungry so just fix something simple but good. Popping a pizza in the oven is something simple and good. After eating, you could go outside and enjoy the nature. It can be sunny, storming or snowing but either way you could enjoy them all. Sitting inside listening...

Words: 577 - Pages: 3

Premium Essay

Cameron Okeke: Safe Spaces Got Me Through College

...2016 one college did the absolutely unthinkable, it attacked safe spaces. The University of Chicago sent a letter to its incoming freshmen telling them that their campus was a hive of intellectual debate, reason, and free speech, scandalous. The blowback from this decision was immense. Tons of Millennials were offended and triggered by the fact that they may have to hear the opinions of people who disagree with them or hear facts, statistics, and jarring world history without trigger warnings, how weak, right? Cameron Okeke on the other hand, paints a different picture of these so called “weak Millennial snowflakes”. In his essay “I’m a black UChicago graduate. Safe spaces got me through college.”...

Words: 1283 - Pages: 6

Free Essay

Summary and Personal Response

...autobiographical non-fiction essay, describing her life growing up on a farm, and its subsequent effect on her life and personality. She is speaking of events in her past, and trying to influence the listener to understand that hard work has benefits, and that what you want may not be what you expected when you get it. In this story, Ms. Hemauer is trying to make the point of how adversity and obstacles, when worked through with support from those around you, not only can be overcome, but can also make you a better person. She is not pleased with her life as a Farm Girl, and she voices her displeasure often throughout the first three quarters of the essay. “I think to myself, ‘A typical ten-year-old child does not have to wake up at five in the morning to do chores!’” (Hemauer, 2011) However, once she realizes the quality of person she has become, she makes an educated and reasoned conclusion that it is due to the work she did helping with chores on her father’s farm. “Though I hated it growing up, working on the farm has taught me many lessons about life, and it has shaped me into the individual I am today.” (Hemauer, 2011) Another point she tries to make as well is that life on a working farm is difficult. She opens with a description of performing her morning chores with her brother and her sisters. This particular morning is described as being very cold and windy due to a blizzard: There is a brisk and bitter wind accompanied by icy snowflakes that feel like needles digging...

Words: 1137 - Pages: 5

Premium Essay

Us History Ip1

...US History Unit 2 Abstract Our country, like every entity will grow and evolve with time. Even a single celled organism will grow and change through time. The United States Federal Government is no different. Throughout the history of the United States, the authority of the Federal Government as grown dramatically. Some may argue the necessity of this growth and some may even pursue a smaller Federal Government in their daily lives. This essay will explore four different periods of time that the Federal Government extended its authority. The Cold War In black and white, grainy letters appear on screen that say, “Fallout, When and how to protect yourself from it”. In the background, beautiful snowflakes fall, representing the nuclear ash that falls after an atomic blast. The scene fades to white, then transitions into a blast, accompanied by a black plume of smoke that reaches the clouds (Defense, 1955). Images like this could sum up the fears and emotions after World War II; the beginning of the Cold War. At the end of World War II, America was the most powerful country in the World. America’s intervention in the war effectively ended it. A major turning point of the war was the bombing of Hiroshima and Nagasaki. This marked the end of the Japanese Empire and the beginning of a new age in the history of Planet Earth. It was the age of the Atomic Bomb. In 1965, American Theoretical Physicist Robert Oppenheimer was quoted on national television...

Words: 845 - Pages: 4

Premium Essay

Essay " Save as Many as You Ruin"

...3En/v Sometimes we do whatever makes us happy, even though it might be rather selfish and might have a consequence for our dearest. This exact scenario is shown in the short story; "Save as many as you ruin" by Simon Van Booy. The story takes place in New York, and even more precise; in a snowy Manhattan. We follow Gerard who is an alone-father with his eight year old daughter Lucy. Gerard loves Lucy as much as a father can love his child. Gerard has been with loads of women, but the one he truly ever loved, and still loves is Laurel, unfortunately is Laurel not the biological mother of Lucy. Issy is Lucy's real mother. Gerard cheated on Laurel, so he could be with Issy. This is the part which can be connected to the introduction of the essay, since Gerard chose to harm Laurel by sleeping with another woman. Gerard haven't met Laurel for eight straight years, which is hard to believe when reading the following quote: "In that moment of recognition he is not consumed by a rushing sensation of love - quite simply a door opens to a room that has never gone away" (l.63) This quote tells us how Gerard has just waited for the only woman he has ever deeply loved, he is not experiencing anything overwhelming nor surprising. Adding the door-metaphor it explains exactly how Gerard has felt about Laurel. The fact Gerard still is enamored in Laurel despite a seven-year break and a relationship between Gerard and Laurel which we must assume isn't in the best of conditions when thinking of...

