A famous fractal called the Sierpinski Triangle starts with one black triangle.  The triangle is then cut into four equal pieces, and the three outer triangles are coloured black and the inner triangle is white.  Each of the three black triangles is again divided into four equal triangles, and the three outer triangles are shaded black. a. Suppose the area of the first triangle is 100 cm2.  Determine the black shaded area in the second triangle.  Do the same for the third triangle. b. Write the general term describing the shaded area of the nth Sierpinski Triangle.

6 years ago

Answered By Navpreet S

Let us suppose given triangle is of length L. In the figure shown below, black triangles are the solid triangles and the unshaded triangles are the ones that have been removed.

Note: The sign ^ indicates raised to the power

Given area of traingle= √3/4 *L² =100 cm2= A1

Next you cut the the traingle in 4 equal halves and then take out the middle traingle.

The area of each triangle can be as A2= √3/4 *(L/2)²

Similarly A3=√3/4 *(L/4)², A4=√3/4 *(L/8)²  

 

In general An = √3/4 *(L/2^{(n-1)^2})= √3/4 *L²* (1/2^{(n-1)^2}) = A1/2^{(n-1)^2} = 100/ 2^{(n-1)^2} 

Attached Whiteboard:

  Play Drawing

---