Transcript Chapter Title

9 The Mathematics of Spiral Growth
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
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Excursions in Modern Mathematics, 7e: 9.1 - 1
Leonardo Fibonacci
In 1202 a young Italian named Leonardo
Fibonacci published a book titled Liber
Abaci (which roughly translated from Latin
means “The Book of Calculation”). Although
not an immediate success, Liber Abaci
turned out to be one of the most important
books in the history of Western civilization.
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Excursions in Modern Mathematics, 7e: 9.1 - 2
“The Book of Calculation”
Liber Abaci was a remarkable book full of
wonderful ideas and problems, but our story
in this chapter focuses on just one of those
problems–a purely hypothetical question
about the growth of a very special family of
rabbits. Here is the question, presented in
Fibonacci’s own (translated) words:
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Fibonacci’s Rabbits
A man puts one pair of rabbits in a certain
place entirely surrounded by a wall. How
many pairs of rabbits can be produced from
that pair in a year if the nature of these
rabbits is such that every month each pair
bears a new pair which from the second
month on becomes productive?
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Fibonacci’s Rabbits
We will call P1 the number of pairs of rabbits
in the first month, P2 the number of pairs of
rabbits in the second month, P3 the number
of pairs of rabbits in the third month, and so
on. With this notation the question asked by
Fibonacci (...how many pairs of rabbits can
be produced from [the original] pair in a
year?) is answered by the value P12 (the
number of pairs of rabbits in month 12). For
good measure we will add one more value,
P0, representing the original pair of rabbits
introduced by “the man” at the start.
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Fibonacci’s Rabbits
Let’s see now how the number of pairs of
rabbits grows month by month. We start
with the original pair, which we will assume
is a pair of young rabbits. In the first month
we still have just the original pair (for
convenience, let’s call them Pair A), so
P1 = 1. By the second month the original
pair matures, becomes “productive,” and
generates a new pair of young rabbits.
Thus, by the second month we have the
original mature Pair A plus the new young
pair we will call Pair B, so P2 = 2.
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Fibonacci’s Rabbits
By the third month Pair B is still too young to
breed, but Pair A generates another new
young pair, Pair C, so P3 = 3.
By the fourth month Pair C is still young, but
both Pair A and Pair B are mature and
generate a new pair each (Pairs D and E). It
follows that P4 = 5.
We could continue this way, but our
analysis can be greatly simplified by the
following two observations:
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Fibonacci’s Rabbits
1. In any given month (call it month N) the
number of pairs of rabbits equals the total
number of pairs in the previous month
(i.e., in month N – 1 ) plus the number of
mature pairs of rabbits in month N (these
are the pairs that produce offspring–one
new pair for each mature pair).
2. The number of mature rabbits in month N
equals the total number of rabbits in
month N – 2 (it takes two months for
newborn rabbits to become mature).
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Fibonacci’s Rabbits
Observations 1 and 2 can be combined and
simplified into a single mathematical
formula:
PN = PN – 1 + PN – 2
The above formula reads as follows: The
number of pairs of rabbits in any given
month (PN) equals the number of pairs of
rabbits the previous month (PN – 1) plus the
number of pairs of rabbits two months back
(PN – 2).
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Fibonacci’s Rabbits
It follows, in order, that
P5 = P4 + P3 = 5 + 3 = 8
P6 = P5 + P4 = 8 + 5 = 13
P7 = P6 + P5 = 13 + 8 = 21
P8 = P7 + P6 = 21 + 13 = 34
P9 = P8 + P7 = 34 + 21 = 55
P10 = P9 + P8 = 55 + 34 = 89
P11 = P10 + P9 = 89 + 55 = 144
P12 = P11 + P10 = 144 + 89 = 233
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Fibonacci’s Rabbits
So there is the answer to Fibonacci’s
question: In one year the man will have
raised 233 pairs of rabbits.
This is the end of the story about
Fibonacci’s rabbits and also the beginning
of a much more interesting story about a
truly remarkable sequence of numbers
called Fibonacci numbers.
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9 The Mathematics of Spiral Growth
