靜宜非線性及相關問題研討會
An algorithm to generate
fractal reptiles
Peng-Jen Lai ( 賴鵬仁 )
Department of mathematics, National Kaohsiung
Normal University
高雄師範大學數學系
20120829
Contents
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Introduction:
1. Tilings 2. Reptiles 3. Fractals
4. Fractal tilings 、Fractal reptiles
Motivation
Survey of methods to generate Fractal reptiles
A geometric algorithm to generate Fractal reptiles
Computer simulation with Maple
Future directions
Reference
Tilings with tiles as regular polygons
The definition of tilings

Definition: [Grünbaum et al. 1987] A plane
tiling is a countable family of closed
topological disk which cover the plane without
gaps or overlaps. More explicitly, the union of
the sets ( which are known as the tiles of ) is to
be the whole plane, and the interior of the sets
are to be pairwise disjoint.
Tiling with non-polygonal tiles
Tiling in non-Euclidean spaces
Escher
作
品
欣
賞
The definition of reptiles(自我複製
磁磚)


The idea reptile was invented by S. W. Golomb
in 1962. More information about the k-rep tile is
presented in [Darst et al 1998, Grunbaum].
Definition: A k-rep tile is defined as any tile T
that can be dissected into k congruent parts
each of which is similar to T.
The following figure shows a famous example
of a 4-rep tile called Sphinx.
Remark: Every reptile can tile the plane!
4-rep tiles
圖7
可四重複製的星狀多邊形
Wiki Fractal figures
碎形天線陣列
Space-filling curves: the Sierpinski curve
drawn with Maple [Rovenski 2000]
Definition of fractals: Using iterated
function system (IFS) [Barnsley]
Definition of fractal tiles
Def: [Feder 1998] A prefractal is an intermidate shape
of generating a fractal using IFS method.
 n ( B) is a prefractal.

A  lim  n ( B) ,
n 
 Def: [Lai 2009] A (kth) prefractal tile is a tile with (kth)
prefractal boundaries.
Now we can define when a fractal tile can tile the plane.
 Def: [Lai 2009] A fractal tile can tile the plane if its
every kth prefractal tile can tile the plane for k  N .

From now on, when we say a fractal tile,
it means a disk-like set with fractal boundary
and being able to tile the plane.

Figure:Fractal tiling with tiles as
Fudgeflakes
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
Def: A fractal reptile is a fractal
tile which is also a reptile.
The aim of this talk: how to design a
fratcal reptile?
Motivation


In the internet we see many fractals which are
claimed to be able to tile the plane. Because of
the complicated boundaries of the fractals, it is
not easy to understand how they can tile the
plane.
It is more difficult to understand why some wellknown fractal is also a reptile.
Wiki
Gosper island is a 7-rep tile
Terdragon is a 3-rep tile
Escher style rules
Parallel translation
D
A
B
C
A
B
Glide-reflection
D
C
To check if a fractal is a tile by Escher style
rules [Lai]
Example: The quadratic Koch curve [Feder]
initiator
generator
How to generate a fractal reptile?

1.
2.
Survey of methods:
[Bandt 1991] Suppose that M represents an
 2 1
M

expansive map ( e.g.,  1 2  ), {y_1,,,y_m} is a
complete residue system for M,
1
and f j ( z)  y j  M z . Then the attractor
set A  mj1 Aj is the union of m tiles that have
disjoint interior and satisfy A_j=f_j(A). Such
tiles are called m-rep tiles. ( 此處之 m-rep tile 是
較廣義的，因為 M 不見得是相似變換。 )
[Bandt et al 2001] They give conditions to
guarantee that a self-affine tile in R^2 is
homeomorphic to a disk.
3. The tiles generated by the IFS method (wavelet) are not
always good-looking. In [Flaherty and Wang 1999] They stated
as follows:
The classification of 2-reptiles by
Ngai et al. [Ngai 2000]
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Twindragon,
Levy dragon,
Heighway dragon,
triangle,
rectangle (e.g. A4 paper),
tame twindragon
Graphs in Wiki
A geometric algorithm to generate fractal
reptile [Lai]
Investigation on the Gosper island
G0
G0
G0
G0
G0
G1
G1
G1
G0
G1
G0
G0
G1
G1
G1
G2
G1
Remark: This view of point is stated in Wiki, but not
generalized for general case.

Criterion 1:
Another example: Greek cross
S0
S0
S1
S0
S0
S0
S0
S1
S1
S1
S2
S1
S1
S3
S3
S3
S3
S3
S3
Computer simulation with Maple

We can use Maple to draw the prefractal of
Greek cross fractal reptile up to several
generation easily.
4th generation of
Greek cross
Fractal reptile
Future directions
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Wavelet 、quasi-crystal and fractal tiles.
Find some conditions to ensure that a fractal
reptile is a topological disk.
How many well-known examples can be
checked by this algorithm?
References
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M. F. Barnsley, Fractals Everywhere, AP Professional, 2ed, 1993.
C. Bandt, self-similar sets 5. Integer matrices and fractal tilings of R^n, Proceedings of AMS 112
(1991), 549-562.
C. Bandt, Y. Wang, Disk-like self-affine tiles in R^2, DiscreteComput. Geom. 26 (2001), 591-601.
T. Flaherty, Y. Wang, Haar-type multiwavelet bases and self-affine multi-tiles, Asian J. Math. Vol.3
No. 2 (1999) 387-400.
R. Darst, J. Palagallo, T. Price, Fractal Tilings in the Plane, Math. Magazine, Vol. 71, No.1.(1998),
p.12-23.
J. Feder, Fractal, Plenum Press,1988.
K. Falconer, Fractal geometry mathematical foundations and applications, John Wiley and Sons,
1990.
B. Grünbaum and G. C. Shephard, Tilings and patterns, W.H.Freeman and Company, 1987.
Ngai, Sirvent, Veerman, Wang, On 2-reptiles in the plane, Geom. Ded. 82 (2000) 325-344.
Peng-Jen Lai, How to make fractal tilings and fractal reptiles, Fractals, 2009.
V. Rovenski, Geometry of Curves and Surfaces with Maple, Birkhauser 2000.
賴鵬仁編著，幾何學講義第二部鑲嵌圖形之幾何結構與碎形幾何學以電腦軟體輔助探
索Learning tilings and fractals with the aid of maple GSP and CorelDraw，白
象文化，ISBN: 978-986-6453-09-0 ( in Chinese).
上課自編教材: 1.Maple快速入門. 2.Maple programing.
還有螢幕操作之錄影檔，有提供給思渤科技(參考思渤科技網頁).
洪維恩著, 數學魔法師Maple6, 基峰出版.
Maple10 Programming Guide, WaterlooMaple.
Thanks for your attention!
北大武日出、雲海、鐵杉，清大登山社朋友潘庭、之中、鳴君等攝2007
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If we use the view point of initiator and generator of boundary.
Then it’s hard to understand why it is a 7-reptile