Fractals and Area summation in Mathematica

[jasondraime]

Homework Statement

Begin with an equilateral triangle T of side length 1
At the middle of each side of T place an equilateral triangle whose side lengths are 1/3
Repeat this process ad infinitum

By summing an appropriate series, show that the area A of the fractal obtained above is finite, find A. Plot the area vs. iteration number, up to iteration 200.
By summing an appropriate series show that the boundary of the fractal has infinite length. What well studied series do we get? Show both theoretically and empirically how we know the series does not converge.

Draw the fractal

Homework Equations

Harmonic Series: 1/n
Telescoping Series: b(n)-b(n+1) where the values in the parentheses are sub-notations
Geometric Series: cr^n

The Attempt at a Solution

So far I am able to create a triangle with side lengths of 1, however I'm already stuck at placing the new triangles within it.

[In]:r = 0;
s = 0;
t = 1;
u = 0;
v = 0.5;
w = 0.866;

[In]:T = Polygon[{{r, s}, {t, u}, {v, w}}]
[Out]:Polygon[{{0, 0}, {1, 0}, {0.5, 0.866}}]

[In]:Graphics[{Directive[Purple, Opacity[.5], EdgeForm[Thickness[.006]]],
tri}, ImageSize -> 200]
[Out]: A purple equilateral triangle

 

[kurt.physics]

with side lengths of 1I'm not sure how to proceed from here, if anyone can provide any help it would be greatly appreciated.