Download Fibonacci and the Golden Ratio

Fibonacci and the Golden Ratio
- Nifty Numbers 2: Convergence of sequences A Historic Note
Leonardo Pisano (1170–1250) is better known as
Fibonacci. Although he was born in Pisa, his family
soon moved to North Africa. He was educated by
the Moors at Bugia in present day Algeria and
therefore learned the Indian number system in the
form that it had been developed within Islam.
When Fibonacci eventually returned to Pisa, he was
welcomed at the court of the Holy Roman Emperor
Frederick II by the leading scholars of the day.
In 1202 Fibonacci published Liber Abaci, the book
Fibonacci
that introduced the Hindu-Arabic place-valued decimal system into Europe. This
was particularly valuable to merchants and accountants who found the Arabic
system much less cumbersome than the Roman system they had been using.
Fibonacci is well known to mathematicians for his famous sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … etc. in which the first two terms are ones
and, after that, each term is defined as the sum of the previous two terms.
Definitions
A sequence is a function whose domain is the set of all integers greater than or
equal to some specified integer (usually the set of positive integers).
Thus, Fibonacci’s Sequence is the infinite set of ordered pairs
(1, 1), (2, 1), (3, 2), (4, 3), (5, 5), (6, 8), (7, 13) … etc.
This is usually simplified as:
F1=1, F2=1, F3=2, F4=3 … etc.
or, more commonly as:
Fn = {1, 1, 2, 3, 5, 8, 13, …}.
It can also be defined using
Copyright 2007, Hartley Hyde
 F = 1,
1