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
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THE FIBONACCI SEQUENCE
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
…
The sequence of numbers shown above is
called the Fibonacci sequence, and the
individual numbers in the sequence are
known as the Fibonacci numbers.
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Fibonacci Sequence
You should recognize these numbers as the
number of pairs of rabbits in Fibonacci’s
rabbit problem as we counted them from
one month to the next.
The Fibonacci sequence is infinite, and
except for the first two 1s, each number in
the sequence is the sum of the two numbers
before it.
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Fibonacci Number
We will denote each Fibonacci number by
using the letter F (for Fibonacci) and a
subscript that indicates the position of the
number in the sequence. In other words, the
first Fibonacci number is F1 = 1, the second
Fibonacci number is F2 = 1, the third
Fibonacci number is F3 = 2, the tenth
Fibonacci number is F10 = 55. We may not
know (yet) the numerical value of the 100th
Fibonacci number, but at least we can
describe it as F100.
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Fibonacci Number
A generic Fibonacci number is usually
written as FN (where N represents a generic
position). If we want to describe the
Fibonacci number that comes before FN we
write FN – 1 ; the Fibonacci number two
places before FN is FN – 2, and so on.
Clearly, this notation allows us to describe
relations among the Fibonacci numbers in a
clear and concise way that would be hard to
match by just using words.
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Fibonacci Number
The rule that generates Fibonacci numbers–
a Fibonacci number equals the sum of the
two preceding Fibonacci numbers–is called
a recursive rule because it defines a
number in the sequence using earlier
numbers in the sequence. Using subscript
notation, the above recursive rule can be
expressed by the simple and concise
formula
FN = FN – 1 + FN – 2 .
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Fibonacci Number
There is one thing still missing. The formula
FN = FN – 1 + FN – 2 requires two consecutive
Fibonacci numbers before it can be used
and therefore cannot be applied to generate
the first two Fibonacci numbers, F1 and F2.
For a complete definition we must also
explicitly give the values of the first two
Fibonacci numbers, namely F1 = 1 and
F2 = 1. These first two values serve as
“anchors” for the recursive rule and are
called the seeds of the Fibonacci sequence.
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FIBONACCI NUMBERS
(RECURSIVE DEFINITION)
■ F1
= 1, F2 = 1 (the seeds)
■ FN
= FN – 1 + FN – 2 (the recursive rule)
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Example 9.1
Cranking Out Large
Fibonacci Numbers
How could one find the value of F100? With a
little patience (and a calculator) we could use
the recursive definition as a “crank” that we
repeatedly turn to ratchet our way up the
sequence: From the seeds F1 and F2 we
compute F3, then use F3 and F4 to compute
F5, and so on. If all goes well, after many
turns of the crank (we will skip the details)
you will eventually get to
F97 = 83,621,143,489,848,422,977
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Example 9.1
Cranking Out Large
Fibonacci Numbers
and then to
F98 = 135,301,852,344,706,746,049
one more turn of the crank gives
F99 = 218,922,995,834,555,169,026
and the last turn gives
F100 = 354,224,848,179,261,915,075
converting to dollars yields
$3,542,248,481,792,619,150.75
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Example 9.1
Cranking Out Large
Fibonacci Numbers
$3,542,248,481,792,619,150.75
How much money is that? If you take $100
billion for yourself and then divide what’s left
evenly among every man, woman, and child
on Earth (about 6.7 billion people), each
person would get more than $500 million!
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Excursions in Modern Mathematics, 7e: 9.1 - 22
Leonard Euler
In 1736 Leonhard Euler discovered a
formula for the Fibonacci numbers that does
not rely on previous Fibonacci numbers.
The formula was lost and rediscovered 100
years later by French mathematician and
astronomer Jacques Binet, who somehow
ended up getting all the credit, as the
formula is now known as Binet’s formula.
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Excursions in Modern Mathematics, 7e: 9.1 - 23
BINET’S FORMULA
N
N






