One of my favorite brain teasers is asking someone to describe a shape with an infinite circumference but finite area. The answer is a fractal (an object or quantity that displays self-similarity) called the Kock Snowflake.

It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The area converges to eight-fifths of the original triangle while the circumference grows infinitely. The figure below shows the first three iterations.

Koch Snowflake

Source: Weisstein, Eric W. “Koch Snowflake.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/KochSnowflake.html

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