- Happy Pi Day (or better yet, \(\pi\) Day!)
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!
- We'll delay our trip up to visit with Mathemalchemy until next
week, to see where we are, and to talk about your projects.
- If you're struggling to create your fractal spiral image, then you
might try the computer lab over in 306. I managed to access the
site from there.
For the moment this is my favorite image:
- Your quiz today concerns the wallpaper groups, but also includes a
novel element (half the quiz score): your Spiral
Fractal to send to me, to create a gallery of fractals.
Please get those to me by Sunday night, so we can look them over in class on Tuesday of next week.
- We started by checking out key for the
wallpaper handout, with the crystallography (modern) names for
the ones that Polya gave to the different groups.
For your quiz you're going to be asked to match a pattern to one of those....
- We created an area fractal (Sierpinski), and a stick fractal --
Cantor's middle third.
- We observed that the length or area, and number of objects usually
grows exponentially -- as a power of a particular base, which
we can usually read off of the generator.
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).
-
In the bulb there is a flower; in the seed, an apple tree; in cocoons, a hidden promise: butterflies will soon be free! (In the bulb there is a flower)
- Let's try a fractal that involves alternating types of shapes:
we'll use triangles and hexagons....
- Let's invent a stick fractal of our own!
- Now let's introduce a little bit of randomness: Sierpinski's
triangle again
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.
From randomness comes order; from simple rules comes complicated objects! But it helps to have a computer do the hard work....
- 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.
- Time now for a quiz....
- Triangle and rectangle paper
- Shodor:
- Snowflake on-line
- Gasket on-line
- The Fractal Microscope (click on the image)
- L-Systems
in PostScript, from which I generated this image:
- Vi Hart
- L-System Fractals
- Nice website with examples of Cantor, Koch, and other fractals
- Here are some more lovely examples
- l-system generator: world's simplest math tool
- I first encountered the L-Systems fractals at britton.disted.camosun.bc.ca/fractals_arcytech/lsystems.html, but that's a dead link. However, I copied the files, and have a local copy here. However, web browsers these days don't seem to like the applet tag in javascript, so this doesn't work anymore in a standard web browser...:(