When one thinks of fractals, what comes to mind may be the pretty, intricate images associated with backgrounds and screensavers. But would you ever think of cancer? The truth is, there is so much more to fractals than what meets the eye. For years, the mathematics behind these "pretty images" has been applied in fields of nature, technology, computer graphics, and most recently - in cancer research. Cancer, like many other aspects of the natural world, cannot be described using the standard Euclidian geometry of smooth shapes. Rather, cancer is highly complex and irregular. Fractal geometry provides a way to quantify the irregularity and raggedness of cancerous tumors, aspects which previously pathologists could only describe in a qualitative manner. And with this new discovery, our understanding of cancer is improving in leaps and bounds. …show more content…
We will look at several examples of different types of fractals to help explain these three characteristics.
2.1 Self Similarity Fractals are self-similar, made up of common patterns that repeat themselves infinitely on different scales. When magnified, a smaller section of the fractal would look identical to the entire fractal. Figures 1 and 2 are some examples of different fractals. These fractals all belong to the class of linear fractals - fractals which are perfectly self-similar, made up of straight regular lines and shapes, and symmetrical. As shown below, all parts encircled in red within each of the fractals are made up of the same patterns, just magnified at different
While the studies at Governor’s School are noticeably more advanced and require more effort than at regular public schools, I see this rigor as the key to my academic success. For me, the classes I take that constantly introduce new thoughts that test my capability to “think outside the box”, are the ones that capture all my attention and interest. For example, while working with the Sierpinski Triangle at the Johns Hopkins Center for Talented Youth geometry camp, I was struck with a strong determination to figure out the secret to the pattern. According to the Oxford Dictionary, the Sierpinski Triangle is “a fractal based on a triangle with four equal triangles inscribed in it. The central triangle is removed and each of the other three treated as the original was, and so on, creating an infinite regression in a finite space.” By constructing a table with the number black and white triangles in each figure, I realized that it was easier to see the relations between the numbers. At Governor’s School, I expect to be provided with stimulating concepts in order to challenge my exceptional thinking.
I have chosen to write about the constellation Cancer (The Crab). I chose Cancer because it is one of only a handful of constellations that I am actually able to identify in the night sky. Cancer is one of the twelve Zodiac constellations; people whose birthdays fall between June 21st and July 22nd have Cancer as their sign. Cancer is the Latin word for crab, and despite the fact that the constellation looks more like a lobster then a crab, it is still referred to as a crab. The constellation is visible from the northern hemisphere from late winter to early spring.
As a cancer clinic volunteer, the daughter of an oncologist, the friend of a breast cancer survivor, and a biological enthusiast, I find the medical field of cancer and its impacts on health fascinating. The human body is so complex, yet, so fragile at the same time and I hope that through this exploration, I will witness how mathematics plays a role in science and more specifically physiology.
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that...
Part 1. (a) Define each, (b) Explain its significance, (c) where indicated with this symbol * provide an example.
Two well-known fractals are named after him the Sierpinski triangle and the Sierpinski carpet, as are Sierpinski numbers and the associated Sierpinski problem. The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Sierpinski who described it in1915.Originally constructed as a curve; this is one of the basic examples of self-similar sets. The Sierpinski triangle has Hausdorff dimension log (3)/log (2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of ½.
Cancer develops when cells in a part of the body begin to grow out of
From the anatomy of a human, the social life of insects, and the way the world functions are all interconnected through complex system science. By taking fractal geometry and implementing it into larger unmanageable scales can help provide further more in depth information pertaining to not just that individual but also the system as a whole.
Statistic images and landscapes, or know as fractal landscapes, and the way that this component works is that these statistic images...
...e amid these interfaces. The contraption will understand these waves as parallel lines alongside equal distances amid them, and cut density for the deeper lines, because the imitated waves come to be softly lesser in number. This aftermath in a stripped outline possessing alternating dark and clear lines at usual intervals [Figure 5].[6]
...on of light and the rays are proportions in the Fibonacci sequence. Fibonacci relationships are found in the periodic table of elements used by chemists. Fibonacci numbers are also used in a Fibonacci formula to predict the distant of the moons from their respective planets. A computer program called BASIC generates Fibonacci ratios. “The output of this program reveals just how rapidly and accurately the Fibonacci ratios approximate the golden proportion” (Garland, 50). Another computer program called LOGO draws a perfect golden spiral. Fibonacci numbers are featured in science and technology.
Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears.
It is constructed by taking an equilateral triangle, and after many iterations of adding smaller triangles to increasingly smaller sizes, resulting in a "snowflake" pattern, sometimes called the von Koch snowflake. The theoretical result of multiple iterations is the creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. Fractals, before that word was coined, were simply considered above mathematical understanding, until experiments were done in the 1970's by Benoit Mandelbrot, the "father of fractal geometry". Mandelbrot developed a method that treated fractals as a part of standard Euclidean geometry, with the dimension of a fractal being an exponent. Fractals pack an infinity into "a grain of sand".
The Golden Ratio is also known as the golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern.
A rectangle is a very common shape. There are rectangles everywhere, and some of the dimensions of these rectangles are more impressive to look at then others. The reason for this, is that the rectangles that are pleasing to look at, are in the golden ratio. The Golden Ratio is one of the most mysterious and magnificent numbers/ratios in all of math. The Golden Ratio appears almost everywhere you look, yet not everyone has ever heard about it. The Golden Ratio is a special number that is equal to 1.618. An American mathematician named Mark Barr, presented the ratio using the Greek symbol “Φ”. It has been discovered in many places, such as art, architectures, humans, and plants. The Golden Ratio, also known as Phi, was used by ancient mathematicians in Egypt, about 3 thousand years ago. It is extraordinary that one simple ratio has affected and designed most of the world. In math, the golden ratio is when two quantities ratio is same as the ratio of their sum to the larger of the two quantities. The Golden Ratio is also know as the Golden Rectangle. In a Golden Rectangle, you can take out a square and then a smaller version of the same rectangle will remain. You can continue doing this, and a spiral will eventually appear. The Golden Rectangle is a very important and unique shape in math. Ancient artists, mathematicians, and architects thought that this ratio was the most pleasing ratio to look at. In the designing of buildings, sculptures or paintings, artists would make sure they used this ratio. There are so many components and interesting things about the Golden Ratio, and in the following essay it will cover the occurrences of the ratio in the world, the relationships, applications, and the construction of the ratio. (add ...