Fractals

In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex

Look at the triangle you made in Step One. What fraction of the triangle did you NOT shade? What fraction of the triangle in Step Two is NOT shaded? What fraction did you NOT shade in the Step Three triangle? Do you see a pattern here? Use the pattern to predict the fraction of the triangle you would NOT shade in the Step Four Triangle. Confirm your prediction and explain. Find another interesting pattern in the fractal called the Sierpinski Triangle. Write a paragraph descibing this pattern.

Step One Step Two Step Three Step Four Step Five

Cynthia Lanius' Lesson Koch Snowflake Fractal, Using JAVA

KOCH Questions What is the perimeter of the snowflake at stage 1? At stage2? Work out the perimeter of the snowflake at each stage. What will the perimeter be after n stages? What about the area of the snowflake - what is the area at each stage? Hint: work with fractions, not decimals

Now think of doing this many, many times. The perimeter gets huge! But does the area? We say the area is bounded by a circle surrounding the original triangle. If you continued the process oh, let's say, infinitely many times, the figure would have an infinite perimeter, but its area would still be bounded by that circle.

An infinite perimeter encloses a finite area... Now that's amazing!!

Chaos What exactly is chaos? The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data.

The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

Langton’s Ant Langton's ant travels around in a grid of black or white squares. If she exits a square, its colour inverts. If she enters a black square, she turns right, and if she enters a white square, she turns left.

Langton’s Ant If she starts out moving right on a white grid, for example, here is how things go:go

Have a go… Use the Langton’s ant applet to have a go, and see what happens? Is the ant’s motion chaotic?

Langton’s Ant The result is a quite complicated and apparently chaotic motion... ……but after about ten thousand moves the ant locks into a cycle of 104 moves which causes it to build a broad diagonal "highway". What's more, the ant seems to always build the highway (though nobody has been able to prove this yet) even if "obstacles" of black are scattered in its path. MoreMore

Langton's ant - Wikipedia, the free encyclopediaLangton's ant - Wikipedia, the free encyclopedia

Conway’s Game of Life The Rules: Cells are either empty or live. A empty cell with exactly three live neighbours becomes a live cell (birth). A live cell with two or three live neighbors stays alive (survival). In all other cases, a live cell dies (overcrowding or loneliness) or remains empty

The rules The Rules For a space that is 'populated': –Each cell with one or no neighbors dies, as if by loneliness. –Each cell with four or more neighbors dies, as if by overpopulation. –Each cell with two or three neighbors survives. For a space that is 'empty' or 'unpopulated' –Each cell with three neighbors becomes populated.

Conway’s Game of Life Life is one of the simplest examples of what is sometimes called "emergent complexity" or "self- organizing systems.“ -the study of how elaborate patterns and behaviors can emerge from very simple rules. It helps us understand, for example, how the petals on a rose or the stripes on a zebra can arise from a tissue of living cells growing together. It can even help us understand the diversity of life that has evolved on earth.