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Section 6.1 Images Viewing a Gallery of Fractals.

Section 6.1 Images Viewing a Gallery of Fractals. Look for patterns. Question of the Day. How can you convince your parents that you’re eating enough broccoli?. Real or fake?. The Sierpinski Triangle. How many Quackers do you see ?. The Fern. Julia Sets.

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Section 6.1 Images Viewing a Gallery of Fractals.

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  1. Section 6.1ImagesViewing a Gallery of Fractals. Look for patterns.

  2. Question of the Day How can you convince your parents that you’re eating enough broccoli?

  3. Real or fake?

  4. The Sierpinski Triangle

  5. How many Quackers do you see?

  6. The Fern

  7. Julia Sets

  8. Section 6.2The Infinitely Detailed Beauty of FractalsHow to create works of infinite intricacy through repeated processes. It’s not where we begin, it’s how we get there.

  9. Question of the Day What do you need to know to predict the world’s population in the future?

  10. Self-Similarity The characteristic of looking the same as or similar to itself under increasing magnification.

  11. A process of repeated replacement. Koch’s Kinky Curve

  12. A process of repeated replacement. Sierpinski Triangle

  13. A process of repeated replacement. Menger Sponge

  14. A process of repeated replacement. Barnsley’s Fern

  15. More on the Sierpinski Triangle! The final bow…

  16. The Chaos Game Whatever number you generate, move halfway from where you are toward that numbered vertex and make a dot. Then roll again to determine the next number… and repeat the process.

  17. Section 6.3Between DimensionsCan the Dimensions of Fractals Fall through the Cracks? Start with the simple and familiar. Look for patterns. Apply patterns to new settings.

  18. Question of the Day How do you turn a triangle into a snowflake?

  19. What is a dimension? 1-dimensional: A line segment 2-dimensional: A filled in square 3-dimensional: A solid cube 4-dimensional: ?

  20. In search of a pattern…

  21. Section 6.4The Mysterious Art of Imaginary FractalsCreating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane Don’t be afraid to look and think before you try to understand.

  22. Question of the Day What’s the story behind this picture?

  23. Julia Sets

  24. Mandelbrot Sets

  25. A process in the plane.

  26. Review of Complex Numbers.

  27. Review of Complex Numbers.

  28. Visualizing Complex Numbers

  29. What is a Mandelbrot Set? An object that captures information about the collection of all (a + bi) – Julia Sets.

  30. Section 6.5The Dynamics of ChangeCan Change Be Modeled by Repeated Applications of Simple Professes? The best predictors of where you will be are where you are now and which way you’re going.

  31. Question of the Day How close is your calculator’s answer to the correct answer?

  32. Repeat, Repeat, Repeat • Start somewhere. • Apply a process and get a result. • Apply the same process to that result to get a new result. • Apply the process again to the new result to get a newer result. • Repeat patiently and persistently, forever.

  33. Conway’s Game of Life • A living square will remain alive in the next generation if exactly two or three of the adjoining eight squares are alive in this generation; otherwise, it will die. • A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.

  34. Play the Game of Life!

  35. Patterns in types of Populations • Explosion • Extinction • Stable • Periodic • Migratory

  36. Population Density Population Density = The change in population density is the change from year n to n+1:

  37. The Rate of Change of Population

  38. Section 6.6Predetermined ChaosHow Repeated Simple Processes Result in Utter Chaos Keep alert for significance even in ridiculous ideas.

  39. Question of the Day What is the dimension of a cloud?

  40. Experiment • Type in a random number. • Multiply it by 180. • Hit the SIN key and write down your answer in the second column next two 2. Keep all decimal places. • Repeat steps 2 and 3 for each new number you generate. Continue to record your results until you have 25 numbers in the second column.

  41. A variation on the experiment • Enter the exact same first decimal number as before and repeat the process five times. • Clear your calculator of the previous result. • For the sixth number use the fifth number rounded to six decimals places. • Now, repeat the process as before until you have 25 numbers in the second column.

  42. Predicting Future Populations The Verhulst Model • = population in year n • = population for the next year.

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