 1   1 5
1 5 
FN  






 5   2 
 2  


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Excursions in Modern Mathematics, 7e: 9.1 - 24
Using a Programmable Calculator
You can use the following shortcut of Binet’s
formula to quickly find the Nth Fibonacci
number for large values of N:
Step 1 Store A  1  5 / 2 in the
calculator’s memory.
Step 2 Compute AN.
Step 3 Divide the result in step 2 by 5.
Step 4 Round the result in Step 3 to the
nearest whole number. This will
give you FN.

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
Excursions in Modern Mathematics, 7e: 9.1 - 25
Example 9.2 Computing Large
Fibonacci Numbers: Part 2
Use the shortcut to Binet’s formula with a
programmable calculator to compute F100.


Step 1 Compute 1  5 / 2. The calculator
should give something like:
1.6180339887498948482.
Step 2 Using the power key, raise the
previous number to the power 100.
The calculator should show
792,070,839,848,372,253,127.
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Example 9.2 Computing Large
Fibonacci Numbers: Part 2
Step 3 Divide the previous number by 5.
The calculator should show
354,224,848,179,261,915,075.
Step 4 The last step would be to round the
number in Step 3 to the nearest
whole number. In this case the
decimal part is so tiny that the
calculator will not show it, so the
number already shows up as a
whole number and we are done.
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Excursions in Modern Mathematics, 7e: 9.1 - 27
Why Fibonacci Numbers Are Special
We find Fibonacci numbers when we count
the number of petals in certain varieties of
flowers: lilies and irises have 3 petals;
buttercups and columbines have 5 petals;
cosmos and rue anemones have 8 petals;
yellow daisies and marigolds have 13 petals;
English daisies and asters have 21 petals;
oxeye daisies have 34 petals, and there are
other daisies with 55 and 89 petals
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Why Fibonacci Number Are Special
Fibonacci numbers also appear consistently
in conifers, seeds, and fruits. The bracts in a
pinecone, for example, spiral in two different
directions in 8 and 13 rows; the scales in a
pineapple spiral in three
different directions in 8, 13,
and 21 rows; the seeds in the
center of a sunflower spiral in
55 and 89 rows.
Is it all a coincidence?
Hardly.
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Excursions in Modern Mathematics, 7e: 9.1 - 30
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Excursions in Modern Mathematics, 7e: 9.1 - 31
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Excursions in Modern Mathematics, 7e: 9.1 - 32
9 The Mathematics of Spiral Growth
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
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Excursions in Modern Mathematics, 7e: 9.1 - 33
Golden Ratio
This number is one of the most famous and
most studied numbers in all mathematics.
The ancient Greeks gave it mystical
properties and called it the divine
proportion, and over the years, the number
has taken many different names: the golden
number, the golden section, and in modern
times the golden ratio, the name that we
will use from here on. The customary
notation is to use the Greek lowercase letter
 (phi) to denote the golden ratio.
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Golden Ratio
The golden ratio is an irrational number–it
cannot be simplified into a fraction, and if
you want to write it as a decimal, you can
only approximate it to so many decimal
places.
For most practical purposes, a good enough
approximation is 1.618.
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THE GOLDEN RATIO
1 5

2
  1.618
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The Golden Property
Find a positive number such that when you
add 1 to it you get the square of the number.
To solve this problem we let x be the desired
number. The problem then translates into
solving the quadratic equation x2 = x + 1. To
solve this equation we first rewrite it in the
form x2 – x – 1 = 0 and then use the
quadratic formula. In this case the quadratic
formula gives the solutions
  1  4 11  1
2
 1 
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2
2
5
Excursions in Modern Mathematics, 7e: 9.1 - 37
The Golden Property
Of the two solutions, one is negative
 1  5 / 2  0.618  and the other is the


golden ratio   1  5 / 2. It follows that 
is the only positive number with the property
that when you add one to the number you
get the square of the number, that is,
2 =  + 1. We will call this property the
golden property. As we will soon see, the
golden property has really important
algebraic and geometric implications.


