Fractals

1. Fractals

2. The patterns

3. Fractals self-similar they are the same from near as from far

4. Benoît B Mandelbrot the man who made geometry an art Mathematician whose fractal geometry helps us find patterns in the irregularities of the natural world Born 20 November 1924 Died 14 October 2010

6. Mandelbrot always had a highly developed visual senseAs a boy, he saw chess games in geometrical rather than logical terms.He shared his father's passion for maps.

7. Fractal • Geometric figure that can be separated into parts, each is similar to the whole. • Fractals • Used in computer graphics • Example - to quickly fill in the leaves on trees or grass; to repeat patterns to make mountains

8. Making a Mountain Using Fractals

9. Fractals have structures that are self-similar over many scales, the same pattern being repeated over and over.

10. Jonathan Jones: The sphere of math has borne few as provocative as the man whose 'fractals' demonstrated the universe's playful irregularity

11. Ice Crystals

12. SnowSelf Similar

13. Snow by René Descartes

20. In you

23. Indian Architect

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31. Fractals Complex Apparently chaotic Yet geometrically ordered shapes That delight the eye and fascinate the mind. They are icons of modern understanding of the universe's complexity.

32. Koch Snowflake One of the earliest Fractal curves to have been described

33. Koch Cube Animation

35. WacławFranciszekSierpiński Polish Mathematician Died 1962 Start with an equilateral triangle in a plane (any closed, bounded region in the plane will actually work). Base is parallel to the horizontal axis Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner Note emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. Holes are an important feature of Sierpinski's triangle Repeat step 2 with each of the smaller triangles

36. Notice that the outline of the figure is an equilateral triangle. Now look inside at all the equilateral triangles. Remember that there are infinitely many smaller and smaller triangles inside. How many different sized triangles can you find? All of these are similar to each other and to the original triangle - self similarity Note the original triangle inside How many copies do you see where the ratio of the outer triangle's sides to the inner ones is 2:1? 4:1? 8:1? Find the pattern

37. WacławFranciszekSierpiński Polish Mathematician Died 1962

41. What happens when you increase the number of rows in a Pascal Triangle? From 3mod to 5mod

42. Fractal Tetrahedron

43. Sierpinski Meets Pascal

44. If each side of the triangle in Figure 1 is 1 inch long, this means the triangle has a perimeter of 3 inches. Suppose you continued the pattern in the diagram until you reached Figure 5. What is the sum of the perimeters of all the white triangles in Figure 5?

45. As the figure numbers increase, the side length of each white triangle is halved, and the number of white triangles is tripled. This means the sum of the perimeters in any particular figure is: 3 × 1/2 or 3/2 the sum in the previous figure The sum of the perimeters in Figure 5 would be 243/16 or 15 3/16 inches.

46. Find a general rule for the sum of the perimeters of the white triangles in each figure. If n is the figure number, and the side length in Figure 1 is 1 inch, then the sum of the perimeters in inches is: =

47. What are the chances that a family of X children will have Y girls?

48. What happens for 3 children? BBB GBB GGB GGG BGB GBG BBG BGG Put this data in a table, where the directions for reading the table are: row number is number of children and column number is number of girls. Leave the impossible entries blank. Each number in this table is the number of ways that many girls can happen. Second 10 in row 5 = 10 different ways to have three girls in 5 children

49. 10 different ways to have three girls in 5 children • GGGBB • GGBGB • GGBBG • GBGBG • GBBGG • BGGGB • BGGBG • BGGGB • BBGGG • BGBGG

50. In combinatorics and counting, we can use these numbers whenever we need to know the number of ways we can choose Y things from a group of X things. • For example, if we need to choose 3 people to work on a problem together and we have 5 people to choose from, there are "5 choose 3" or 10 different ways to do this -- Using the fifth row, third entry in the triangle. • This is the same idea as for the "number of girls in a family" problem.