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Transcript of Fractals
and Sarah Bray Fractals What are fractals? Fractals are shapes with infinite detail and perimeter. The more it is magnified the more complex it becomes. At all levels of the fractal it is self-similar, meaning it is basically the shape as that of another level. This idea formed the concept of fractal geometry, where lines are not the basic building blocks of shapes. All fractals are graphed in the complex plane. Fractals were originally developed to help study nature. The complex patterns found by scientists were too complex to describe using 'regular' math, so they started using fractals. ("Introduction to Fractal Geometry"). Koch Snowflake Perhaps the simplest of all fractals is the Koch Snowflake, which was created by Swedish researcher Neils Fabio Helge von Koch (born 1870, died 1924). The lower image depicts the creation of Koch Snowflake in a few simple steps. When creating a Koch Snowflake, each side of the triangle is divided into three sections. In the middle section a smaller triangle is added and the process is repeated again, this time dividing each of the now six triangles into three sections. The Koch Snowflake can be continued for an infinite amount of times, each level displaying exact similarity ("Koch Snowflake Fractal"). Self-Similarity There are 4 different types of self-similarity a fractal can display. Exact similarity is when the fractals looks exactly the same on all scales. Likewise, if a fractal is only somewhat similar on all scales it is considered to exhibit Quasi similarity. Statistical similarity is a fractal that repeats numerical values so that patterns are maintained. Finally, a fractal that is similar in a Qualitative way is used when referring to a fractal pattern in time ("Introduction to Fractal Geometry"). Koch Snowflake- Equations a Julia Set -a Number of sides
n= 3 x 4
Length of sides
L= n x 3
P=L x n The Julia set was created by French researcher Gaston Julia (born 1893, died 1978). Drawn entirely by hand, the Julia Set, showcasing Quasi similarity, became the most complex fractal of its time. After researching fractals for years, Gaston Julia published a book featuring his findings and, most importantly, the Julia Set ("Fractal Start Page"). Julia Set- Equations R = P(z)/Q(z)
R is a rational function
P and Q are polynomials with common divisors
Doesn't approach infinity after applying R many times The Mandelbrot Set was discovered by Yale professor Benoit Mandelbrot (born 1924, died 2010). He used a computer to expand the equation of the Julia Set and he was able to create a fractal representing all possible Julia Sets ("Fractal Start Page"). Mandelbrot Set- Equations Quadratic occurrence equations
z = z + C
C is number of steps to get r = 2
'Kidney bean' part
4x=2cos(t) - 2cos2(t)
4y=2sin(t) - 2sin2(t)
center at (-1, 0)
radius of 1/4 n+1 n 2 max Why are fractals important? Medical Science Nature is scale invariant, so formations like coastlines, fault lines, and oil fields can be measured through the use of fractals. Through these measurements made by fractals, scientists can predict where reserves of oil could possibly be and how big they can be ("Formulas to Find Oil Fields").
A new version of the Big Bang Theory states that the universe consists of many spirals of different sizes, which are self-similar and have infinite detail and perimeter. In this way, some theorists think that our universe is one giant fractal ("Fractal Universe"). Military Identify man-made structures ("Applications").
Tracking submarines ("Applications").
Fractals are plugged into a machine that run equations on the natural surroundings and if something does not add up, it's probably a man-made structure. This holds true especially on coastlines, which are fractal in nature. If the coastline and the fractal equation used for that coastline are not exactly the same, there is normally a man-made structure on the coastline ("Applications"). Nature Everything in nature is self-similar. From seashells to germs, everything in nature is iterative, recursive, and generally infinite but with a set perimeter, like fractals ("17 Captivating"). Therefore fractals are used in the study and understanding of many different natural processes, like weather and migration patterns ("Applications"). Computer Animations Writing Computer model using fractals for wind Bone fractures break in a self-similar manner, so doctors use fractals to "graph" the fracture and understand how the bone broke ("Fractals in Physiology").
Cancer grows in a fractal-like pattern, so doctors use fractals to graph where and how big the cancer call has ("Fractals in Physiology").
