The interconnection between number theory, algebra, geometry

and calculus is shown through Fibonacci sequence,

golden section and logarithmic spiral. In this two-part

article, we discuss how simple growth models based on

these entities may be used to explain numbers and curves

abundantly found in nature

Mallik, A. K.

English

Publications of the IAS Fellows

29RESONANCE ç September    2004
GENERAL   ç ARTICLE
The interconnection between number theory, algebra, ge-
ometry and calculus is shown through Fibonacci sequence,
golden section and logarithmic spiral.  In this two-part
article, we discuss how simple growth models based on
these entities may be used to explain numbers and curves
abundantly found in nature.
Introduction
According to Kronecker, a famous European mathematician,
only natural numbers, i.e., positive integers like 1, 2, 3, … are
given by God or belong to the nature. All other numbers like
negative numbers, fractional numbers, irrational numbers, tran-
scendental numbers, complex numbers, etc., are a creation of the
human mind. Of course, all these other numbers are created using
the natural numbers. We are so used to natural numbers that we
may fail to notice some interesting patterns in them. For example,
let us observe the simple yet beautiful regularity of appearance of
all the consecutive natural numbers in the following equations:
1+2 = 3
4+5+6 = 7+8
9+10+11+12 = 13+14+15,
and it continues in this fashion ad infinitum.
In this article, we shall see, how by playing around with natural
numbers some other curious numbers and curves are generated,
and discuss their appearances in nature. Of late, such an approach
is becoming popular to propose mathematical models for the
growth of living organisms and other patterns commonly found
in nature.
From Natural Numbers to Numbers and
Curves in Nature – I
A K Mallik
A K Mallik is a Professor
of Mechanical Engineer-
ing at Indian Institute of
Technology, Kanpur. His
research interest includes
kinematics, vibration
engineering.  He partici-
pates in physics and
mathematics education at
the high-school level.
Keywords
Fibonacci sequence, golden
section, logarithmic spiral,
mathematical models.
30 RESONANCE ç September   2004
GENERAL  ç ARTICLE
Fibonacci Sequence
In 1202, Italian mathematician Leonardo of Pisa, nicknamed
Filius Bonacci or Fibonacci, used the natural numbers to con-
struct the following sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... . (A)
In this sequence, called the Fibonacci sequence, each number
(from the third onwards) is the sum of its immediate two prede-
cessors. Incidentally, Fibonacci is also credited with the expo-
sition of Hindu-Arabic numerals to Europe through his book
Liber Abaci.
The ratios of two consecutive numbers in the Fibonacci sequence
are seen to generate the following sequence:
1, 0.5, 0.666…, 0.6, 0.625, 0.615…, 0.619…, 0.61818.. (B)
This sequence is seen to be alternating in magnitude but converg-
ing. We may write the Fibonacci sequence as
2121 ,1,1 -- +=== nnn FFFFF for 3³n .
Thus for 3³n , 
1
2
1
1
-
-
-
+=
n
n
n
n
F
F
F
F
. (1)
As  ,¥®n  let j®
+1
lim
n
n
F
F
. So from (1)
j
j
+= 1
1
  or,
.012 =-+ jj (2)
Taking only the physically meaningful root,
....618034.0
2
15 =-=j (3)
Note that j is independent of F 1 and F 2. Some easy to prove but
31RESONANCE ç September    2004
GENERAL   ç ARTICLE
useful identities involving the Fibonacci sequence are given in
Box 1.
Golden Section
The sequence of the ratio of two successive numbers in the
Fibonacci sequence converges to a very special (as we shall see)
irrational, fractional number which was well known to the early
Greek mathematicians as ‘Golden Section’. To the Greeks, who
were predominantly geometers, this Golden Section was a harmo-
nious, almost mystical, constant of nature. To interpret the
Golden Section geometrically, we consider a line AB of unit
length (Figure 1a). Let C be a point on and inside AB, such that
(AB/AC) = (AC/BC).   If AC =j , then
j
j
j -
=
1
1 ,
or  012 =-+ jj .
This is the same as equation (2). Figure 1b explains how the point
C is obtained geometrically. In this figure AB = 1,
BD (perpendicular to AB) = ½, DE = DB and AC = AE.
The reciprocal of Golden Section is 
2
15/1 +== jy . It
may be noted that
( )jjjjy -=+== 1/1/1 . (4)
Figure 1a. Geometric inter-
pretation of Golden Sec-
tion.
Figure 1b. Geometric de-
termination of Golden Sec-
tion.
.
1
1
1.........
53
1
42
1
31
1
.1
2.........23
2
2
2
1
.212.........531
.12.........321
+
-=+++
+=++++
=-++++
-+=++++
nFnFFFFFFF
nFnFnFFFF
nFnFFFF
nFnFFFF
Box 1.
32 RESONANCE ç September   2004
GENERAL  ç ARTICLE
It is easy to show that  y  is the only number for which the number,
its integer part and its fractional part are in geometrical progres-
sion. (See Box 2).
Using the values of
÷
÷
ø
ö
ç
ç
è
æ -
=
4
5210
36sin o  and ÷
÷
ø
ö
ç
ç
è
æ +
=
4
5210
72sin o , it is easy to
show that the ratio of the side to the diagonal of a regular pentagon
is also equal to the Golden Section (see Figure 2). Furthermore, in
this figure it can be shown that, (AQ/AD) = (QD/AQ) = j .
Golden Section, being an irrational number, can be expressed as
an infinitely continued fraction as shown below:
...1
11
1
1
11
1
+
+
+
+
=j .
(5)
Terminating this infinitely continued fraction at successive steps
we generate exactly the sequence (B) which converges ultimately
to the Golden Section. Also remembering that a continued
fraction never ceases, (5) can be rewritten as j = (1/1+j), which
defines the Golden Section as seen earlier. It is interesting to note
that the reciprocal of Golden Section can also be expressed as an
Figure 2. Golden Section in
regular pentagon.
Box 2.
Let the integer part be n and the fractional part be m, then
( ) .
2
15
..,0
impliesThis.
22
n
m
einmnm
m
n
n
mn
-=
=-+
=
+
Since m is a fraction, n = 1, therefore the number m + n = 
2
15 +
.
33RESONANCE ç September    2004
GENERAL   ç ARTICLE
infinite series using only the digit 1 as
...1111/1 ++++== jy .
At this stage it is worthwhile to note the continued fraction repre-
sentations of some other fractional irrational numbers as shown
below.
...2
12
1
2
12
1
12
+
+
+
+
=-
          
