Curiosities · Foundation · 5 min
A window into fractals
A repeated rule can make an intricate geometric curve or the boundary of a complex dynamical system.
WHAT HAPPENS IF…Does the curve’s length settle to a finite value?
Preparing your experiment…
Each stage replaces one segment by four segments, each one-third as long. The number of segments is 4^n, their length is 3^(−n), and total length grows as (4/3)^n. The Koch view draws a finite construction. In the Mandelbrot view, iterate z → z²+c from zero. Once |z| exceeds 2, that orbit escapes. A point that has not escaped through a finite cap is not thereby proved to belong to the Mandelbrot set.
A detail that matters
The screen’s finite resolution cannot display infinitely many stages.
Keep the connection going
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