See a simple pattern repeating across scales, a "fractal," drawn with recursive code. As it grows, the perimeter length lengthens hugely, but the area doesn't increase much after the first few iterations!
Such self-similarity across scales is called a fractal. Fascinatingly, the fractal's irregular perimeter grows in length towards infinity, while the area growth levels off. How long is the length of a coastline? The zigzags of coastlines have this fractal property, at smaller and smaller scales, down to the microscopic level. The wiggles fold into larger wiggles. That means the area doesn't have to grow much, but length increases. This quality extends a coastline's length towards infinity, but our measuring tools are crude so there's a limit there. This fractal here also has a beautiful symmetry, and looks like a snowflake. Originally it was called a Koch curve (see "On a Continuous Curve Without Tangents" by Helge von Koch, 1904). We repeat (iterate) the process for 7 levels. At level 0, it just draws a triangle. At level 1, it interrupts each side of the triangle with a smaller triangle. Then for additional levels, it draws the same pattern at smaller and smaller scales. If we repeated for more levels, the perimeter length grows astronomically, towards infinity.
