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.

### 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 *n*th iteration, then:

M x

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