Conference PaperPDF Available

New Gosper Space Filling Curves

School of Administration and Informatics, Department of Liberal Art, Research Institute of Education,
University of Shizuoka, Nagano Prefecture College, Tokai University,
52-1 Yada, Shizuoka-shi, Shizuoka, 8-49-7 Miwa, Nagano-shi, Nagano, 2-28-4 Tomigaya, Shibuya-ku, Tokyo,
422-8526 JAPAN 380-8525 JAPAN 151-8677 JAPAN
We investigate several properties of a beautiful space
filling curve known as the Gosper curve, and try to
construct curves having the same properties. We found
17 new Gosper curves through computer searches, and an
infinite series of curves that are natural extensions of the
original Gosper curve.
KEY WORDS: Gosper curve, Space filling curves,
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]. The
Gosper curve is said to be a very beautiful and complex
monster curve because it has several special properties
that other space filling curves such as the Peano curve and
the dragon curve do not have.
In this paper, we point out several properties of the
Gosper curve and try to construct curves having the same
properties. Then, we show 17 new Gosper curves that we
found through computer searches. We also show an
infinite series of curves that are natural extensions of the
original Gosper curve.
In section 2, we describe the Gosper curve. In section 3,
we propose a construction method for new Gosper curves.
In section 4, results are shown and discussed.
The Gosper curve is a recursive curve 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
(a) (b) (c)
Fig. 1 Gosper curve.
The arrows of generator in the curve obtained after
replacing the initiator to the generator any times are
located on the edge of a regular triangular lattice. The
degree of the generator arrow at the lattice point is two in
the interior of the Gosper curve and is one at the both
ends of initiator, i.e., the Gosper curve is the path from
the root to the tip of the initiator without any branches
visiting all interior lattice points. To our knowledge, the
Gosper curve is the only space filling curve to have these
properties. Therefore, in this paper we define the Gosper
curve in terms of these properties. We then try to find new
Gosper curves which satisfy our definition other than the
Gosper curve found by William Gosper. We call the
number of arrows included in the generator the size of
generator or Gosper curve itself.
We attach an equilateral triangle to the initiator arrow
and to the generator arrow so that the triangle has the
arrow as its edge and so that the direction of the arrow is
counterclockwise in the triangle as shown in Fig. 2 (a).
We call the equilateral triangle attached to each arrow a
(a) (b)
Fig. 2 Flags.
In the recursive procedure constructing Gosper curve,
the flag for the initiator is replaced by the seven flags for
the generator. The area of the flags is conserved during
the replacement. Further, the flags fill the triangular
lattice alternatively as shown in Fig. 2(b). From these
simple flag properties, the following procedure to
construct the generator of the Gosper curve can be
(P1) Chose the size of the Gosper curve N.
(P2) Prepare an initiator arrow of length
and attach the
initial flag to it.
(P3) Prepare N flags with side length NL/ to be used
in the generator. The N flags should satisfy
conditions (P4) to (P7). Bellow, we call the set of N
flags “generator flags”.
(P4) The generator flags should be connected at their
(P5) Two vertices of generator flags should coincide with
the ends of the initiator. From this condition the size
of Gosper curve is restricted to
xyyxN ++= 22 , ,...2,1,0, =yx (1)
Therefore the possible sizes of Gosper curves are
(P6) The shape of the generator flags is symmetric under
a 3/2
rotation around its geometrical center.
(P7) If two coincident generator flags exist and one is
translated by a distance L in a direction parallel to
one of the three edges of a triangle in the initiator
flag, then the two generator flags should not overlap.
(P8) By taking only one edge from each flag and by using
the N edges of the generator flags, construct a path
without branches from the root to the tip of the
(P9) Convert the edges in the path to the arrow so that the
direction of the arrow is counterclockwise in the flag.
We then obtain the generator arrow.
Using this procedure, we obtain the seven generator
flags shown in Fig.3 for 7N. The generator flags
which corresponds to the William Gosper’s curve is the
only generator flags satisfying conditions (P8) and (P9).
3=N 4=N 7=N
Fig. 3 Generator flags for 7N.
As we have already mentioned, the smallest Gosper
curve obtained by the procedure given in the previous
section is the curve with 7=N found by William Gosper.
For 197
N, there are two curves for 13
and 19
N. We show these curves together with their
generators and generator flags in Fig. 4 and Fig. 5,
Fig. 4 A new Gosper curve for
= 13
Fig. 5 A new Gosper curve for
= 19.
For 3719
N, there are seven and eight curves for
N and 37, respectively. The seven curves for
N are constructed by the three generator flags
shown in Fig.6 (31a), (31b) and (31c). Gosper curves
which have the same size can be distinguished by using
the flag and a serial number, for example (31a-1). It is
notable that five new Gosper curves can be produced
from the generator flags (31b). The curves produced from
the same generator flags have similar shapes.
Fig. 6 New Gosper curves for 31=N.
Eight curves for 37=N are constructed from the four
generator flags shown in Fig.7 (37a), (37b), (37c) and
(37d). The generator flags (37a), (37b), (37c) and (37d)
construct 1, 2, 3 and 2 curves, respectively. Although we
did not search for curves with 37>N because of the
limitation of our computer power, we expect that there
exist many big and complex Gosper curves.
Fig. 7 New Gosper curves for 37=N.
