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Fractals. The patterns. Fractals. self-similar t hey are the same from near as from far. 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 - PowerPoint PPT Presentation
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Fractals
The patterns
Fractalsself-similarthey are the same from near as from far
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 1924Died 14 October 2010
Mandelbrot always had a highly developed visual sense
As a boy, he saw chess games in geometrical rather than logical terms.
He shared his father's passion for maps.
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
Making a MountainUsing Fractals
Fractals have structures that are self-similar over many scales, the same pattern being repeated over and over.
Jonathan Jones: The sphere of math has borne few as provocative as the man whose 'fractals' demonstrated the universe's playful irregularity
Ice Crystals
SnowSelf
Similar
Snow by René Descartes
In you
Indian Architect
Indian Architect
EuropeanArchitect
EuropeanArchitect
EuropeanArchitect
EuropeanArchitect
AfricanArchitect
Fractals
ComplexApparently chaotic Yet geometrically ordered shapes That delight the eye and fascinate the mind.
They are icons of modern understanding of the universe's complexity.
Koch Snowflake
One of the earliest Fractal curves to have been described
Koch Cube Animation
Wacław Franciszek Sierpiński Polish MathematicianDied 1962
1. Start with an equilateral triangle in a plane (any closed, bounded region in the plane will actually work).
2. Base is parallel to the horizontal axis3. 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
4. Note emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original.
5. Holes are an important feature of Sierpinski's triangle
6. Repeat step 2 with each of the smaller triangles
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
Wacław Franciszek Sierpiński Polish MathematicianDied 1962
What happens when you increase the number of rows in a Pascal Triangle?
From 3mod to 5mod
Fractal Tetrahedron
Sierpinski Meets Pascal
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?
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.
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:
132
n
n
1
332
n
=
What are the chances that a family of X children will have Y girls?
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
1. GGGBB 2. GGBGB 3. GGBBG 4. GBGBG 5. GBBGG 6. BGGGB 7. BGGBG 8. BGGGB9. BBGGG10.BGBGG
10 different ways to have three girls in 5 children
•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.
