1) The Chaos Game
One of the most interesting fractals arises from what Michael Barnsley has dubbed ``The Chaos Game'' [Barnsley]. The chaos game is played as follows. First pick three points at the vertices of a triangle (any triangle works---right, equilateral, isosceles, whatever). Color one of the vertices red, the second blue, and the third green.
Next, take a die and color two of the faces red, two blue, and two green. Now start with any point in the triangle. This point is the seed for the game. (Actually, the seed can be anywhere in the plane, even miles away from the triangle.) Then roll the die. Depending on what color comes up, move the seed half the distance to the appropriately colored vertex. That is, if red comes up, move the point half the distance to the red vertex. Now erase the original point and begin again, using the result of the previous roll as the seed for the next. That is, roll the die again and move the new point half the distance to the appropriately colored vertex, and then erase the starting point.
Figure 1: Playing the chaos game with three vertices
Now continue in this fashion for a small number of rolls of the die. Five rolls are sufficient if you are playing the game ``by hand'' or on a graphing calculator, and eight are sufficient if you are playing on a high-resolution computer screen. (If you start with a point outside the triangle, you will need more of these initial rolls.)
After a few initial rolls of the die, begin to record the track of these traveling points after each roll of the die. The goal of the chaos game is to roll the die many hundreds of times and predict what the resulting pattern of points will be. Most students who are unfamiliar with the game guess that the resulting image will be a random smear of points. Others predict that the points will eventually fill the entire triangle. Both guesses are quite natural, given the random nature of the chaos game. But both guesses are completely wrong. The resulting image is anything but a random smear; with probability one, the points form what mathematicians call the Sierpinski triangle and denote by S
I have made a geogebra file which generalize this up-to seven vertices
here are some cases
1) n = 3 and proportion is 0.5
2) n = 4 proportion = 0.7
3) n = 4 prop = 0.54 (vertices arranged as of square)
2) Julia Set
Julia set fractals are normally generated by initializing a complex number z = x + yi where i^2 = -1 and x and y are image pixel coordinates in the range of about -2 to 2. Then, z is repeatedly updated using: z = z^2 + c where c is another complex number that gives a specific Julia set. After numerous iterations, if the magnitude of z is less than 2 we say that pixel is in the Julia set and color it accordingly. Performing this calculation for a whole grid of pixels gives a fractal image.
Checkout the goe-gebra build
here's a screenshot of the program
3) Sunflower and golden ratio
Consider you have a flower, and you are trying to place seeds on the face of the flower in such a way that you can fit as many as possible. If you place a seed, and then rotate the flower face a certain amount, and place another seed, and then repeat this process, what would be the ideal amount to rotate the flower face?
For example, if you place a seed, rotate the flower 1/2 turn, place another seed and repeat, then you will get two parts of the flower with all the seeds. If you do 1/3 of a turn then you will get 3 lines of seeds, and if you do 1/10 of a turn you get 10 lines.
The best number to rotate the flower for optimal seed placement is the golden ratio, where the entire face gets covered more or less evenly without the seeds clustering in any one spot. As the video demonstrates, this is because the golden ratio is a highly irrational number, meaning it is not very well approximated by any whole number or even any fraction of whole numbers.
As it turns out, when you map the seeds of a sunflower on the face of the flower, it's as though the flower were playing this very game, placing seeds to near mathematical perfection. Make sure to watch the video for a visual explanation and more information about the awesome power of the golden ratio—and why it is the most irrational of numbers.
Geogebra build
here's a screenshot of the program
コメント