Start with an equilateral triangle, a pencil, a ruler and a die. Label one vertex of the triangle “1 and 2”, the next “3 and 4” and the final one “5 and 6”. Mark a dot on one of the vertices of the equilateral triangle. Now roll the die. This tells you which corner of the triangle to move towards. Mark your next dot half way between your last dot and the corner of the triangle your die identified. Then roll the die again and mark your next dot half way between your last dot and the corner of the triangle your die identified. Repeat this until you discover the beautiful result!
You may like to write a spreadsheet to do this for you rather than constructing it by hand. I wrote a spreadsheet that did 10 000 trials and look at the result:
You get the famous fractal, Sierpinski’s Triangle!
If you extend the idea into three-dimensions, adding a fourth vertex directly above the centre of the equilateral triangle, the points form a Sierpinski Tetrahedron!