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

Excursions in Modern Mathematics, 7e: 9.1 - 38
Fibonacci Numbers - Golden Property
We will use the golden property 2 =  + 1
to recursively compute higher and higher
powers of . Here is how:
If we multiply both sides of 2 =  + 1 by ,
we get
3 = 2 + 
Replacing 2 by  + 1 on the RHS gives
3 = ( + 1) +  = 2 + 1
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Fibonacci Numbers - Golden Property
If we multiply both sides of 3 = 2 + 1 by ,
we get
4 = 22 + 
Replacing 2 by  + 1 on the RHS gives
4 = 2( + 1) +  = 3 + 2
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Fibonacci Numbers - Golden Property
If we multiply both sides of 4 = 3 + 2 by ,
we get
5 = 32 + 2
Replacing 2 by  + 1 on the RHS gives
5 = 3( + 1) + 2 = 5 + 3
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Fibonacci Numbers - Golden Property
If we continue this way, we can express
every power of  in terms of :
6 = 8 + 5
7 = 13 + 8
8 = 21 + 13 and so on.
Notice that on the right-hand side we always
get an expression involving two consecutive
Fibonacci numbers. The general formula
that expresses higher powers of in terms of
and Fibonacci numbers is as follows.
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POWERS OF THE
GOLDEN RATIO
 N = FN + FN – 1
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Ratio: Consecutive Fibonacci Numbers
We will now explore what is probably the
most surprising connection between the
Fibonacci numbers and the golden ratio.
Take a look at what happens when we take
the ratio of consecutive Fibonacci numbers.
The table that appears on the following two
slides shows the first 16 values of the ratio
FN / FN – 1.
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Ratio: Consecutive Fibonacci Numbers
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Ratio: Consecutive Fibonacci Numbers
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Ratio: Consecutive Fibonacci Numbers
The table shows an interesting pattern:
As N gets bigger, the ratio of consecutive
Fibonacci numbers appears to settle down
to a fixed value, and that fixed value turns
out to be the golden ratio!
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RATIO OF CONSECUTIVE
FIBONACCI NUMBERS
FN / FN – 1 ≈ 
and the larger the value of N, the better
the approximation.
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9 The Mathematics of Spiral Growth
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
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Gnomons
The most common usage of the word
gnomon is to describe the pin of a sundial–
the part that casts the shadow that shows
the time of day. The original Greek meaning
of the word gnomon is “one who knows,” so
it’s not surprising that the word should find
its way into the vocabulary of mathematics.
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Gnomons
In this section we will discuss a different
meaning for the word gnomon. Before we
do so, we will take a brief detour to review a
fundamental concept of high school
geometry–similarity.
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Similar Figures
We know from geometry that two objects
are said to be similar if one is a scaled
version of the other. (When a slide projector
takes the image in a slide and blows it up
onto a screen, it creates a similar but larger
image. When a photocopy machine reduces
the image on a sheet of paper, it creates a
similar but smaller image.) The following
important facts about similarity of basic twodimensional figures will come in handy later
in the chapter:
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Similar Figures - Triangles
Two triangles are similar if and only if the
measures of their respective angles are the
same. Alternatively, two triangles are similar
if and only if their sides are proportional. In
other words, if Triangle 1 has sides of length
a, b, and c, then Triangle 2 is similar to
Triangle 1 if and only if its sides have length
ka, kb, and kc for some positive constant k.
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Similar Figures - Squares - Rectangles
Two squares are always similar.
Two rectangles are similar if their
corresponding sides are proportional.
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Similar Figures - Circles - Disks - Rings
Two circles are always similar.
Any circular disk (a circle plus all its interior)
is similar to any other circular disk.
Two circular rings are similar if and only if
their inner and outer radii are proportional
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Gnomon
In geometry, a gnomon G to a figure A is a
connected figure that, when suitably
attached to A, produces a new figure similar
to A.
By “attached,” we mean that the two figures
are coupled into one figure without any
overlap.
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Gnomon
Informally, we will describe it this way: G is
a gnomon to A if G & A is similar to A. Here
the symbol “&” should be taken to mean
“attached in some suitable way.”
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Example 9.3
Gnomons to Squares
Consider the square S. The L-shaped figure
G is a gnomon to the square–when G is
attached to S as shown, we get the square S’.
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Example 9.4
Gnomons to Circular
Disks
Consider the circular disk C with radius r. The
O-ring G with inner radius r is a gnomon to C.
Clearly, G & C form the circular disk. Since all
circular disks are similar, C’ is similar to C.
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Example 9.5
Gnomons to Rectangles
Consider a rectangle R of height h and base
b. The L-shaped figure G can clearly be
attached to R to form the larger rectangle.
This does not, in and of itself, guarantee that
G is a gnomon to R.
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Example 9.5
Gnomons to Rectangles
The rectangle R’ [with height (h + x) and base
(b + y)] is similar to R if and only if their
corresponding sides are proportional, which




