On the Boundary of the Harter-Heighway dragon curve

TL;DR

This paper uses an L-system to establish recurrence formulas for the boundary length of the Harter-Heighway dragon curve, confirming longstanding conjectures and deriving related sequence formulas for binary strings and matrices.

Contribution

It introduces a novel application of L-systems to prove recurrence relations for the dragon curve's boundary length, resolving historical conjectures.

Findings

Recurrence formulas for boundary length are proven.

Longstanding conjectures from 1975 are confirmed.

Formulas for related binary and ternary sequences are derived.

Abstract

In this article we apply an L-system to prove a recurrence formula for the length of the boundary of iterands of the well known Harter-Heighway dragon curve, a space filling curve with fractal boundary. This leads to finding formulas for related sequences of certain binary strings and ternary matrices. This proves some long standing conjectures for the recurrence relation for the number of terms in the boundary of the dragon curve, first stated in unpublished work Daykin and Tucker in 1975.