Fractals Of The World

1. Fractals Of The World By Leslie Ryan

2. Common Terms • Iteration- To repeat a pattern multiple times, usually with a series of steps. • Reflection- An image that is thrown back from light, heat, or sound. Like an image you see in a mirror. • Infinite- A design or something that repeats itself and does not stop, there is no end. • Chaos Theory- A type of math that has to do with complex systems, so complex that if you were to change one small thing it could end up making a huge problem. • Recursive- A whole pattern made up of the same repetitive patterns. • Self-Similarity- Has an exact or almost exact piece of the same bigger pattern and repeats. • Repetition- Something that is made by repeated steps, or designs, usually looks like there is not an end to the pattern.

3. What Is A Fractal? A fractal is typically something that continuously repeats itself or something that creates multiple and identical patterns within the one object. What Is Not? Something that is not a fractal is something lacking in unique patterns and lack of repetition within the object.

4. Fractals In Nature From the food we eat, to the flowers we smell, and the ground we walk on, this whole world is full of fractals! Everything has it’s own pattern and designs that make up that object. Nature just happens to have quite a bit of things that have a repetitive pattern. The pictures on this slide are main examples of fractals within nature.

5. PYTHAGORAS TREE The math used to make this type of fractal is based on right triangle geometry, specifically the Pythagorean Theorem. It was invented by Albert E. Bosman, a Dutch mathematics teacher, in 1942. It is named after the ancient Greek mathematician Pythagoras, as shown on the right. Pythagoras Trees: The math used to make this type of fractal is based on right triangle geometry, specifically the Pythagorean Theorem. Pythagoras Trees: The math used to make this type of fractal is based on right triangle geometry, specifically the Pythagorean Theorem. Invented by Albert E. Bosman, a Dutch mathematics teacher, in 1942. It is named after the ancient Greek mathematician Pythagoras, as shown on the left. Invented by Albert E. Bosman, a Dutch mathematics teacher, in 1942. It is named after the ancient Greek mathematician Pythagoras, as shown on the left. Invented by Albert E. Bosman, a Dutch mathematics teacher, in 1942. It is named after the ancient Greek mathematician Pythagoras, as shown on the left. Pythagoras Trees: The math used to make this type of fractal is based on right triangle geometry, specifically the Pythagorean Theorem.

6. MANDELBROT SET The type of math used for this fractal is algebra. More specifically polynomials and the quadratic equation. Benoit Mandelbrot is the mathematician that the Mandelbrot set is named after, mainly because he was the one who studied and popularized it.

7. JULIA SET Both the Julia and Mandelbrot sets are drawn with the same function. The type of math is Algebra, dealing more with polynomials and quadratic equations. Formula Used: This fractal is named after, Gaston Julia, a French mathematician who studied them and later wrote a paper on the type of fractal.

8. DRAGON FRACTAL The math used to construct a dragon fractal is referred to as the Lindenmayer System. The Lindenmayer System is not exactly math based but instead is the process of how something grows, typically plants. The formulas behind this kind of fractal are as followed: The three pictured men to the right are the NASA physicists that first investigated the dragon fractal into depth. Bruce Banks, John Heighway,and William Harter.

9. SIERPINSKITRIANGLE This fractal originally starts as a type of curve but it is mathematically generated so it can reproduce at any size, big or small. The basic idea is you take out one triangle and continue to take them out till you have your desired effect. The formula used to create this kind of fractal is as followed: The polish mathematician, Wacław Franciszek Sierpiński, described this fractal in 1915. That is why the fractal is named after him today.

10. NEWTON FRACTAL The math behind this fractal is polynomials and it also uses Newton’s method. A method used for finding better approximate numbers for zeros. The following formula is the one used to make this kind of fractal. Sir Isaac Newton, an English mathematician, helped create the method used to help make the pattern with a Newton fractal.

11. MORE FUN FRACTALS

12. Software Programs Software Programs or Fractal Generators are used to help create more complex fractals. Fractals that typically can not be done by hand. They usually let you pick your own colors and what pattern you would like to follow, or if you know what your doing you can create a type of fractal all your own.

13. What You Can Learn There is a lot you can learn from fractals. Some people specifically study fractals in college. The math behind a lot of fractals is confusing and typically not understandable by people who haven’t studied fractals for awhile. A type of math you can learn from fractals is the Chaos Theory. You can also learn a lot about your own artistic ability by creating an iteration of designs and combining different color patterns.

14. My Thoughts & Ideas My thoughts on fractals are not the nicest. I think fractals are complicated. A lot of terms used in the descriptions of fractals were hard to understand. Most of the math used to create fractals is also very difficult. I would never pursue a career dealing with fractals.

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