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    <p>If you would like a high-quality approximation of the Sierpinski triangle, you can use an approach called the <a href="http://en.wikipedia.org/wiki/Chaos_game" rel="noreferrer">chaos game</a>. The idea is as follows - pick three points that you wish to define as the vertices of the Sierpinski triangle and choose one of those points randomly. Then, repeat the following procedure as long as you'd like:</p> <ol> <li>Choose a random vertex of the trangle.</li> <li>Move from the current point to the halfway point between its current location and that vertex of the triangle.</li> <li>Plot a pixel at that point.</li> </ol> <p>As you can see <a href="http://en.wikipedia.org/wiki/File:Sierpinski_chaos_animated.gif" rel="noreferrer">at this animation</a>, this procedure will eventually trace out a high-resolution version of the triangle. If you'd like, you can multithread it to have multiple processes plotting pixels at once, which will end up drawing the triangle more quickly.</p> <p>Alternatively, if you just want to translate your recursive code into iterative code, one option would be to use a worklist approach. Maintain a stack (or queue) that contains a collection of records, each of which holds the vertices of the triangle and the number n. Initially put into this worklist the vertices of the main triangle and the fractal depth. Then:</p> <ul> <li>While the worklist is not empty: <ul> <li>Remove the first element from the worklist.</li> <li>If its n value is not zero: <ul> <li>Draw the triangle connecting the midpoints of the triangle.</li> <li>For each subtriangle, add that triangle with n-value n - 1 to the worklist.</li> </ul></li> </ul></li> </ul> <p>This essentially simulates the recursion iteratively.</p> <p>Hope this helps!</p>
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    1. POGenerating the Sierpinski triangle iteratively in Mathematica?
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    1. COAt first I simply wanted to translate the code but the chaos game approach seems really interesting!! I will try it out when I get home! Thank you very much, this was very helpful!
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    2. COThanks again, I wrote it with the Chaos Game approach! I have added it to my post in case you are interested in seeing how approached it.
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