Words: 913 - Pages: 4

Premium Essay

Summary Of Culture Jamming

...hoaxing and demonstrates its methods using con artist Joey Skaggs’ example. Skaggs’ objective is to reveal how ‘the media has been turned into a government propaganda machine, manipulating us into believing we’ve got to go to war in the Middle East’ (p. 51). He cannot call a press conference to discuss this, so he demonstrates the mechanism of creating a believable fake news story by creating one himself. He begins by ‘gett[ing] someone from an out-of-state newspaper to run a story on something sight unseen’ (p. 50) – in one case, reporting about a non-existent brothel for dogs. Then Skaggs reproduces that story ‘in a second mailing’ (p. 50), and journalists believe it because it is something that has appeared in print. Consequently, ‘a snowflake becomes a snowball and finally an [uncontrollable] avalanche’ (p. 50). (Skaggs’ fake story about the Cathouse for Dogs evoked ASPCA’s (American Society for the Prevention of Cruelty to Animals) angry response and even received an award ‘for best news broadcast of the year’ (p. 50). After Skaggs’ fake news story appears in media, he holds a conference where he reveals his deception. By revealing his own hoax, he demonstrates the mechanism of media hoax creation in general and then he talks about media as a propaganda machine. Accounts of case studies in Part II expose controversies associated with culture jamming, such as filmmaker Emile de Antonio illegally obtaining archival footage for his documentary Millhouse: A White Comedy as ‘he...

Words: 1191 - Pages: 5

Premium Essay

Oppression In A Thousand Splendid Suns

...“A Thousand Splendid Suns” written by Afghan author Khaled Hosseini is an example of the portrayal of the lives of women in Afghanistan. The challenges and oppression in the lives of two Afghan women; Laila and Mariam are shown Throughout the novel, which provided or facilities on the relationship between both these women. This novel is a good representation of the cruelty experienced by the Afghan women in their everyday life. “A Thousand Splendid Suns” is a significant piece of text that bring the predicament of the women who have lived under the debilitating shadow of patriarchal oppression on women. Using the vales, the authors comprehension kills and the contributing factors to emphasize key aspects of of this essay. In countries...

Words: 1235 - Pages: 5

Free Essay

Bbc1

...1. Which sentence uses verbs correctly? Choose 1 answer - ANSWER D – A. Angry cats are wanting to be let outside. –WRONG TENSE should be “want” B. The hungry dogs have ate their breakfast. - WRONG TENSE should be “eaten” C. Yesterday, the teacher prepares the assignment. - WRONG TENSE should be “prepared” D. The children are ready to go to school. 2. Which sentence has the correct subject-verb agreement? Choose 1 answer ANSWER A – ignore the stuff between the commas A. The children, hushed by their teacher, try to keep quiet. B. The books, read by the student, is returned to their shelf. INCORRECT – plural “books” doesn’t match singular “is returned” C. The librarian, anxious to find the books, browse the stacks. Singular “librarian” does not match plural “browse” D. The student, bothered by the uproar, request less noise. Singular “student” does not match plural “request” 3. Which sentence has the underlined word spelled correctly? Choose 1 answer ANSWER A – B is spelled correctly but it’s the wrong use. A. The principal sent the student a letter of congratulations. B. The principle ate lunch with the students. INCORRECT – wrong version of word for sentence C. The prinsipal signs the final grade reports. INCORRECT – no ‘s’ D. The princepal earned minimal interest. INCORRECT – no ‘e’ 4. Classify each sentence according to whether it exhibits or does not exhibit parallelism. Select your...

Words: 6519 - Pages: 27

Premium Essay

Week 5 Psy/405

...personality. Not only did phycologist have to consider what makes up the human personality; these same brilliant minds came up with several various approaches to the human personality. Each one of the following positions or theories all touch basis on different developments of a person’s personality. Whether it is the dispositional approach that theorist state are key traits in development that make an individual unique to him or her, or the humanist/existential approach which focuses more on the evidence for the stability of genetically influenced personality, more specifically evolution and how it applies to the issues of ambition and leadership. Here we will describe the differences between dispositional personality and humanistic. In this essay we will discuss the role of personality in affecting situational behavior. Following this, examine the personality characteristics attributed to each theory in our approach. Finally, we will explain the interpersonal relational aspects associated with the theories selected. . A number of different theories have emerged to explain different aspects of personality. Some theories focus on explaining how personality develops while others are concerned with individual differences in personality. In the following we will review the differences in dispositional and humanistic and the role of personality in the affecting situational behavior. We will discuss why they are each different from one another and who was the founder of each theory....

Words: 1446 - Pages: 6