It is possible to have an object with the properties of finite area and infinite perimeter. In this object, the sides enclose a finite region but the sides are made up of infinite pieces.

Check out the Koch snowflake. It starts with an equilateral triangle. Then, on the first iteration, each side is broken so that a smaller triangle can be put on. Then, each side is broken again so that yet a smaller triangle can be put on. This is done infinitely so that you get…

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