Fractals

Fractals

Project Type: Presentation (pptx)

Downloads: 0 - 10 Wednesday 15th March 2017 Report

Fractals - Overview

------------ PAGE 1 ------------ Joe Czupryn MTH 491 ------------ PAGE 2 ------------ I. Introduction II. Dimension III. Brief history IV. Specific fractals and their properties V. Appearances and applications ------------ PAGE 3 ------------ applications ------------ PAGE 4 ------------ ƒí Self- similarity – when broken into smaller and smaller pieces, the new pieces look exactly the same as the original ƒí Dimension - how much an object fills a space ------------ PAGE 5 ------------ ƒí S represents the scaling factor and is always a natural number. ƒí N represents the number of smaller, self- similar figures ( for a scaling factor S) needed to create the larger figure. ------------ PAGE 6 ------------ a b c ------------ PAGE 7 ------------ c ------------ PAGE 8 ------------ c ------------ PAGE 9 ------------ ƒí 1600s - Gottfried Leibniz ƒí 1883 - Georg Cantor ƒí 1904 – Helge von Koch ƒí 1915 – Vaclav Sierpinski ƒí Early 1900s – Gaston Julia and Pierre Fatou ------------ PAGE 10 ------------ ƒí Polish- born, French mathematician ƒí Fractals: Form, Chance and Dimension ( 1975) ƒí The Fractal Geometry of Nature ( 1982) ------------ PAGE 11 ------------ ƒí In the n th step, 3 ( n- 1) triangles will be removed. ------------ PAGE 12 ------------ removed. ------------ PAGE 13 ------------ ƒí ä will be used to refer to the side length of the equilateral triangle. ƒí In the n th iteration, 3 * 4 ( n- 1) triangles are added. ------------ PAGE 14 ------------ added. ------------ PAGE 15 ------------ ƒí Area of an equilateral triangle = ƒí Formula for a geometric series with common ratio r: ------------ PAGE 16 ------------ r: ------------ PAGE 17 ------------ Using geometric series ------------ PAGE 18 ------------ ƒí Each iteration increased the length of a side to ( 4/ 3) its original length. ƒí Thus, for the n th iteration, the overall perimeter is increasing by ( 4/ 3) n . Divergent Sequence ------------ PAGE 19 ------------ ƒí The perimeter is then considered to be infinite! ƒí How does this apply to Mandelbrot’s “ How long is the Coast Line of Britain?” problem? ------------ PAGE 20 ------------ ƒí Trees and plants ƒí In the human body: „X Blood vessels „X Alveoli in the lungs ------------ PAGE 21 ------------ ƒí Used by Boeing to generate some of the first 3- D computer generated images ƒí Currently being used to make antennas smaller in cell phones ------------ PAGE 22 ------------ ƒí Fractal patterns exist in a healthy human heartbeat ƒí May give doctors a way to detect small tumors/ early stages of cancer
Back to top of page