Fractals are all around us, in our lungs, kidneys, blood vessels, plants, in the clouds, in the tress, in the landscape, and even in our very heartbeats. But what exactly is a fractal? In simple terms a fractal is something that has self-similarity. That means that as you zoom in and out the object looks the same.
Fractals are all around us in nature. Fractals are in the clouds, in the trees and in our lungs, livers and veins. A team of people went into a protected forest and cut down a few trees (with permission of course!) and took measurements of the trees. Based on those measurements they could see that there were constant repeating patterns of where the branches split off and the thickness of the branches. They saw that this extended not just for single trees, but for the entirety of the forest.
Classical mathematics ran on the basic assumption that everything is regular and has smooth edges. That is until Benoit Mandelbrot came around. He saw mathematical equations as pictures in his head. After teaching in France for a time, he went and worked for IBM. There was an issue with transmitting data over phone lines and decided to plot the noise data. The graph was the same regardless if the time interval was a minute an hour, a day or even a week. Benoit Mandelbrot was one of the first to start experimenting with fractals. He created a set of numbers now known as the Mandelbrot set by using a computer to run an equation millions of times and turned the numbers into points on the graph. He discovered that when you zoomed in on the series after it had been plotted, the images created would be repeated. He later went on to write a book on fractals appearance in nature.
A programmer by the name of Loren Carpenter came across ...
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... has small bumps or grooves in it that can be created using fractals.
Fractals can also be used in animation. As stated earlier it’s possible to create mountains and also entire planets using fractals. Fractals were first introduced to the film industry in Star Trek II: The Wrath of Khan, where the entire planet that they flew over was created using fractals. In one scene of the new Star Wars movie; Anakin and Obi Wan are facing off inside of an active volcano. The lava in the scene was created using fractals. The artist add swirls to the 3D model of the initial lava spout and shrunk them. They repeated the process over and over again, layering each one until the entire background was made of fractal swirls.
Fractals have become an important part in video game creation today. It can be used to create terrain, forests, entire worlds, textures and special effects.
(http://www.encyclopedia.com/doc/1O7-densityfrequencydominance.html) Biodiversity is the number of richness or the number of species in a local area. This happens when someone can look at a species, in order to indicate a degree of uncertainty. This can happen by calculating the number of species given, where the individual is picked at random from the community. In other words, if the diversity is high, then oneself will have a poorer chance of correctly calculating the species of the next individual picked at random. (http://www.tiem.utk.edu/~gross/bioed/bealsmodules/shannonDI.html) This experiment was a way to find out the diversity of the school parking lot and the possibility to identify the type or model of the student’s, faculties and guest
James Gleick was quoted by Yeongmahn You, where he stated that “fractal means self-similarity; self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern”. In other words self-similarity is a repetition of the detail that present from the smallest to the largest scale, therefor creating a hidden pattern of order that has structure and regularity (Gleick 1987:103).
Divisionism, which involved breaking down light and color into a series of stippled dots and stripes. Fracturing the picture plane into segments to achieve an ambiguous sense of depth. Futurist paintings were exhibited in Milan. The paintings featured threadlike brushstrokes and highly keyed
When viewing Georges Seurat’s, A Sunday on La Grande Jatte (Fig. 31-37), perception is changed vastly depending on the viewer 's proximity to the piece. At close range, all that is visible is a mass array of countless circular dots and tiny lines in a vast range of colors. Greens, blues, reds, oranges, yellows, white, browns, black and purples are all visible in a multitude of intensities. The Divisionism technique utilized causes this piece to appear as an abstract collection of colors when viewed at close range. Yet when distance is between the piece and the viewer, these seemingly sporadic dots come together to create a complete and detailed scene. Primarily consisting of biomorphic shapes, Seurat’s incorporates in every inch of the canvas
"15 Uncanny Examples of the Golden Ratio in Nature." Io9. N.p., n.d. Web. 10 Mar. 2014.
Complex science provides an opportunity to break down something known to be a very large structure in order to see answers from a more individual point of view. It can become very beneficial in the long run depending on how it is applied. However, that’s just it you do not need to know how the entire global economic system works in order to benefit from it. When Benoit Mandelbrot created fractals they were used in various ways, to measure nature an...
It’s also hard to not follow the pattern of black zigzag like lines that stretch across it, but if you can get your eyes to look past it and see the different accentuating colors and shapes, it’s really an amazing work of art. I can only imagine how it would look at its full scale of 8 feet by 20 feet. That’s just an incredible size, and to see it in that size would allow for someone to see all the minute details and color flecks that are bound to be there, that you can’t see from a computer or television screen.
In fact, a rectangle with side lengths φ is said to be a golden rectangle, which is a result of the assumption that h=1 [2]. The number phi has even gripped theologians to ...
The recursive sequence of numbers that bear his name is a discovery for which Fibonacci is popularly known. While it brought him little recognition during the course of his life, is was nearly 100 years later when the majority of the mathematicians recognized and appreciated his invention. This fascinating and unique study of recursive numbers possess all sorts of intriguing properties that can be discovered by applying different mathematical procedures to a set of numbers in a given sequence. The recursive / Fibonacci numbers are present in everyday life and they are manifested in the everyday life in which we live. The formed patterns perplex and astonish the minds in real world perspectives. The recursive sequences are beautiful to study and much of their beauty falls in nature. They highlight the mathematical complexity and the incredible order of the world that we live in and this gives a clear view of the algorithm that God used to create some of these organisms and plants. Such patterns seem not have been evolved by accident but rather, they seem to have evolved by the work of God who created both heaven and
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
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 a strange ratio that scientists have found all throughout nature, architecture, art, and various other places. Some say that the Golden Ratio could only have been made possible by God while others believe it is merely a coincidence. This “Golden Number” has been thought of as the most pleasing to the eye and many tests have been done to see whether humans’ perception of beauty is affected by the appearance of this phenomenon.
There is always room in mathematics, however, for imagination, for expansion of previous concepts. In the case of Pascal’s Triangle, a two-dimensional pattern, it can be extended into a third dimension, forming a pyramid. While Pascal himself did not discover nor popularize it when he was collecting information on the Triangle in the 17th century, the new pattern is still commonly called a Pascal’s Pyramid. Meanwhile, its generalization, like the pyramid, to any number of dimensions n is called a Pascal’s Simplex.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.