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pftrails:self-crossing_curves

Self-crossing plane-filling trails

On this page we see plane-filling trails visualizing three self-crossing plane-filling curves: the Peano Ballet, the Peano Railroads, and Ventrella's Groovy Crosser. These curves have complicated structures. Visualizations as plane-filling trails may not suffice for such curves: a combination of a plane-filling trail image with a colour gradient drawing may work best.

The Peano Ballet

Definition: Δ(1,0,0,1,1)(0,1,0,1,-1)(1,0,0,1,1)

Simply reflecting each segment in the definition of the balanced Peano curve results in this string of dancers, each of which are composed of many smaller copies of themselves.

ballet.jpg

Peano Railroads

Definition: Δ(1,0,0,-1,1)(1,0,0,1,-1)(0,1,0,-1,-1)

Ventrella's Groovy Crosser

Definition: Δ(0,0,-1,-1,-1)(1,0,0,1,1)(0,1,0,1,1)(0,0,1,1,1)(0,-1,0,-1,-1),(1,0,0,1,1),(0,-1,0,-1,-1)

I found this curve in Ventrella's book1) Showing its structure clearly remains a challenge.


1) J. Ventrella. Brainfilling curves: a fractal bestiary, page 110. Self-published / Eyebrain Books, 2012.
pftrails/self-crossing_curves.txt · Last modified: 2020/02/17 08:15 by administrator