Day 18r, MAT115

Your new assignment is to read the article Pi Day: How One Irrational Number Made Us Modern, which appeared in the New York Times today.

It features so much:

• irrational numbers (of course, since $\pi$ is irrational)
• regular polygons
• infinity
• Archimedes! (My favorite mathematician)

My colleague Marla Lemmon was spotted wearing fractal-themed earrings today in honor of pi day!

For the moment this is my favorite image:

Please get those to me by Sunday night, so we can look them over in class on Tuesday of next week.

For your quiz you're going to be asked to match a pattern to one of those....

And that even though the material may appear to be completely removed, there may be some "dust" left over (Cantor's middle third fractal, for example, has endpoints of intervals that are never removed).

The Chaos game - generating fractals using random movement!

• Once again, simple rules lead to complex objects. This time, however, we don't proceed methodically, but rather haphazardly - and yet we still produce the complex images.

  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.

  • with 10 points
  • with 100 points
  • with 500 points
  • with 1000 points
  • with an infinite number of points!
  From randomness comes order; from simple rules comes complicated objects! But it helps to have a computer do the hard work....

1. Snowflake on-line
2. Gasket on-line
3. The Fractal Microscope (click on the image)

• Vi Hart draws fractals