Fractals were technically discovered by Benoît Mandelbrot in 1975. I say technically because you can’t really discover something if you are making it up. Mandelbrot basically put fractals into words. He explained them as being geometric shapes that when divided into parts, each part would be a smaller replica of the whole shape. Benoit came up with the word "fractal" as the new scientific term for these patterns. The word comes from the Latin word “fractus”. The latin word fractus means "broken" or “fractured”. Fractals have been naturally occurring for as long as we can imagine, but no one realized it until Mandelbrot.
Mandelbrot began his research when he found fractals in nature. However, they are also found in mathematical equations and within modern industries such as cell phones and medicine. The main attribute of a fractal is its repeating pattern at any scale (zoom level). Additionally, fractals are known for expanding and evolving symmetry. They can be called a self similar pattern if the replication is the same at every level. In my opinion, the easiest fractal to understand is the Sierpinski Triangle fractal (1st picture). It is created by drawing a triangle, then separating it into 4 triangles, and then removing the middle one. You’re probably wondering why fractals are important and why they are worth knowing. Now that you know the most basic one, we can go deeper into fractals. Many people would find no interest in fractals and do not know that they are naturally occurring in day-to-day life. We find fractals in 3 main areas: Nature, Geometry, and Algebra. 
First, we will look at naturally occurring fractals. Fractals in nature often range in scale. There are fractals inside the human body such as blood vessels and within neurons, in the branching of trees and sprouting of plants, and also in weather such as lighting, hurricanes, and rivers. A large portion of fractals in nature are in the spiral pattern, which appears more often than other patterns in nature. The spiral is found inside of tree’s trunks, hurricanes (3rd picture), and agave cactus’ (2nd picture).
Second, there are an infinite amount of fractals in geometry, you can even make your own. The most popular one is the Sierpinski Triangle (1st picture). This triangle is simply created by removing the middle triangle from the triangle(s) from the prior generation. Every generation increases the amount of triangles by a factor of 3: 1,3,9,27,81,243,729 to infinity. Another popular geometric fractal is the koch curve (4th picture). You start with the first bold line, which has 4 line segments. For the next generation of the pattern, you simply replace each of the 4 line segments with the original 4 line segment. You can keep doing this forever. Geometric fractals are never ending because we can perfect them, unlike fractals in nature.
The third main fractal is the algebraic fractal, which are the hardest to understand in my opinion. The most popular algebraic fractal is the Mandelbrot set founded by Mandelbrot in 1980. The equation is found below: you plug in a constant for C and any number you want for Z old. When you get Z new, you plug it back into the equation and you repeat this step. We are interested in C and how different values of C affect the equation. Eventually, you will see one of two patterns. Either the answers will be going towards infinity or the value of the equation will stay between a fixed set of numbers. Values of C that are found in the black cause the equation to stay finite whereas all values of C outside the black cause the equation to go towards infinity. This was hard for me to comprehend at first but Mandelbrot simply found every value of C that makes the equation repeatedly stay finite instead of going to infinity. At the end of this he got a fractal. At first, I was spectacle to how this was a fractal, but as you see in the zoomed in image, it begins to repeat itself.
The title of this blog post really caught my eye. The word fractals sounded familiar when I first read it, but once I started reading the article, I discovered that this was something completely new to me. I thought it was cool how fractals occur in nature. It reminded me of my blog post. Symmetry and fractals are very closely linked, yet they each have their own cool properties. I thought it was interesting how a person can create his or her own fractals. The fact that possibilities for different fractals are infinite in geometry makes sure that everyone can create their own amazing, unique fractal.
ReplyDeleteFractals, such a simple concept with so many complex forms. With Frank providing pictures of different kinds of fractals, it made it much easier to visualize what these patterns look like and how they take different forms. After looking at the triangle picture, you immediately see what Frank is talking about when he explains what a fractal is. It is just a repeating pattern of the same shape within its original shape.The next two pictures are good examples of where we can see fractals in everyday life. One point I found interesting was that of the blood vessels and illness containing fractals. It seems they are truly everywhere. I thought the algebraic fractal was a bit confusing to understand, but the rest was very practical and applicable to everyday life. I have noticed fractals before, but never thought anything of them because it seemed to just be part of a pattern or design. I find fractals very common in designs and patterns of bed comforters, sheets and pillows. I believe I have a fractal design on my comforter at home and didn't realize until now. This topic is one that changes the way you look at certain things, such as a pattern being randomly designed, to being a fractal design.
