Table of Contents
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.
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Peano Railroads
Definition: Δ(1,0,0,-1,1)(1,0,0,1,-1)(0,1,0,-1,-1)
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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.
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