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On the fractal nature of Penrose tiling
P. Ramachandrarao*, Arvind Sinha and D. Sanyal
National Metallurgical Laboratory, Jamshedpur 831 007, India
An earliest preoccupation of man has been to find
ways of partitioning infinite space into regions having
a finite number of distinct shapes and yielding beauti-
ful patterns called tiling. Archaeological edifices, every-
day objects of use like baskets, carpets, textiles, etc.
and many biological systems such as beehives, onion
peels and spider webs also exhibit a variety of tiling.
Escher’s classical paintings have not only given a new
dimension to the artistic value of tiling but also
aroused the curiosity of mathematicians. The gener-
ation of aperiodic tiling with five-fold rotational
symmetry by Penrose in 1974 and the more recent
production of decorated pentagonal tiles by Rosemary
Grazebrook have heightened the interest in the subject
among artists, engineers, biologists, crystallographers
and mathematicians
1–5
. In spite of its long history, the
subject of tiling is still evolving. In this communica-
tion, we propose a novel algorithm for the growth of a
Penrose tiling and relate it to the equally fascinating
subject of fractal geometry pioneered by Mandelbrot
6
.
The algorithm resembles those for generation of fractal
objects such as Koch’s recursion curve, Peano curve,
etc. and enables consideration of the tiling as cluster
growth as well. Thus it clearly demonstrates the dual
nature of a Penrose tiling as a natural and a non-
random fractal.
T
HE
aperiodic tilings have many interesting properties
which can be illustrated with respect to the most discussed
quasi-periodic tiling: the Penrose tiling
7
. These can be
infinite in number. The tilings exhibit local isomorphism
which ensures that every finite region in any one tiling is
contained somewhere inside every other and that too infi-
nitely many times. The tilings can be generated from one
another by the methods of inflation or deflation. For
example, in deflation a cluster of tiles is subdivided into
smaller pieces following specific procedures. Performing
such operations iteratively, one can generate an aperiodic
tiling with a much larger number of smaller tiles. These
procedures endow the tiling with the property of self-
similarity. These properties have suggested, right from the
time of the discovery of Penrose tilings, that the tiling is
fractal in nature
2
. If so, it should be possible to generate
the tiling by an algorithm that is characteristic of fractal
generation and consisting of the three steps of initiation,
generation and cascade at all scales. Till date, there are no
methods of generating a Penrose tiling in a fashion similar
to the generation of a non-random fractal object. We
postulate, in this communication, such an algorithm and
arrive at the Hausdorff fractal dimension.
Let us consider a figure in the form of two straight
edges AB and BC of equal length, a, and intersecting each
other at an angle of 108° (Figure 1 a). The point of inter-
section, B, is called a vertex. Divide the side AB in the
ratio
Ï„
: 1, as seen from A, and replace a short segment
with an isosceles triangle having sides BE and ED of
length a/
Ï„
. The triangle should be erected such that
*For correspondence. (e-mail: dnml@csnml.ren.nic.in)
Figure 1. Proposed algorithm for the generation of a Penrose tiling.
a, the initiator; b, the first generation (n = 1); c, the second generation
(n = 2); and d, Penrose tiling in a thick rhomb: n = 7.
c
d
b
a
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CURRENT SCIENCE, VOL. 79, NO. 3, 10 AUGUST 2000
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the angle at B is divided in the ratio 2
:
1 and
âˆ
EDB =
âˆ
EBD = 72° each.
Ï„
is the golden mean and has
the value of (1 + )/2. Next join the point E to C which
lies at a distance of a/
Ï„
as well. We may now erase all
other lines which do not have a length a/
Ï„
(Figure 1 b).
These steps generate three line segments of equal length
(DE, EB and EC) and two new vertices D and E with
âˆ
ADE and
âˆ
BEC being 108°. It may be noted that one of
these angles is exterior to the apex of the isosceles tri-
angle erected, while the other is exterior to its base. These
two angles are once again divided as before by erecting
two isosceles triangles of side a/
Ï„
2
such that two adjacent
angles of 36° each are formed at the apex and two adja-
cent angles of 72° form at the base of the previously
erected isosceles triangle. All vertices at a distance of a/
Ï„
2
will then be joined and all segments whose length is
not a/
Ï„
2
will be erased (Figure 1 c). The two isosceles tri-
angles erected in this operation are EFG and DHI. It may
be seen that the rhombi BDHI and BIEF are the thin and
thick rhombi used by Penrose to generate an aperiodic
tiling. The newly formed vertices with an included angle
of 108° are at F, G, H and I. The angle EIB is also 108°.
