introduction introduction: mandelbrot set. fractal geometry ~*beautiful mathematics*~ fractal...
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Introduction
Introduction: Mandelbrot Set
Fractal Geometry
~*Beautiful Mathematics*~
FRACTAL GEOMETRY
Ms. Luxton
What is a fractal?• A fractal is any pattern that reveals greater
complexity as it is enlarged• A “self-similar” pattern meaning they are the
same from near as it is from far.
Fractals you may see every day:
A little bit of history:
• The mathematics behind fractals began to take shape in the 17th century
• The first geometric definitions of a fractal were created about 100 years ago – they were called Monsters!!!
The Sierpinski Triangle Fractal
• Level 1. Using a sheet of blank typing paper, a ruler, a pencil, and a colored pencil, draw a large equilateral triangle on your paper.
Base
Place this information into the following table:
Level Perimeter Area
1 3 1
2
3
4
• Level 2. Place a light mark on the midpoint of each side of your triangle (use your ruler). Then draw 3 lines that connect the marks. You should see 4 new smaller congruent triangles. Shade in the middle triangle. We will call this level 2.
Motif
Level Perimeter Area
1 3 1
2 4.5 or 9/2 .75 or ¾
3
4
Level 3. You will notice that there are 3 new white triangles, one in each corner. Apply the motif process (removing the middle triangle) to each of these 3 triangles. We will call this level 3
Level Perimeter Area
1 3 1
2 4.5 or 9/2 .75 or ¾
3 6.75 or 27/4 .5625 or 9/16
4
Repeat at least one more level.When we repeat a process over and over, we can use the term that each repetition is called an "iteration," based upon an iteration rule.
“Continuous process of removals”
What will the perimeter be after level 5?What will the area be after level 5?
Level Perimeter Area
1 3 1
2 4.5 or 9/2 .75 or ¾
3 6.75 or 27/4 .5625 or 9/16
4 10.125 or 81/8 0.421785 or 27/64
Level 5Level Perimeter Area
1 3 1
2 4.5 or 9/2 .75 or ¾
3 6.75 or 27/4 .5625 or 9/16
4 10.125 or 81/8 0.421785 or 27/64
5 14.625 or 234/16 or 81/256
• Can you find a formula that will help you to find out what the perimeter will be at any level?
𝑹𝒆𝒄𝒖𝒓𝒔𝒊𝒗𝒆 𝑭𝒐𝒓𝒎𝒖𝒍𝒂 :𝑷𝒏=𝟑𝟐𝑷𝒏−𝟏
𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑅𝑢𝑙𝑒:𝑃 (𝑛)=3( 32)𝑛−1
Can you find a formula that will help you find out what the area will be at a any level?
Recursive Formula: An = (3/4) An-1
Funtion Rule: A(n) = (3/4)n-1
What will the perimeter be as the number of levels n get very large?What will the area be as the number of levels n get very large?
Level Perimeter Area1 3 1
2 4.5 or 9/2 .75 or ¾
3 6.75 or 27/4 .5625 or 9/16
4 10.125 or 81/8 0.421785 or 27/64
5 14.625 or 234/16 or 81/256
"Self-similarity" is one of the key characteristics of a fractal. Can you see how if you look at one portion of the Sierpinski Triangle, it resembles the original Sierpinski Triangle?
Why is this important?
The Coastline Paradox
Benoit Mandelbrot
• The Fractal Geometry of Nature (1982)• Clouds are not spheres, mountains are not
cones, coastlines are not circles, and bark is not smooth, nor does lightening travel in a straight line. -- Benoit Mandelbrot
“Chaos Theory”
Benoit Mandelbrot
• Mandelbrot was able to continue the earlier mathematicians’ work and create new and beautiful fractals using a recently invented powerful tool: the computer.
• The Mandelbrot Set
Fractals are used in many ways:
An environmentalist may need to estimate how much of the coastline has been affected by an oil spill.
A biomedical engineer may want to calculate how much surface area is covered by the bronchial tubes in the human lungs.
A computer graphic artist may want to create a computer generated landscape of the earth.
An economist may use fractals in the study of the behavior of the stock market.
A geologist may use fractals to help model earthquake data.
Fashion designers and artists use fractals to create new designs.
Ecologists can predict the way trees and plants can grow and can create ecosystems.
Writing Prompts:
• Why are fractals important?
• Describe a fractal you see on a daily basis. How do you know this is a fractal?