Summary: On the coordinate functions of
L´evy's dragon curve
Pieter C. Allaart and Kiko Kawamura
University of North Texas
August 16, 2005
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.,
227-247, 249-291 (1938)] is a well-known self-similar planar curve with
non-empty interior. We derive an arithmetic expression for the coordi-
nate functions of L´evy's dragon curve, and show that the 3
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 space-filling curves of Peano and Hilbert. The
proof is based on deriving suitable uniform upper bounds for the sizes of
the graphs' level sets.
L´evy's dragon curve D (Figure 1) was introduced and studied in 1938 by P.