ReplyDeleteThe title of this post caught my eye because recently in my Netflix show (White Collar) a fractal was created and unlocks a secret about a sunken submarine, crazy I know. I had no idea what a fractal was, however, when the show introduced it. After reading this post I can identify it as a Koch curve, in case anyone was concerned. I think it is very interesting how often a fractal can be found in nature, it kind of reminds me of the golden ratio. The fact that fractals can fractals found in blood vessels scares me because that means I will probably have to learn about them in the future as a pre med student. They seem a little difficult to understand. Nonetheless, fractals are an interesting topic and I look forward to learning more about them.
ReplyDeleteI find that fractals come up quite a bit in this class. For starters, when we watched the video of that girl doodling in math class each and every one of her “doodles” seemed to be fractals that can be linked to this blog post. Secondly, in the current project that Jo and I are working on we found that the fractal triangle is relevant when graphing all of the different possibilities of playing the Tower of Hanoi. All in all I find fractals to be extremely interesting because, like shown in the video, it’s a form of doodling with math tied into it.
ReplyDeleteThe title of this post really caught my eye, because this topic is a completely new concept for me, because I have never heard of fractals before and I was very interested in learning. I also found it very interesting that this concept was created and I find it interesting that someone could just make something up like this. The concept is actually quite intriguing and seems pretty difficult to wrap my mind around how something like this was invented. Overall, this blog post was very well written and the pictures caught my eye and really made the blog post more fun to read.
ReplyDeleteI’ve heard of fractals before but I have never really known what they were or what they were used for. Like Molly said, the term reminds me of the golden ratio in Monica’s post and Jo’s post about symmetry. I didn’t know that they could be found in nature and in blood vessels is interesting to me and makes me want to learn more. I think Frank did a good job of breaking down 3 areas where they could be found instead of skewing off and going into different realms. The part at which he mentioned the Koch Curve relates back to me and Monica’s final project which I thought was interesting.
ReplyDeleteFractals are a very interesting part of nature. People have seen them all over the place but didn’t know exactly what they were, but the way Mandelbrot was able to describes them makes it clear to all how obvious they are in life. The image used by Frank with the triangles broken down into smaller and smaller triangles is one of my favorite images to look at, it just looks so elegant and clean. The spiraling pattern a fractal takes in nature such as in a hurricane is fascinating because I would never have thought that so much math and geometry went into something as devastating and magnificent as a naturally occurring hurricane. When I think of the math behind something I think of a math problem not the weather.
ReplyDeleteIt's surprising to me to see how repetition in nature can be talked about as a mathematics topic. This post reminded me of that video we watched in class one Friday. The one were the girl kept repeating drawing the same patterns and they ended up continuing infinitely. The one example that you used that I found really interesting was the one where you have the line segment with a triangle in the middle of it. This one caught my eye because I find it interesting how there will keep building onto each other but never cross paths. Overall this was a very interesting post and topic and was written really well.
ReplyDeleteI really like the idea of fractals and how they are neat and concise, but I'm still a bit confused about how they are present in things like rivers and lightning, for example. Not to use my own project as an example, but its the easiest for me to see - snowflakes are a great model of how fractals are present in nature mostly because you can clearly see the symmetry. Additionally, we have proof of how these flakes are formed, but I still can't picture how fractals are a part of things like leaves and plant sprouts. All in all, I liked this post because I could relate to it, and it was very well written / eloquent!
ReplyDeleteFractals are really interesting and it amazes me how such things appear in nature! The example of the plant is beautiful, I never knew fractals could be so prominent in our life and I've never even known what they were. Fractals looks like it can lead to some really cool art even if it's not intended. To be honest, I don't really enjoy the math side of the fractals. I much more prefer looking at the fractals and applying it to the world around me. Now, I’ll probably be looking for fractals around all around me! Some seem much harder to notice because I feel like it should be symmetrical but if a hurricane is an example than that’s really impressive.
ReplyDeleteI have honestly never thought about fractals in a complex way. The word itself seems so complicated but its only shapes being broken down into smaller shapes. While reading this post I had no idea that fractals were as common as Frank made them out to be, however it is extremely interesting that fractals can be found all around nature. My favorite of the pictures included was the cactus fractal because it looks beautiful. I would agree that the algebraic fractals were the hardest to understand, but they looked the coolest out of the three different types.
ReplyDeleteAt a first glance of this blog I had no clue what fractals were so it intrigued me to read more about it. When I started reading and read that this was just a made up thing I wondered why it was so special like I can even make up something, but reading it more it become more interesting to me. Fractals were not a hard concept to understand which I liked that and I am familiar with them I just never knew that this is what they were called. I had no idea however the importance of them just like Frank didn't. I thought it was neat in how they were involved in things that are used everyday that I never knew about. I wonder how they are using this concept and finding illnesses with it I think that would be something to further look into.
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