 F2 = 1,
 F =F +F
n
n−1
n−2

Page 1 of 8
Fibonacci and the Golden Ratio
ClassPad Time
Switch on your ClassPad o and tap on the Spreadsheet icon
If your spreadsheet has data from a previous investigation, check if you need to
save this before you clear it from the screen.
To clear the screen, tap on the File menu and select New.
You will be asked if you are sure and you then tap OK.
Typing the column Headings
Tap on the cell A1. We are going to type the letter “n” into this cell.
To do this:
•
press the k button;
•
select the 0 keyboard;
•
tap the icon for the letter “n”;
•
check your typing in the formula bar and then tap the s key or press E.
Tap on the cell B1. We are going to type the label F(n) into this cell.
To do this:
•
select the 0 keyboard;
•
to get the capital letter F, tap the V key and then the F;
•
You can type the “(” using the buttons on your ClassPad
or you can move to the 0 keyboard to tap the “(” icon.
•
Select the icon for the letter “n” as before.
•
Then tap the closing bracket “)” much as you typed “(”.
check your typing in the formula bar and then tap the s key or press E.
Filling the Domain
In the column A we are going to list the first 50 natural numbers.
Tap on cell A2 and type the number 1 into the Formula Bar.
Then press E.
At cell A3 type the formula:
Then press E.
=A2+1
Keep the cell A3 highlighted and from the Edit Menu select Fill Range.
You will need to edit the Range in the dialogue box to read A3:A50 and press OK.
When you tap OK the dialogue box vanishes and you find that column A has been
filled with the sequence of natural numbers all the way down to A50. All you did
was to add one to the cell immediately above each entry.
Copyright 2007, Hartley Hyde
Page 2 of 8
Fibonacci and the Golden Ratio
The Fibonacci Numbers
In column B we are going to calculate 50 terms of the Fibonacci Sequence.
Tap on cell B2 and type the number 1 into the Formula Bar.
Then press E.
Tap on cell B3 and type the number 1 into the Formula Bar.
Then press E.
At cell B4 type the formula:
Then press E.
=B2+B3
Keep the cell B4 highlighted and from the Edit Menu select Fill Range.
You will need to edit the Range in the dialogue box to read B4:B50 and tap OK.
When you tap OK you are presented with the first 50
Fibonacci Numbers. Have a look. They get big very
quickly, don’t they? As you look down the list you will
see that many of the larger numbers have been
unnecessarily displayed using Scientific Notation. To
change this, tap on the column header A. From the
Edit Menu select Number Format and then choose
Normal 2. Now they look better.
Save your spreadsheet as “Fib1”.
That Ratio
In column C we are going to calculate the ratio that
each term makes with the previous term.
At cell C2 type the formula: =B3/B2. Then press E.
From the Edit Menu select Fill Range and edit the
Range in the dialogue box to C2:C50 and tap OK.
Look down the column. Notice that the numbers are
getting closer and closer to 1.618033989 …
Now where have you seen that number before?
Checkpoint
Copyright 2007, Hartley Hyde
Page 3 of 8
Fibonacci and the Golden Ratio
Convergence
When a sequence of numbers gets closer and closer to a particular value we say
that it converges.
In this case, the sequence {Fn+1 /Fn} converges to the Golden Ratio
ϕ = 1.618 ...
Drag your stylus across cells A2:A10. Copy and paste these cells to F2:F10.
Drag your stylus across cells C2:C10. Copy and paste these cells to G2:G10.
Drag your stylus across the range of cells F2:G10.
Tap on the graphing Tool Bar and choose the scatter
graph icon shown in this screen. This graphs the
function Fn against values of n on the horizontal axis.
This graph shows that the sequence oscillates above
and below the value of
ϕ until the oscillation is so
small we can no longer see it on the graph. If you
examine the values in column C carefully you will
notice that this oscillation continues right down until
the last decimal place your ClassPad can display.
Save your work as “Fib2”.
The Swiss Cross
The figure above is the same shape as the cross on a Swiss Flag and that used as
an icon for the Red Cross Society. You may not have time to construct the figure
during this lesson, we will leave that to your teacher’s judgment, however, here is
a brief explanation of how it can be constructed in case you wish to draw it later.
Draw a vertical line segment AB. Highlight AB, move to the Measurement Bar and
constrain its length to eight units.
Find the midpoint C, hide AB and join line segments AC and CB.
Copyright 2007, Hartley Hyde
Page 4 of 8
Fibonacci and the Golden Ratio
Find the midpoints D and E of line segments AC and CB.
Draw perpendiculars to AB at D and C.
Hide AC and CB and join AD, DC and CE. Draw line segments DF and CG.
To stabilise the figure, constrain the slope of CG to zero.
Use the Measurement Bar to constrain the lengths of DF and CG to match DC.
Draw the line FG. Continue placing and matching the lengths of line segments
until you have finished the figure.
Finally, you need to mess up the appearance of the cross by joining line segments
MI, EI and ME as shown below.
Euler’s Formula
From the Swiss Cross diagram we observe that:
In rt-∠ EMI,
In rt-∠ EDM,
In rt-∠ EAI,
DM = 2√5 units
DM = 2 units
AI
EM = 2√5 units
DE = 4 units
AE = 6 units
(
)
⇒ tan α + β =
1
=1
1
1
⇒ α + β = tan−1 
1
⇒ tan α =
2 1
=
4 2
1 
⇒ α = tan−1 
 2
= 2 units
⇒ tan β =
2 1
=
6 3
1
⇒ β = tan−1 
 3
1 
1
1
⇒ tan−1  = tan−1  + tan−1  … Euler’s Formula
1 
 2
 3
Copyright 2007, Hartley Hyde
Page 5 of 8
Fibonacci and the Golden Ratio
Wetherfield’s Notation
{n}= tan
−1
means the angle (marked
()
1
n
θ) that you find in a right-
angled triangle that has an adjacent side of n units
and an opposite side of 1 unit like this one.
This angle is so useful in computing science it is given its own notation.
Using this notation we can restate Euler’s Formula as {1} = {2} + {3}.
ClassPad Time
Switch on your ClassPad o and tap on the Spreadsheet icon
If your spreadsheet has data from a previous investigation, check if you need to
save this before you clear it from the screen.
Load the file Fib1.
Tap on the cell C1. We are going to type the letters “{Fn}” into this cell:
•
press the k button;
•
select the 0 keyboard, tap S and tap “{”;
•
tap I, tap the E icon to the left of the letter “a” and tap “F”;
•
tap the E icon to release upper case and tap the letter “n”;
•
tap S and tap “}”;
check your typing in the formula bar and then tap the s key or press E.
At cell C3 type the formula: =tan-1(1/$B3) then press E.
You can find “tan-1(” in the ( keyboard but don’t confuse it with “tanh-1(”.
Copyright 2007, Hartley Hyde
Page 6 of 8
Fibonacci and the Golden Ratio
Looking for Patterns
Tap on the cell D1. Type the label
You will find the
Σ{Fn}.
Σ icon under the 9 keyboard and the - option, however,
this icon expects you to insert a formula. It is better to use the 0 keyboard,
request the Greek M keyboard and then tap Caps which is just above the
.
When you typed the =tan-1(1/$B3) formula at C3 you may have noticed the $B3
address. This is an absolute address that forces the formula to look in Column B.
That means that we can copy and paste cells from column C into column D and
they will not lose their address references back to column B.
Then press E.
At cell D2 type the formula: =4*sum(D3:D50)
This formula will now add up any cells we paste into column D.
Start by copying cell C4 and pasting it into D4.
We have put {2} into D4.
Then copy C5 and paste it into D5.
We have put {3} into D5.
We know that {2} + {3} = {1}
Euler’s Formula
And from the Swiss Cross this extends to:
{2} + {3} = {1} = tan
−1
()= α + β = 45° = π4
1
1
So when you added {2} and {3} you got a quarter of
π.
Therefore when you typed the formula: = 4*sum(D3:D50)
you were ready to work out 4 × ({2}+{3}) which is π.
Save the file Fib3.
There are many more such patterns. Try these:
{2} + {5} + {8} =
{2} + {5} + {13} + {34} + {55}=
Checkpoint
You have found that from the Fibonacci Sequence we can derive both
φ and π.
Some folk are obsessed with the patterns yet to be discovered in this sequence.
Copyright 2007, Hartley Hyde
Page 7 of 8
Fibonacci and the Golden Ratio
Checkpoints
I wouldn’t recommend that students build the Swiss Cross during the lesson.
It may make an interesting homework exercise.
Checkpoint 1
At this point we are only looking for recognition of Golden Ratio
ϕ = 1.618 ...
and that should be prior learning.
Checkpoint 2
Students should obtain the answers
{2} + {5} + {8} =
π/4.
and {2} + {5} + {13} + {34} + {55}=
π/4.
They may forget the factor of 4 that was built into the formula.
We are not concerned about how these equations might be derived.
If students are curious, the derivations can be found on the net.
The important part of this arithmetic is that {n} converges much more quickly for
larger values of n.
This is very important for the development of computer algorithms.
The students are probably not ready for any detail about this, but we have
dislodged a little bell that may one day ring.
The main purpose of the lesson is to show that some nifty numbers like
ϕ and π
turn up in the most unlikely contexts.
Copyright 2007, Hartley Hyde
Page 8 of 8