Answer to Question #62997 in Physics for dagmawi

Question

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller
equilateral triangles. Leave the middlesmall triangle blank, but for each of the
other 3 small triangles, draw lines connecting the midpoints of the sides to create
4tinytriangles.Again leave each middle tiny triangle blank and draw the lines to
divide the others into 4 parts. Find the infinite series for the total area left blank
if this process is continued indefinitely. (Suggestion: Let the area of the original
triangle be 1; then the area of the first blank triangle is 1/4.) Sum the series to find
the totalarea left blank. Is the answer what you expect?Hint:What is the“area”
of a straight line? (Comment: You have constructed afractal called theSierpi´nski
gasket. A fractal has the property that a magnified view of a small part of it looks
very muchlike the original.)