Self-avoiding space-filling folding curves in dimension 3

TL;DR

This paper extends the concept of self-avoiding space-filling folding curves from two dimensions to three dimensions, constructing an example in $R^{3}$ with similar properties.

Contribution

The paper introduces a new example of self-avoiding, space-filling folding curves in three-dimensional space, expanding the known class of such fractal curves.

Findings

Constructed a 3D self-avoiding space-filling folding curve.

Demonstrated properties similar to 2D folding curves in 3D.

Provided insights into fractal structures in higher dimensions.

Abstract

Various examples of folding curves in $R^{2}$ have been considered: dragons and other square curves, terdragons and other triangular curves, Peano-Gosper curves based on hexagons. They are self-avoiding. They form coverings of $R^{2}$, by one curve or by a small number of curves, which satisfy the local isomorphism property. They were used to define some fractals. We construct an example with similar properties in $R^{3}$.