Deterministic geometrical construction of the Sierpinski
triangle
The most known way to generate the Sierpinski triangle
is a geometrical construction from any triangle where parts are removed
following a deterministic algorithm.
You can begin with any triangle, here pictures show an
equilateral triangle because it is more pleasant to eyes.
Using the midpoints of each side of the triangle,
create a new triangle that you remove from the original. This leaves three
new triangles similar to the original but with dimensions exactly onehalf
the dimensions of the original triangle.
For each of the three remaining triangle, iterate
the former process : you get nine new triangles, onefourth the dimension
of the original triangle.
Continue the iterations as indicated on the following
figure.
After some iterations, you recognize the Sierpinski
triangle arising from thew successing removals.
Iteration number

Number of remaining triangles

Dimension compared to the original

1

3

1/2

2

9

1/4

3

27

1/8

4

81

1/16

n

3^n

1/(2^n)