The procedure described above can be repeated to divide
all the 108° angles by erecting isosceles triangles of side
1/
Ï„
of the side on which they are being erected. We also
join the newly formed vertices to already existing vertices
at a distance equal to the side of the triangle erected. The
procedure can be repeated ad infinitum. With each com-
plete operation of the algorithm, the edges in the pattern
are reduced by a factor of
Ï„
. The pattern of a Penrose
tiling begins to emerge as we proceed and all the known
distinct vertices appear when the length scale reaches
a/(
Ï„
6
). Such a complete Penrose tiling shown in Figure
1 d, is obtained after reflecting the structure derived at
n = 7 in a mirror placed along AC. The ratio of the edge
length of a tile (a
n
) to the edge of the rhomb tiled (a) will
always be
Ï„
n
, where n represents the number of times the
scale length has been reduced by a factor of
Ï„
. All further
discussions refer to the tiling within this rhomb. Many
others have repeatedly stressed the self-similar nature of
the Penrose tiling. The present algorithm is the first to
employ an iterative procedure commonly employed to
generate deterministic fractals. This novel algorithm
which has an initiator (lines at 108° to each other), a
generator (erection of isosceles triangles with sides 1/
Ï„
times the previous length scale and erasing all sides of
other length scales) and cascading is typical of pro-
cedures which generate patterns with fractal dimensions.
This procedure also obviates the need for assembling tiles
according to matching rules of Penrose. In addition, it
permits us to utilize various procedures developed for
determining the fractal dimension of patterns arising from
such an algorithm.
Conventionally, the estimation of the fractal dimension
of a non-random fractal based on an iterative scheme of
the above three steps is given as,
,
)
/
1
log(
)
log(
lt
0
f
l
N
D
l
→
=
where N is the number of edges of length l which occur
within the fractal object. The discovery of quasi-crystals
has conclusively demonstrated that aperiodic packing of
atoms occurs in nature. In a Penrose tiling, the closest
distance of separation of two atoms can only be along the
short diagonal of a thin rhomb. For the present algorithm,
if the edge length of a rhomb is taken as the reference,
when n tends to infinity, the edge length tends to zero and
the rhomb collapses to a point. Thus it would not be app-
ropriate to evaluate the fractal dimension of the tiling in
the limit when n tends to infinity. Therefore, when one
considers the Penrose tiling from a crystallographic point
of view, a limit has to be placed on the value of n, the
number of recursions of the algorithm. The recursions
have to be limited only to the extent of achieving dis-
tances of separation of the order of atomic spacings
between the vertices in the tiling. From the point of view
of the present algorithm, one achieves atomic distances of
separation from a starting separation, a, of 1 cm in 39
recursions. Hence, the formula for D
f
can be modified as,
,
)
log(
)
log(
lt
1
f
−
′
→
=
n
n
n
N
D
Ï„
where n
′
is the physical limit on the number of recursions.
One can obtain the edge length and the number of vertices
generated in a tiling in terms of the Fibonacci numbers.
These have been estimated as:
N
vertices
= F
2n +1
+ 4F
n
,
,
2
8
2
vertices
edges
n
n
A
F
N
N
+
−
=
where F represents the respective Fibonacci numbers in a
sequence starting with zero. A
n
is an even integer depen-
dent on n. Based on an analysis using the edge length in
the above formula for D
f
, we arrive at a value of the
Hausdorff dimension as 1.974 using the size of n
′
as 39.
This value is close to the fractal dimension obtained
for the growth of two-dimensional clusters of atoms.
However, it may be noted that limit of D
f
as n
→
∞
exists
and its value is 2, implying a non-fractal space filling
structure.
Quasicrystals which incorporate aperiodic tilings, like
Penrose tiling, are known to grow by the clustering pro-
cess. Recently Lord et al.
5
have shown that the clustering
process is very similar to the way quasicrystals actually
grow. A cluster is conceived to grow by accretion of suc-
cessive shells around an initial seed, such as a 12- or 13-
5
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CURRENT SCIENCE, VOL. 79, NO. 3, 10 AUGUST 2000
366
atom icosahedron. The clusters grow as they bond to each
other by sharing the atoms. One of the most commonly
discussed cluster growth models is the diffusion-limited
aggregation (DLA) of atoms
8–10
. DLA exhibits random
branching during growth and can be shown as a non-
deterministic fractal, governed by the sticking probability.
