Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle. Class 8

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle. Class 8

Question 1.

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle. Class 8

Solution:

Note : Do as directed as drawn in the textbook.

Question 2.

Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle. Class 8

Solution:

Number of holes : 0, 1, 4, 13,..., [latex]\frac{3^n-1}{2}[/latex]

Number of remaining triangles : 1, 3, 9, 27, 3ⁿ.

Question 3.

Find the area of the region remaining at the nth step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit. Class 8

Solution:

For Sierpinski carpet:

Area of remaining region at step 1

= 1 × [latex]\frac{8}{9}[/latex] = [latex]\frac{8}{9}[/latex]

Area of remaining region at step 2

= [latex]\frac{8^2}{81}[/latex] = [latex]\left(\frac{8}{9}\right)^2[/latex]

Area of remaining region at step 3 = [latex]\left(\frac{8}{9}\right)^3[/latex]

Area of remaining region at step n = [latex]\left(\frac{8}{9}\right)^n[/latex]

For For Sierpinski triangles :

Area of remaining region at step 1 = [latex]\frac{3}{4}[/latex]

Area of remaining region at step 2 = [latex]\frac{3^2}{16}[/latex] = [latex]\left(\frac{3}{4}\right)^2[/latex]

Area of remaining region at step 3 = [latex]\left(\frac{3}{4}\right)^3[/latex], ...........

Area of remaining region at step n = [latex]\left(\frac{3}{4}\right)^n[/latex].