Friday, February 05, 2021

[vrfxvbda] differentiable fractal

turn a regular n-gon polygon into a regular 2n-gon by chopping off corners in just the right way.  repeat the operation infinitely.  this results in a curve consisting of "all corners".  if this limit curve were like the Koch snowflake, it would be a fractal that is continuous everywhere but differentiable nowhere -- impossible to determine the tangent anywhere.

however, the limit curve for regular polygons is of course a circle, differentiable everywhere.  under what conditions does an infinitely repeated fractal-like construction counterintuitively result in a differentiable curve?  when it exists, how do we find that differentiable curve and compute its derivatives?

subdivision surfaces in 3D computer graphics also counterintuitively result in smooth surfaces.

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