The Koch snowflake
Press the iteration buttons 0 to 6, or tap the picture to go to the next one. Each step replaces the middle third of every edge with a bump: a smaller equilateral triangle pointing outward. The bumps added at the current step are orange. Watch the two numbers under the picture: the perimeter is multiplied by 4/3 every time and keeps growing, while the area creeps up and stalls below 8/5 of the first triangle.
What's going on
Perimeter. An edge of length L becomes four edges of length L/3, total 4L/3. So every step multiplies the whole perimeter by 4/3. After n steps it is (4/3)ⁿ times the original, and (4/3)ⁿ grows without limit: by step 6 it is more than 5 times round, by step 50 it would be more than a million times round, though the snowflake still fits on this page.
Area. Step 1 adds 3 triangles, each 1/9 of the original, so 1/3 is added. Step 2 adds 12 triangles each 1/81 of the original, 12/81 = 4/27 in total. Each step adds 4 times as many triangles, each 1/9 the size, so the added area is multiplied by 4/9 each time. The additions form a geometric series 1/3 + 4/27 + 16/243 + … whose sum is (1/3)/(1 − 4/9) = 3/5. Total area: 1 + 3/5 = 8/5 of the first triangle, and never more.
A region with finite area and infinite boundary. Painting it takes a finite tin of paint; walking round it takes forever. The snowflake's edge is more than a line but less than a surface: its dimension is log 4 / log 3 ≈ 1.26.