by
b
requires that

.
h
h x
b y
 .
This can be simplified to
h x
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Example 9.5
Gnomons to Rectangles
There is a simple geometric way to determine
if the L-shaped G is a gnomon to R–just
extend the diagonal of R in G & R. If the
extended diagonal passes through the
outside corner of G, then G is a gnomon; if it
doesn’t, then it isn’t.
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Example 9.6
A Golden Triangle
Let’s start with an isosceles triangle T, with
vertices B, C, and D whose angles measure
72º, 72º, and 36º, respectively. On side CD
we mark the point A so that BA is congruent
to BC. (A is the
point of intersection
of side CD and the
circle of radius BC
and center B.)
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Example 9.6
A Golden Triangle
Since T’ is an isosceles triangle, angle BAC
measures 72º and it follows that angle ABC
measures 36º. This implies that triangle T’
has equal angles as triangle T and thus
they are similar
triangles.
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Example 9.6
A Golden Triangle
“So what?” you may ask. Where is the
gnomon to triangle T? We don’t have one yet!
But we do have a gnomon to triangle T’ – it is
triangle BAD, labeled G’. After all, G’ & T’
is a triangle similar to T’.
Note that gnomon G’ is an
isosceles triangle with
angles that measure 36º,
36º, and 108º.
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72-72-36 and 36-36-108 Triangles
We now know how to find a gnomon not
only to triangle T’ but also to any 72-72-36
triangle, including the original triangle T:
Attach a 36-36-108 triangle, G, to one of the
longer sides of T.
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72-72-36 and 36-36-108 Triangles
If we repeat this process indefinitely, we get
a spiraling series of ever increasing 72-7236 triangles.
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72-72-36 and 36-36-108 Triangles
It’s not too far-fetched to use a family
analogy: Triangles T and G are the
“parents,” with T having the “dominant
genes;” the “offspring” of their union looks
just like T (but bigger). The offspring then
has offspring of its own (looking exactly like
its grand-parent T), and so on ad infinitum.
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Golden Triangles
Example 9.6 is of special interest to us for
two reasons. First, this is the first time we
have an example in which the figure and its
gnomon are of the same type (isosceles
triangles). Second, the isosceles triangles in
this story (72-72-36 and 36-36-108) have a
property that makes them unique: In both
cases, the ratio of their sides (longer side
over shorter side) is the golden ratio.These
are the only two isosceles triangles with this
property, and for this reason they are called
golden triangles.
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Excursions in Modern Mathematics, 7e: 9.1 - 69
Example 9.7
Square Gnomons to
Rectangles
Consider a rectangle R with sides of length l
(long side) and s (short side), and suppose
that the square G with
sides of length l is a
gnomon to R.
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Excursions in Modern Mathematics, 7e: 9.1 - 70
Example 9.7
Square Gnomons to
Rectangles
If so, then the rectangle
R´ must be similar to R,
which implies that their
corresponding sides
must be proportional
(long side of R´ / short
side of R´ = long side
of R / short side of R):
ls l

l
s
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Excursions in Modern Mathematics, 7e: 9.1 - 71
Example 9.7
Square Gnomons to
Rectangles
After some algebraic manipulation the
preceding equation can be rewritten in the
form
2
l
l
    1
s
s
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Excursions in Modern Mathematics, 7e: 9.1 - 72
Example 9.7
Square Gnomons to
Rectangles
Since (1) l/s is positive (l and s are the lengths
of the sides of a rectangle), (2) this last
equation essentially says l/s that satisfies the
golden property, and (3) the only positive
number that satisfies the golden property is ,
we can conclude that
l

