Sequences and Fractals
- 1. We started with this problem… A small forest of 4000 trees is under a new forestry plan. Each year 20% of the trees will be harvested and 1000 new trees are planted. Will the forest size ever stabilize? If so, how many years and with how many trees? Yes, in approximately 36 years with 5000 trees. Look at one decimal place to see if it stabilizes. The starting value Percentage of the trees left after the harvest, which is 80% Using your calculator, press the Y= button and enter this…
- 2. Fractals The Koch Snowflake Each side of the triangle is divided three times and the middle section forms the base of a new equilateral triangle outside the original one. This process is repeated. The Sierpinski Triangle This fractal also starts with an equilateral triangle. To draw the fractal, you find the midpoint of each side of the original triangle, and then draw three segments connecting the midpoints. There are now four triangles inside the original triangle. The middle triangle is not shaded, and the process is repeated with the other three shaded triangles.
- 3. Draw a Fractal Use pencil and paper (metric graph paper if possible) to draw the fractal described below. • Draw a square with 8-cm sides in the middle of the paper. • Position the paper horizontally (in landscape format). Extend the diagram to the left and right by drawing a square on each side of the original square -- touching the original square. The sides of the new squares should be half as long as the side lengths of the original square. • Repeat the previous step three times. Your fractal should now have five generations, including the original square.
- 4. The Rectangle ... Draw a rectangle that measures 12 cm by 8 cm, and shade the inside of the rectangle. Construct the midpoints of each side of the rectangle, and then draw a quadrilateral by joining these points. Shade the quadrilateral white. Now continue the process by finding the midpoints of the quadrilateral, drawing the rectangle, and shading it the same colour as the first rectangle. Draw six generations. (The initial rectangle is the first generation.)
- 5. The Square ... Create a fractal that begins with a large square 20 cm on each side. Each pattern requires that the square be divided into four equally sized squares, that the bottom-left square be shaded, and the process continues in the upper-right square. Repeat the process four times.
- 6. Create a fractal where a square is inscribed in a circle. The diameter of the original circle is 16 cm. Shade the area between the circle and the square. Inscribe a circle inside the resulting square, and then inscribe a new square inside that circle, and shade the area between the new circle and square. Repeat this process three times. The Circle-Square ...