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August 26

Koch Snowflake

Hi! If I were to speculate what proportion of the lines in a Koch snowflake are parallel, would I be correct to say 1/6, because each new iteration of the fractal has sides that are 60 degrees more than the previous? And would the answer be the same for a Sierpinski carpet?

~~~~ Duomillia (talk) 00:32, 26 August 2023 (UTC)[reply]

For the Koch snowflake, indeed it would be the case that at each iteration of the construction, the length of all sides parallel to any given side of the snowflake (including the given side itself) would make up 1/6-th of the total length (with the exception of the very first iteration, which is just a triangle.) It should be noted that this isn't a property of the actual Koch snowflake itself, which is the limit of this construction, since the Koch snowflake doesn't include any actual lines. Assuming that, for the Sierpinski carpet, you're referring to the boundary, then the same property holds at all iterations of the construction as well, though this time the actual carpet (i.e. the limit of all the iterations) does actually have lines. GalacticShoe (talk) 01:08, 26 August 2023 (UTC)[reply]

Do you mean Sierpiński carpet (1/4) or Sierpiński triangle (1/3)? —Tamfang (talk) 16:27, 30 August 2023 (UTC)[reply]