s
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Excursions in Modern Mathematics, 7e: 9.1 - 73
Rectangles-Squares and Gnomons
We can summarize all the above with the
following conclusion:
A rectangle with sides of length l and s (long
side and short side, respectively) has a
square gnomon if and only if
l
 .
s
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Excursions in Modern Mathematics, 7e: 9.1 - 74
Golden and Fibonacci Rectangles l   .
s
A rectangle whose sides are in the
proportion of the golden ratio is called a
golden rectangle. In other words, a golden
rectangle is a rectangle with sides l (long
side) and s (short side) satisfying l/s = . A
close relative to a golden rectangle is a
Fibonacci rectangle–a rectangle whose
sides are consecutive Fibonacci numbers.
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Excursions in Modern Mathematics, 7e: 9.1 - 75
Example 9.8
Golden and Almost
Golden Rectangles
This rectangle has l = 1
and s = 1/.
Since l/s = 1/(1/) = ,
this is a golden rectangle.
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Excursions in Modern Mathematics, 7e: 9.1 - 76
Example 9.8
Golden and Almost
Golden Rectangles
This rectangle has l =  + 1
and s = .
Here l/s = ( + 1)/ .
Since  + 1 =  2,
this is a golden rectangle.
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Excursions in Modern Mathematics, 7e: 9.1 - 77
Example 9.8
Golden and Almost
Golden Rectangles
This rectangle has l = 8 and s = 5. This is a
Fibonacci rectangle, since 5 and 8
are consecutive Fibonacci
numbers. The ratio of the sides
is l/s = 8/5 = 1.6 so this is not a
golden rectangle. On the other
hand, the ratio 1.6 is
reasonably close to so we will
think of this rectangle as
“almost golden.”
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Example 9.8
Golden and Almost
Golden Rectangles
This rectangle has l = 89 and s = 55 and is a
Fibonacci rectangle. The ratio of the sides is
l/s = 89/55 = 1.61818…, in theory this is not a
golden rectangle. In practice, this
rectangle is as good
as golden–the ratio
of the sides is the
same as the golden
ratio up to three
decimal places.
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Excursions in Modern Mathematics, 7e: 9.1 - 79
Example 9.8
Golden and Almost
Golden Rectangles
This rectangle is neither a
golden nor a Fibonacci
rectangle. On the other
hand, the ratio of the sides
(12/7.44 ≈ 1.613) is very
close to the golden ratio.
It is safe to say that, sitting
on a supermarket shelf,
that box of Corn Pops
looks temptingly golden.
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Excursions in Modern Mathematics, 7e: 9.1 - 80
Golden Rectangles
From a design perspective, golden (and
almost golden) rectangles have a special
appeal, and they show up in many everyday
objects, from posters to cereal boxes. In
some sense, golden rectangles strike the
perfect middle ground between being too
“skinny” and being too “squarish.”
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Excursions in Modern Mathematics, 7e: 9.1 - 81
Golden Ratio Hypothesis
A prevalent theory, known as the golden
ratio hypothesis, is that human beings have
an innate aesthetic bias in favor of golden
rectangles, which, so the theory goes,
appeal to our natural sense of beauty and
proportion.
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9 The Mathematics of Spiral Growth
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
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Spiral Growth in Nature
In nature, where form usually follows
function, the perfect balance of a golden
rectangle shows up in spiral-growing
organisms, often in the form of consecutive
Fibonacci numbers. To see how this
connection works, consider the following
example, which serves as a model for
certain natural growth processes.
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Example 9.9
Stacking Squares on
Fibonacci Rectangles
Start with a 1 by 1 square. Attach to it a 1 by
1 square. Squares 1 and 2 together form a 1
by 2 Fibonacci rectangle. We will call this the
“second generation” shape.
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Example 9.9
Stacking Squares on
Fibonacci Rectangles
For the third generation, tack on a 2 by 2
square (3). The “third-generation” shape is
the 3 by 2 Fibonacci rectangle.
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Example 9.9
Stacking Squares on
Fibonacci Rectangles
Next, tack on a 3 by 3
square, giving a 3 by 5
Fibonacci rectangle.
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Example 9.9
Stacking Squares on
Fibonacci Rectangles
Tacking on a 5 by 5 square results in an 8 by
5 Fibonacci rectangle. You get the picture–we
can keep doing this as long as we want.
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Excursions in Modern Mathematics, 7e: 9.1 - 88
Example 9.9
Stacking Squares on
Fibonacci Rectangles
We might imagine these growing Fibonacci
rectangles as a living organism. At each step,
the organism grows by adding a square (a
very simple, basic shape). The interesting
feature of this growth is that as the Fibonacci
rectangles grow larger, they become very
close to golden rectangles, and become
essentially similar to one another. This kind of
growth–getting bigger while maintaining the
same overall shape–is characteristic of the
way many natural organisms grow.
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Excursions in Modern Mathematics, 7e: 9.1 - 89
