The perimeter grows without bound, while the area stays finite
If you repeatedly add a pointed bump to the middle of each side of an equilateral triangle, you get a "Koch snowflake." Surprisingly, its perimeter grows to infinity, while its area remains bounded by a finite value no greater than 8/5 times the area of the original triangle. Use the slider to build it yourself.
Why does this happen? With each iteration, the number of segments becomes 4 times larger, while the length of each segment becomes 1/3 as long. So the perimeter is multiplied by 4/3 each time (and grows without bound because 4/3 > 1), while the newly added area consists of increasingly smaller pieces, so the total cannot exceed a certain finite value.
Iterations0
Perimeter (start = 3)
3
Area multiplier
1.000
💡 Shapes that are "in between" 1D (a line) and 2D (a surface) like this are called fractals. The Koch snowflake has a dimension of about 1.26 — it is more complex than a line, but it does not fill an entire plane.