If you take a triangle and draw three its mid-lines, you will get
    splitting of your triangle in four smaller similar triangles.


    By doing this way, you get splitting of an equilateral triangle  into
    the union of four adjacent smaller equilateral triangles.


    So, having 4 equilateral triangles, you can unite them to get a 
    larger equilateral triangle, twice as large as the original triangles.


    Having 16 equilateral triangles, you construct first 4 larger 
    equilateral triangles, using quadruples of triangles;

    then you unite these 4 larger equilateral triangles to construct
    even larger final equilateral triangle.