Next, we compare the three curves with sizes 7
19=N and (37a-1). We found that the generator of the
curve for 19=N contains the generator of the curve for
7=N, and that the generator for (37a-1) contains the
generator for 19=N. Similarly, the generator flags of the
curve for 19=N are obtained by adding a layer of flags
around the generator flags of the curve for 7=N, and the
generator flags for (37a-1) are obtained by adding a layer
around the generator flags for 19
N. Thus the curve for
19=N and the curve (37a-1) are natural extensions of
the Gosper curve for 7=N.
It is possible to make the natural extension of the
Gosper curve for 7=N by adding a layer several times.
We explain the extension method as an example of the
curve for 37=N. First, we add a layer of flags around
the generator flags for 37=N as shown in Fig. 8. The
resulting generator consists of 61 flags. Thus we will
obtain the Gosper curve of the size 61=N. Next, we
choose the ends of initiator to be the leftmost vertex on
the button edge and the rightmost vertex, then we draw a
dotted line from the ends of the initiator to the ends of the
curve already drawn for 37=N as shown in Fig. 8.
After converting each segment of the dotted line to the
arrow, we obtain the generator of Gosper curve for
N. It is clear that bigger curves can be obtained by
continuing this procedure.
Fig. 8 Natural extension of the Gosper curve
= 61
As mentioned above, by adding n layers to the
generator flags for 7
N, we obtain a generator with size
133 2++= nnN ,...)3,2,1( =n. (3)
Thus we can obtain an infinite series of Gosper curve with
the size given by equation (3). The shape of the curves
obtained looks like snail.
In this paper, we pointed out several properties of the
monster curves found by William Gosper, and constructed
new monster curves having the same properties. We
found that the smallest size of the new monster curves is
13 and obtained an infinite series of natural extensions of
the Gosper curve. These new recursive monster curves are
beautiful and will be useful for computer graphics.
We should note that Gosper curve for 137
<N may
exist since the procedure (P1)—(P9) is not a necessary
and sufficient condition for our definition of the Gosper
curve. Finding a procedure corresponding to the necessary
and sufficient conditions and applying the procedure to
the other regular lattices are topics of future research.
[1] M. Gardner, In which “monster” curves force
redefinition of the word “curve”, Scientific American
235 (December issue), 1976, 124-133.
[2] B. B. Mandelbrot, The fractal geometry, (New York ,
W. H. Freeman and Company, 1977).
... Each number in this trace is between 0 and S 2 − 1 where S is the scale factor for the fractal coordinate system, and identifies a sub-tile in bottom-to-top, left-to-right order, so that given within=tile coordinates (t x , t y ), the trace number n t = St y +t x . As a concrete example, the tile 1/(−1, −1) in the fractal coordinate system with scale 4 has trace coordinates 2/ [5]: it can be found by going up to the origin tile 2/(0, 0) and then descending into the sub-tile at (t x , t y ) = (1, 1) within that tile. Similarly, the tile 0/(2, 3) in scale 4 would have trace coordinates 2/ [14,3], whereas the tile 0/(2, 2) has trace coordinates 1/ [15]. ...
... It also generates this path incrementally: you can ask for any part of the path and it will give it to you without generating the rest of the path (although technically, it does generate some necessary constraints on the rest of the path in the process). In fact, these paths also form a family of space-filling curves [5,18] which incorporate a natural level of visual variety. (1) Each 5 × 5 region is traversed by a path that enters at one edge and exits at another, randomly chosen from all possible 5 × 5 paths which enter and exit at those coordinates (taking advantage of symmetries, there are less than two thousand such paths, and we simply store all of them in a cache indexable by entry/exit points). ...
... There are also threads of related work in mathematics (e.g., [5,15]), and physics (e.g., [15]) although this work does not directly apply any recent ideas from those fields. In particular, the discrete fractal coordinates presented here are far less complex than the notion of fractal coordinates present in [15]. ...
... Alternative space-filling fractal curves that map various other shapes. The Gosper curve, for instance, is often used to map larger hexagonal surfaces and its pattern is seen in Fig 7.4 [47]. ...
... Gosper fractal (space-filling) curve pattern[47] -increasing resolution images from left to right. ...
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... Since the early classics introduced by Peano, Hilbert, Polya, and Sierpinski, newer plane-filling curves have been discovered by several others, including Gosper [16], Mandelbrot [9], McKenna [10], Dekking [5], Fukuda [6], Ventrella [14], Arndt [1], Bandt [3], and others. As new curves are added to the repertoire, there is opportunity to make comparisons and identify common themes. ...
... The filling density of the covered area can be easily controlled by the order of the curve that is generated recursively like a fractal. A subset of SFCs, such as Hilbert, Gosper and Fukuda-Gosper [10] also possesses a self-avoidance capability. They forms a single closed path that will be interrupted in one or more parts in case a crack appears on its surface. ...
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... Thus, for a given curve a resolution or space sensitivity can be defined. We consider hereafter the Gosper (N = 2) space filling curve [15] (Fig. 3). Accordingly, the space sensitivity is given by: To comply with the metal matter of the hip, generally made of titanium alloy, the RFID transponder to be connected with the crack electrode is a cavity-backed square loop as in [17]. ...
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... A. Topological properties of SFC A SFC is univocally described in terms of initiator (d I ) and generator [32] (Fig. 13.a). The initiator is the starting path that coincides with the distance between the start and the end point of the curve and accordingly defines the size of the cell. ...
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"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
In which “monster” curves force redefinition of the word “curve”
  • M Gardner
M. Gardner, In which “monster” curves force redefinition of the word “curve”, Scientific American 235 (December issue), 1976, 124-133.
The fractal geometry
  • B B Mandelbrot
B. B. Mandelbrot, The fractal geometry, (New York, W. H. Freeman and Company, 1977).