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 one-half the dimensions of the original triangle.
For each of the three remaining triangle, iterate the former process : you get nine new triangles, one-fourth 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)