Notwithstanding this randomness, DLA manifests, irres-
pective of the magnitude of the sticking probability, some
important regularities in an average sense
10
, e.g. (i) the
most probable value of the angle between neighbouring
branches is found to be 36°, and (ii) Fibonacci numbers
are known to occur in DLA clusters. In Penrose tilings
also, these two parameters, viz. the angle of 36° and Fibo-
nacci numbers, govern the pattern generation. However
unlike those in DLA, the angles in a tiling are exact multi-
ples of 36° and the number of tiles, edges and vertices are
precise functions of the Fibonacci numbers as shown
above. It remains to be seen as to how a deterministic
algorithm for Penrose tiling, like the one generated in the
present study, and diffusion controlled probabilistic DLA
algorithm of Whitten and Sander
9
bear these important
similarities.
1.
Penrose, R., Bull. Inst. Math. Appl., 1974, 10, 266–271.
2.
Gardner, M., Sci. Am., January 1977, 110–121.
3.
Stewart, I., Sci. Am., July 1999, 96–98.
4.
Mackay, A. L., Physica A, 1982, 114, 609.
5.
Lord, E. A., Ranganathan, S. and Kulkarni, U. D., Curr. Sci.,
2000, 78, 64–72.
6.
Mandelbrot, B. B., Fractal Geometry of Nature, W.H. Freeman,
San Francisco, 1983.
7.
Penrose, R., Aperiodicity and Order (ed. Jaric, M. V.), 1989,
vol. 2, pp. 69–110.
8.
Kaye, B., Chaos and Complexity, VCH, New York, 1993.
9.
Whitten, T. A. and Sander, L. M., Phys. Rev. Lett., 1981, 47,
1400–1403.
10.
Stewart, I., New Sci., August 1992, 14.
ACKNOWLEDGEMENTS. We are grateful to Professors N. Kumar,
S. Ranganathan, G. Ananthakrishna, Shrikant Lele, G. V. S. Sastry and
Dr R. K. Mandal for helpful discussions.
Received 21 February 2000; revised accepted 1 June 2000
Participatory approach in varietal
improvement: A case study in finger
millet in India
B. T. S. Gowda*, B. H. Halaswamy*, A. Seetharam*,
D. S. Virk
â€
and J. R. Witcombe
â€
*Project Coordination Cell, AICRP on Small Millets, GKVK,
Bangalore 560 065,India
â€
Centre for Arid Zone Studies, University of Wales,
Bangor LL 57 2UW, UK
Crop improvement research has made a significant
contribution in the last 5 decades through the deve-
lopment and release of a large number of varieties in
all important crops for general cultivation. It is gener-
ally felt that the modern plant breeding has catered
more to the needs of rich farmers who could afford
high management under irrigated situations. In con-
trast, subsistence farmers growing millets and other
minor crops in unfavourable environments use low
levels of inputs and have not been benefitted by high
yielding variety (HYV) technology. In the present
case study, the usefulness of the participatory app-
roach for identifying cultivars for harsh environments
and acceptable to resource-poor farmers has been
demonstrated. The study carried out in Chitradurga
district using six finger millet varieties with 150 farmers
pointed out the effectiveness of such an approach in
identifying cultivars for meeting the requirement of
the resource-poor farmers under real farm situations.
Another important outcome of the farmer participa-
tory varietal trial has been the identification of a most
suitable variety for growing in a specific niche as a
second crop which otherwise would have been diffi-
cult under the variety evaluation system confining to
research stations.
C
ROP
improvement research has been in progress since
the beginning of this century. However, progress in plant
breeding has been phenomenal in the last 5 decades in
India. The coordinated crop improvement programme was
first conceived in maize during 1956 and later on ex-
panded to all the other important crops either individually
or in groups under the National Agricultural Research
System (NARS). The conceptualization of land grant sys-
tem of agricultural research, education and extension and
the establishment of agricultural universities further helped
in developing broad-based agricultural research activities
to address the agricultural research needs of different
states. As a result, a large number of varieties were bred
and released for general cultivation in all important crops
in the country. The successful adoption of these varieties
along with improved cultivation practices helped in a 4-
fold increase in food grain production, from 50 million
tonnes to more than 200 million tonnes. However, much
*For correspondence. (e-mail: btsg@uasblr.kar.nic.in)
author