FLATLAND the Movie, Area of a Fractal Plant

FLATLAND the Movie, Area of a Fractal Plant

Arlene Square wants to add a special plant to her garden, but she is not sure if she has enough room for it to fit.

The plant starts as a square of side length 1 meter. Each year, the plant grows larger in a very unique way. The plant sprouts an additional square for every edge that it had for the previous year. The length of the sides of the new square is always 1/3 of the length of the edge that it sprouted from.

Arlene needs to know the length of the sides for each year and the number of smaller squares that have grown for each year. With this information, she can determine the total area that the plant is occupying.

To do this, Arlene decides to draw a picture for what the plant looks like after a few years. She starts by drawing the four squares that sprout after the first year. On the next page is the picture that she drew.

1. In the first year, the square plant has four edges. In the second year, the plant has 20 edges. Find a formula for en, the number of edges of the plant in the n-th year.

en = ____________________

2. In the picture, draw what the plant looks like in the second year. Make sure you add one square for each of the 20 edges. Note that, in this year, the plant starts to intersect itself and has holes contained on the inside. The lines bordering these holes are still considered edges of the plant. Draw what the plant looks like in the third year. Make sure you add one square for each edge.

3. In the first year, the length of each individual edge of the square plant is 1. In the second year, the length of each individual edge of the square plant is 1/3. Find a formula for ln , then length of each individual edge in the n-th year

ln = ____________________

4. In the first year, the area is 1 square meter. In the second year, the plant grows by 4*(1/3)2. Find a formula for gn, the amount that the plant grows in the n-th year. (note that gn is defined only for n >= 2.

gn = ________________________

5. In the first year, the plant occupies a total of one square meter. In the second year, the plant occupies a total of 1 + 4(1/9) square meters. Find a formula for the total area An that the plant occupies in the n-th year.
To simplify your answer, note that you can write your expression in the following form:
1 + a(1 + x + x2 + ...+ xn) = (1/xn+1)/(1-x)

which is valid provided x not equal 1, to your previous expression to get an answer that is much more simplified.

6. Fill in the following table. Use your calculator to give a decimal approximation (correct to 3 decimal places) of the values for An.

From the values in the table, take a guess as to the area as n approaches infinity: _________

Is there any correlation between your answer you just gave and the picture of the square plant that you drew?