**Generalized Gosper Space Filling Curves
and Infinite Series of Them**
Jin Akiyama, ** Hiroshi
Fukuda**, Hiro Ito, and Gisaku Nakamura School of Administration and
Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526,
Japan
**Abstract**
The Gosper curve is a space
filling curve discovered by William Gosper, an American computer
scientist, in 1973, and was introduced by Martin Gardner in 1976
[1,2]. It is constructed by recursively replacing
a dotted arrow, called the initiator, by seven arrows, called
generator, as shown in Fig. 1(a), Fig. 1(b) and Fig. 1(c)
illustrate the curves obtained by replacing the initiator by
generator once and twice respectively.
The Gosper curve is said to be monster curve since it is a path
from the root to the tip of the initiator without any branches
visiting all interior lattice points on a regular triangular
lattice. In 2001[3], we found
a lot of such monster curves,
namely, generalized Gosper curves, by computer search. Their
shapes are similar to the original Gosper curve as shown in Fig.
2. In this work, we report on the following three new results.
Figure2:Generalized Gosper curve with *N* = 13,43,91.
*Size of Curves*
We investigate the necessary conditions for the size of
the curves *N*, the number of arrows included in the generator. We
can show *N* has to satisfy
*N*=6*n*+1, *n* = 1,2,3,....
and
*N*= *x*^{2}+ *y*^{2}+ *xy*,x=0,1,2,...., y=0,1,2,....,
*(x,y)* (0,0).
Therefore possible sizes for generalized Gosper curves are
*N*=7,13,19,25,31,37,43,49,61,67,73,79,91,97,....
*Number of Curves*We did computer search for the
generalized Gosper curves for *N* 61. Then we could determine
the number of generalized Gosper curves *C*_{N} for given *N*. We
obtained *C*_{N} = 1,1,1,0,7,8 and 24 for *N* = 7,13,19,25,31,37 and 43, respectively. For *N* > 43,
calculations are not finished
and we found *C*_{49}134+75 and *C*_{61}485.
*Infinite Series of Curves*
We discussed the natural extension of the Gosper curve in [3]. It is a simple
procedure to generate infinitely many curves with *N* = 3*k*^{2}+3*k*+1,
*k*=1,2,3,... from Gosper curves as shown in Fig. 3, where
we can find simple relation between successive generators in *k*.
We apply this simple procedure for other curves, and so far as our
investigation is concerned, more than three quarters of Gosper
curves can generate infinitely many curves. For example, the curve
with *N*=13 shown in Fig.2 generates infinite series with
*N* = 9*k*^{2}+3*k*+1,. Two curves with *N*=43 and 91 in Fig.2 are those
for *k*=2 and 3 in this series.
Figure3:Infinite Series of Gosper curve with *N* = 3*k*^{2}+3*k*+1
**References**
1.M. Gardner, In which 'monster' curves force redefinition of the
word 'curve', Scientific American **235** (1976) 124--133.
2.B. B. Mandelbrot, The Fractal Geometry, W. H. Freeman and Company,
New York, 1977.
3.H. Fukuda, M. Shimizu and G. Nakamura, New Gosper Space Filling
Curves, Proceedings of the International Conference on Computer
Graphics and Imaging (CGIM2001),(2001) 34--38 .
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