...2
11
1
2
11
1
13
+
+
+
+
=-
...5
4
4
3
3
22
1
1
1
2
+
+
+
+
+
=-e .
Looking at these continued fraction expressions, one suspects
that the Golden Section enjoys a special status, employing only
the digit 1, even among the irrational fractional numbers. Num-
ber theorists call it the ‘worst’ irrational fraction, implying that it
is most badly ‘approximable’ by a rational number. This approxi-
mation is carried out by truncating the continued fraction. One
can easily verify that at every step this approximation is worst for
the Golden Section amongst the four irrational fractions men-
tioned above.
Golden Angle
One can easily extend the concept of Golden Section to a unit
circle instead of a straight line of unit length. This is shown in
Figure 3. The Golden Angle is defined as the smaller angle q,
where,
q
q
q
-
=
-
o
o
o 360
360
360
Figure 3. Golden Angle.
34 RESONANCE ç September   2004
GENERAL  ç ARTICLE
or,   ( )[ ]o0 36015.137 jq -=» .     (6)
Golden Rectangle
Let us now define a Golden Rectangle ABDC
of sides of length having the ratio 1:1+j
(Figure 4). If this rectangle is subdivided
into a square of unit side (AFEC), then the
remaining rectangle FBDE is also a Golden
Rectangle (reduced in scale and rotated
through a right angle). This process can be
continued indefinitely to generate similar figures with a reduc-
tion in the scale at every step by a factor j. The continued
similarity implies that diagonals AD and BE serve also as diago-
nals of all the nested rectangles of the same orientation (i.e.,
rotated through 180o) as seen in Figure 4. The point (denoted by
O in Figure 4) of intersection of AD and BE is the limit point or
pole around which all the figures are nested.
Golden Spiral
A very remarkable curve can be drawn using these nested Golden
Rectangles as shown in Figure 5. This curve passes through the
points C, F, H, … of Figure 4. These points divide the longer
sides of the Golden Rectangles in the Golden Section. Of course,
as will be seen later, a similar curve could have also been drawn
through three successive corners of each Golden Rectangle.
From the similarity of the nested rectangles, it is not hard to see
that the curve remains similar everywhere but for its size.
With the limit point O (Figure  4) as the origin, the equation of
the curve in polar coordinates is of the form
.qaer = (7)
Since the curve intersects any radial reference line infi-
nite times, one can define   r=1 at q = 0  arbitrarily, with
q   going from –¥  (at the origin) to ¥. As the value of  q
(measured in the counterclockwise direction) increases
Figure 4. Golden Rectangle
and its nesting.
Figure 5. Golden Spiral (left-
handed) in a set of nested
Golden Rectangles.
35RESONANCE ç September    2004
GENERAL   ç ARTICLE
by p/2, the curve (also the value of r) expands by a factor
÷
÷
ø
ö
ç
ç
è
æ +
=
2
15
/1 j . So the value of   
2
15ln2 +=
p
a defines the value
of the exponential growth rate with rotation. The curve spiral-
ling out in the counterclockwise direction, as shown in Figure 5
with the positive value of a mentioned above, is called a left-
hand spiral. Figure 6 shows a right-hand spiral with
2
15ln2 +-=
p
a .   Notice that the nesting of the Golden
rectangles in Figures 5 and 6 are carried out in opposite
directions. In Figure 6, the spiral is drawn through
three successive corners of each Golden Rectangle (as
mentioned earlier) and circumscribes the set of nested
rectangles.
Logarithmic Spiral
The spiral of the form r = egq, studied extensively by
mathematician Jakob Bernoulli, is called an equiangu-
lar or logarithmic spiral,  g > 0 indicates a left hand
spiral g < 0  a right hand spiral and g = 0 refers to a  unit
circle. Some remarkable properties of this logarithmic
spiral are mentioned here.
(i) At all points on the curve, the radius vector
OP makes an equal angle  a  to the curve (i.e.,
to the tangent to the curve at P) and hence
the name equiangular spiral. It can be readily
