Koch Snowflake
Start with a triangle. Every step, replace each straight edge with four shorter
edges that bump outward — the perimeter grows without bound while the shape stays
inside a fixed area.
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How it works: start with an equilateral triangle (3 edges). At every
iteration, every straight edge in the current shape is replaced by 4 shorter edges:
split the edge into thirds, then instead of the middle third, draw two sides of an
equilateral triangle pointing outward (the "bump"). Do that to all edges
simultaneously and repeat. Each iteration multiplies the segment count by 4 and
shrinks each segment's length to 1/3 of the previous iteration's — so after 6
iterations there are 3 × 46 = 12,288 tiny segments. The perimeter
grows by a factor of 4/3 every round and diverges to infinity as iterations continue,
yet the whole shape stays trapped inside a fixed bounding circle — a real (if
idealized) example of a curve with infinite length enclosing a finite area, and one of
the first shapes ever described as a "fractal."