Read

• Marc Frantz's Lesson 6: Fractal Geometry in your course packet.
• Chapter 4 of Chaos and Fractals : new frontiers of science by Peitgen et al. in your packet.  You can skip the part about Hausdorff dimension if you want.

• Comments:  the ``fractal'' dimension is what we called the self-similarity dimension.
• I didn't tell you about fixed points yet, but they're not hard.  A fixed point of a linear transformation T (x,y) is the solution to the equation T(x,y) = (x,y).  For example, the Sierpinski triangle is generated by three transformations:
• T₁ (x, y) = (x/2, y/2 + 1/2)
• T₂ (x, y) = (x/2, y/2)
• T₃ (x, y) = (x/2 + 1/2, y/2)
The fixed point of T₁ is found by solving
x = x/2
y = y/2 + 1/2
which gives x = 0 and y = 1, so the point (0, 1) is fixed by T₁.  You should be able to show that the fixed points of T₂ and T₃ are (0, 0) and (1, 0) respectively.

Using Devaney's notation, we would say that Sierpinski's Triangle is generated by the IFS with contraction factor beta = 1/2 and fixed points (0, 0), (1, 0), and (0, 1).  That is, there are three transformations; each transformation ``smushes'' the fractal by 1/2 size around one of the fixed points.

• Here's a copy of Pascal's Triangle.
• Here's a blank triangle for you to color..  Isn't math fun??