Day 18, 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!

I'll be meeting with them after class, so there will be no office hours today until 1:30 or so (but I will be available until about 4:50, when I have my next class).

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.

• Informal definition: A world within a world!

  there's a perfect copy of the fractal contained within itself -- and perhaps infinitely many!

• We saw examples of fractals:
  • A0 paper, which we discovered had side relative ratio of $\sqrt{2} \approx 1.4142$. It, too, can spiral! Not all spirals lead to gold, but those using squares do.....

    By the way, $\sqrt{2}$ is irrational, like \(\pi\), and is the number said to have killed a man....

  • "Russian Dolls"
  • Fractals in nature (e.g. Romanesco broccoli, tea oil in lakes,...)
  • Fractals in graphs (e.g. Fibonacci's immortal, incestuous rabbits)
  • Fractals in the duality of Platonic solids
  • World's Most Famous Fractal discovered/created by the late Benoit Mandelbrot
  • Spiral homework fractals, e.g.

    You'll be making one of very own Spiral Fractal as half of this week's quiz grade.

1. It's a process or graph that possesses "self-similarity".

2. It often possesses a simple and recursive (or iterative) definition -- that is, we do something, and then we "do it again!"

1. Start with a straight stick.
2. Replace the middle third of the stick with a little "tent" -- two third-length sticks.
3. Do it again and again!

This fractal becomes infinitely long, but in a confined space! Very strange.... but this type of strange behavior is typical of fractals.

There is a lovely summary of the system in this article.

• How many sticks are there at each iteration of the process? (Each time we "do it again", we have another iteration).
• How long is the fractal becoming?
• How long will it be in the end?

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! Then all hell broke loose.

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

• Vi Hart draws fractals