The Sierpinski Triangle

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Source: Wolfram Mathworld

Constructing a Sierpinski Triangle

STEP 1:

Start with an equilateral triangle.

STEP 2:

Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.

STEP 3:

Now cut out the triangle in the centre.

STEP 4:

Repeat steps 2 and 3 with each of the remaining smaller triangles.

Properties

 

If we let  be the number of black triangles after iteration n,  be the length of a side of a triangle, and  be the fractional area which is black after the nth iteration, then:

M x