Figure 6. Golden Spiral
(right-handed) circumscrib-
ing a set of nested Golden
Rectangles.
Figure 7. Rectification of a
logarithmic or equiangular
spiral.
36 RESONANCE ç September   2004
GENERAL  ç ARTICLE
seen that gqa /1tan ==
dr
dr , i.e., ga 1cot -=  is a con-
stant (independent of the point P). For another ap-
proach to obtain the same result see Box 3.
(ii) The evolute of a logarithmic spiral is also a logarithmic
spiral. This implies that the locus of the centre of
Box 3.
(i) The property of equiangularity can be proved by the conformal (angle preserving) property of the
mapping by the analytic function w=ez   for the complex variable z . If  z = x + iy, then w = u + iv = R (cos
F  +  i sin F )  with  R = ex  and  F = y. So y = constant, i.e., lines parallel to the x-axis are mapped into F
= constant , i.e., radial lines in the u–v plane. The line, passing through the origin and making an angle a
to the x-axis in the x–y plane, i.e., with the equation y = (tan a)x, is mapped on to the logarithmic spiral
R = e (F/tan a). Thus by the conformal property of the transformation the radial lines makes equal angle a
to the logarithmic spiral R = e (F/tan a) at every point. For the curve R = egq, cot a  =  g.
(i) In polar coordiantes  (r, q) , the radius of curvature
2
22
2
2/32
2
2
qq
q
r
d
rd
r
d
dr
r
d
dr
r
-+
+
=
÷
ø
öç
è
æ
ú
û
ù
ê
ë
é
÷
ø
öç
è
æ
.
For  R = egq, gqgr e21+= . So the radius of curvature subtends a right angle at the pole and the equation
of the evolute comes out as ( ) gqgpg eer 2/-= .
(iii) ( ) ( )ò ò
¥-
+=+=ò= ÷
ø
öç
è
æq q
q
q dr
d
dr
rddrdss 2
2
22 .  For gqer =
 ( ),cos/1
2
a
g
g gq res =
+
=  since  .cot ag =
In 1645, Evangelista Torricelli obtained this result without using integral calculus, which was yet to arrive!
37RESONANCE ç September    2004
GENERAL   ç ARTICLE
curvature at all points of a logarithmic spiral forms
another logarithmic spiral. For proof, see Box 3.
(i i i) The pedal curve, i.e., the locus of the foot of the
perpendicular from the pole to the tangents of a loga-
rithmic spiral is another logarithmic spiral.
(iv) The caustic, i.e., the envelope formed by rays of light
emanating from the pole and reflected by the curve, is
also a logarithmic spiral.
(v) The length of the spiral from a point P (at a distance r
from the pole) to the pole is s =  r/cos a. In Figure 7,
s = PT. For proof, see Box 3.
(vi) The process of inversion by ( )rr /1® changes a loga-
rithmic spiral into its mirror image, i.e., into the same
spiral but of opposite hand.
Jakob Bernoulli was so enchanted with this ‘Spira Mirabilis’
(marvellous spiral), which retains its shape under so many
operations, that he wished to have this spiral engraved on his
tomb with the inscription ‘Eadem mututa resurgo’ (‘Though
changed, I shall arise the same’).  However, the engraver (a non-
mathematician, of course) made a mistake by engraving the
Archimedean spiral  r = gq [1]!
Address for Correspondence
A K Mallik
Depa!rtment of Mechanical
Engineering
Indian Institute of Technology
Kanpur  208 016, India.
Email: akmallik@iitk.ac.in
Suggested Reading
[1] E Maor, e: The story of a Number , Universities Press (India) Limited,
Hyderabad, 1999.
[2] http://www.goldenmeangauge.co.uk/golden.htm
[3] http://www.ma.utexas.edu/~lefcourt/SP97/M302/projects/lefc062/
FLOWERS.html.
[4] I Stewart, Nature’s Numbers – Discovering Order and Pattern in the
Universe , Phoenix Paperback, London, 1996
[5] P Ball, The Self-Made Tapestry – Pattern Formation in Nature, Oxford
University Press, Paperback, Oxford, 2001.
[6] http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
[7] C S Ogilvy, Excursions in Geometry, Dover Books, New York, 1990
[8] T A Cook, The Curves of Life, Dover Books, New York, 1979.


From natural numbers to numbers and

curves in nature – I

http://repository.ias.ac.in/93328/1/from_natural.pdf

From natural numbers to numbers and curves in nature – I

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