All structures of the body are self-similar to their overall appearance, so fractals are used to understand the human body. An example of this would be the lung to the right ("Fractals in Physiology"). "The Sarah-Kirsten Set" Works Cited ("Koch Snowflake Fractal") ("Fractal Equations") ("Fractal Equations") ("Sekino's") ("Sekino's") ("Sekino's") ("Sekino's") ("Sekino's") ("Sekino's") ("Sekino's") ("Wilkinson") ("Wind Energy Simulations") Fractal equations are used to compress files on computers. They are especially used for compressing images so the pictures do not lose their definition ("Applications").
Fractal equations are used to form models, especially for self-similar models like wind turbulence and stress loading on oil rigs ("Applications"). ("17 Captivating") ("Applications")
"17 Captivating Fractals Found in Nature | WebEcoist." 17 Captivating Fractals Found in Nature | WebEcoist. N.p., n.d. Web. 20 Mar. 2013. <http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/>.
"Applications." Fractal Applications. N.p., n.d. Web. 20 Mar. 2013. <http://math.youngzones.org/Fractal webpages/fractal_applications.html>.
"Formulas to Find Oil Fields." Times Higher Education. N.p., n.d. Web. 21 Mar. 2013. <http://www.timeshighereducation.co.uk/160443.article>.
"Fractal Equations." Fractal Programming -. N.p., n.d. Web. 20 Mar. 2013. <http://www.fractalsciencekit.com/program/maneqn.htm>.
"Fractal Start Page." Fractal Start Page. N.p., n.d. Web. 20 Mar. 2013. <http://math.youngzones.org/Fractal%20webpages/fractal_start.html>.
"Fractals in Physiology." Fractal Geometry Panorama. Yale University, n.d. Web. 20 Mar. 2013. <http://classes.yale.edu/fractals/panorama/Biology/Physiology/Physiology.html>.
"Fractaluniverse.org." Fractal Universe. N.p., n.d. Web. 21 Mar. 2013. <http://www.fractaluniverse.org/v2/?page_id=2>.
"How To Write A Novel Using The Snowflake Method." How To Write A Novel Using The Snowflake Method. N.p., n.d. Web. 20 Mar. 2013. <http://www.advancedfictionwriting.com/art/snowflake.php>.
" "How To Write A Novel Using The Snowflake Method." How To Write A Novel Using The Snowflake Method. N.p., n.d. Web. 20 Mar. 2013. <http://www.advancedfictionwriting.com/art/snowflake.php>.
"Introduction to Fractal Geometry." Introduction to Fractal Geometry. N.p., n.d. Web. 20 Mar. 2013. <http://www.fractal.org/Bewustzijns-Besturings-Model/Fractals-Useful-Beauty.htm>.
"Koch Snowflake Fractal | Koch Snowflake Fractal |Khan Academy." Khan Academy. N.p., n.d. Web. 20 Mar. 2013. <https://www.khanacademy.org/math/geometry/basic-geometry/koch_snowflake/v/koch-snowflake-fractal>.
"Sekino's Fractal Gallery." Sekino's Fractal Gallery. N.p., n.d. Web. 20 Mar. 2013. <http://www.willamette.edu/~sekino/fractal/gallerymain.htm>.
"Studying Mandelbrot Fractals." Math Forum: Suzanne Alejandre. Drexel University, n.d. Web. 20 Mar. 2013. <http://mathforum.org/alejandre/applet.mandlebrot.html>.
"Wilkinson Microwave Anisotropy Probe." (WMAP). N.p., n.d. Web. 20 Mar. 2013. <http://map.gsfc.nasa.gov/>.
"Wind Energy Simulations." NREL: Computational Science -. N.p., n.d. Web. 20 Mar. 2013. <http://www.nrel.gov/energysciences/csc/projects/wind_energy_simulations>. Works Cited Part 2 All computer animations use fractal algorithms to create details that are high-definition. Fractal equations are used for background landscapes, weather, and skin textures. An example of this would be that fractals were used for raindrops on the dinosaur scales in Jurassic Park ("Applications"). Some authors use a method to plan their novels, called the “Snowflake Method,” which is based off of the Koch Snowflake. First the idea of the novel is put into one main sentence, which is then written into three sentences. Each of these sentences are expanded into three different paragraphs. Eventually the novel idea will be fully plotted and authors normally stop plotting after writing a ten-page summary of their novel, their characters, and the like. A first draft of a novel that uses the “Snowflake Method” will normally be better than a draft that’s been edited thrice without the Koch Snowflake planning beforehand (“How to Write”). "a"= number of 'levels to the fractal