Example 9.10 Growth of a “Chambered”
Fibonacci Rectangle
Let’s revisit the growth process of the
previous example, except now let’s create
within each of the squares being added an
interior “chamber” in the form of a quartercircle. We need to be a little more careful
about how we attach the chambered square
in each successive generation, but other than
that, we can repeat the sequence of steps in
Example 9.9 to get the sequence of shapes
shown on the next two slides.
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Excursions in Modern Mathematics, 7e: 9.1 - 90
Example 9.10 Growth of a “Chambered”
Fibonacci Rectangle
These figures depict the consecutive
generations in the evolution of the chambered
Fibonacci rectangle.
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Excursions in Modern Mathematics, 7e: 9.1 - 91
Example 9.10 Growth of a “Chambered”
Fibonacci Rectangle
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Example 9.10 Growth of a “Chambered”
Fibonacci Rectangle
The outer spiral formed by the circular arcs is
often called a Fibonacci spiral.
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Excursions in Modern Mathematics, 7e: 9.1 - 93
Gnomon Growth
Natural organisms grow in essentially two
different ways. Humans, most animals, and
many plants grow following what can
informally be described as an all-around
growth rule. In this type of growth, all living
parts of the organism grow simultaneously–
but not necessarily at the same rate. One
characteristic of this type of growth is that
there is no obvious way to distinguish
between the newer and the older parts of
the organism. In fact, the distinction
between new and old parts does not make
much sense.
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Gnomon Growth
Contrast this with the kind of growth
exemplified by the shell of the chambered
nautilus, a ram’s horn, or the trunk of a
redwood tree. These organisms grow
following a one-sided or asymmetric growth
rule, meaning that the organism has a part
added to it (either by its own or outside
forces) in such a way that the old organism
together with the added part form the new
organism. At any stage of the growth
process, we can see not only the present
form of the organism but also the
organism’s entire past.
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Gnomon Growth
All the previous stages of growth are the
building blocks that make up the present
structure. The other important aspect of
natural growth is the principle of selfsimilarity: Organisms like to maintain their
overall shape as they grow. This is where
gnomons come into the picture. For the
organism to retain its shape as it grows, the
new growth must be a gnomon of the entire
organism. We will call this kind of growth
process gnomonic growth.
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Example 9.11 Circular Gnomonic
Growth
We know from Example 9.4 that the gnomon
to a circular disk is an O-ring with an inner
radius equal to the radius of the circle.
We can thus have circular
gnomonic growth by the
regular addition of Orings.
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Example 9.11 Circular Gnomonic
Growth
O-rings added one layer at a time to a
starting circular structure preserve the
circular shape through-out the structure’s
growth. When carried to three dimensions,
this is a good model for the way the trunk of a
redwood tree grows. And this is why we can
“read” the history of a felled redwood tree by
studying its rings.
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Example 9.12 Spiral Gnomonic
Growth
The figure shows a diagram of a cross
section of the chambered nautilus. The
chambered
nautilus builds its
shell in stages,
each time adding
another chamber
to the already
existing shell.
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Example 9.12 Spiral Gnomonic
Growth
At every stage of its growth, the shape of the
chambered
nautilus shell
remains the
same–the
beautiful and
distinctive
spiral.
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Example 9.12 Spiral Gnomonic
Growth
This is a classic example of gnomonic
growth–each new chamber added to the shell
is a gnomon of the entire shell. The
gnomonic growth of the shell proceeds, in
essence, as follows: Starting with its initial
shell (a tiny spiral similar in all respects to the
adult spiral shape), the animal builds a
chamber (by producing a special secretion
around its body that calcifies and hardens).
The resulting, slightly enlarged spiral shell is
similar to the original one.
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Example 9.12 Spiral Gnomonic
Growth
The process then repeats itself over many
stages, each one a season in the growth of
the animal. Each new chamber adds a
gnomon to the shell, so the shell grows and
yet remains similar to itself. This process is a
real-life variation of the mathematical spiralbuilding process discussed in Example 9.10.
The curve generated by the outer edge of a
nautilus shell–a cross section is called a
logarithmic spiral.
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Complex Gnomonic Growth
More complex examples of gnomonic
growth occur in sunflowers, daisies,
pineapples, pinecones, and so on. Here, the
rules that govern growth are somewhat
more involved, but Fibonacci numbers and
the golden ratio once again play a
prominent role.
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