Sierpinski triangle
The Sierpinski triangle, also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e., it is a mathematically generated pattern that can be reproducible at any magnification or reduction. [Source: Wikipedia].
Which is the area of the Sierpinski triangle?
The answer is 0.
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1 solution
The n+1-esim triangle is made by 3 triangles obtained by scaling the n-esim one by 1/2: the area of the n+1 esim triangle is 3*(1/2)^2= 3/4 the area of the previous one. Iterating the process, the area of the n esim triangle is (0.75)^n times the area of the starting one. As n tends to infinity, this area tends to zero.