 
Summary: On the coordinate functions of
L´evy's dragon curve
Pieter C. Allaart and Kiko Kawamura
University of North Texas
August 16, 2005
Abstract
L´evy's dragon curve [P. L´evy, Les courbes planes ou gauches et les
surfaces compos´ees de parties semblables au tout, J. Ecole Polytechn.,
227247, 249291 (1938)] is a wellknown selfsimilar planar curve with
nonempty interior. We derive an arithmetic expression for the coordi
nate functions of L´evy's dragon curve, and show that the 3
2
dimensional
Hausdorff measure of the graph of each coordinate function is strictly pos
itive and finite. This complements known dimensional results concerning
the coordinate functions of spacefilling curves of Peano and Hilbert. The
proof is based on deriving suitable uniform upper bounds for the sizes of
the graphs' level sets.
1 Introduction
L´evy's dragon curve D (Figure 1) was introduced and studied in 1938 by P.
