A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.
Previously pre-posted (please peruse):
• Fractal Fourmulas
A square can be divided into four right triangles. A right triangle can be divided into a square and two more right triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.
All roads lead to Rome, so the old saying goes. But you may get your feet wet, so try the Sierpiński triangle instead. This fractal is named after the Polish mathematician Wacław Sierpiński (1882-1969) and quite a few roads lead there too. You can create it by deleting, jumping or bending, inter alia. Here is method #1:
Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.
Here is method #2:
Pick a corner at random, jump half-way towards it, mark the spot, repeat.
And here is method #3:
Bend a straight line into a hump consisting of three straight lines, then repeat with each new line.
Each method can be varied to create new fractals. Method #3, which is also known as the arrowhead fractal, depends on the orientation of the additional humps, as you can see from the animated gif above. There are eight, or 2 x 2 x 2, ways of varying these three orientations, so eight fractals can be produced if the same combination of orientations is kept at each stage, like this (some are mirror images — if an animated gif doesn’t work, please open it in a new window):
If different combinations are allowed at different stages, the number of different fractals gets much bigger:
• Continuing viewing Tri Again.