Golden Ratio 5wc5g

The Golden Ratio - Nifty Numbers 1: The

α β formulae -

The Golden Ratio

A point C divides a line AB in the “extreme and median ratio” if the ratio of the longer portion (p) to the shorter (q) is the same as the ratio of the whole (p+q) to the longer portion (p). In our notation:

ϕ=

p+q p = p q

The “extreme and median ratio” is a translation of the term Euclid used for this ratio. It is also known as the Golden Ratio, Divine Proportion, Golden Section, Golden Mean and the mean of Phidias. Many artists down the ages have considered that a rectangle which has its length to height proportion in the Golden Ratio is the most harmonious shape for use in architecture, picture frames and body proportions. Phidias The ratio is usually given the Greek symbol

ϕ (Phi) in honour of the great Greek

sculptor Phidias who made extensive use of the ratio when deg buildings such as the Parthenon and the Propylaea on the Acropolos in ancient Athens. Phidias also carved the statue of Zeus at Olympia. This statue was one of the Seven Wonders of the World.

Unfortunately, his name is seldom mentioned without reference to his sudden disappearance when he was accused of pilfering some of the gold with which he was building the statue of Athena. The Parthenon

Copyright 2007, Hartley Hyde

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The Golden Ratio The Roots of a Quadratic Equation Consider the quadratic equation

ax2+bx+c = 0

Which can be written in the form

x + 2

bx c + =0 a a

Let the equation have roots (or solutions)

α and β,

bx c + ≡ (x − α )(x − β ) a a bx c ⇒ x + + ≡ x − (α + β )x + αβ a a ⇒x + 2

2

2

By equating coefficients, we get two relationships between the roots

α and β

and the coefficients of the quadratic a, b, and c:

α+β = − and

αβ =

b a

c a

… 1 Sum of Roots

… 2 Product of Roots

From the Golden Ratio Definition to a Quadratic Start with the definition

Dividing by

q gives us

Substitute

ϕ=

p q

ϕ=

p p+q = q p

p p + q q ⇒ϕ = p q ⇒ϕ =

ϕ +1 ϕ

⇒ ϕ = ϕ +1 2

Multiply by

ϕ

⇒ ϕ − ϕ −1 = 0 2

ClassPad Time Switch on your ClassPad o and tap on the Main icon J If your screen 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 Edit Menu and select Clear All and then Clear All Variables. At each step you will be asked if you are sure and you then tap OK.

Copyright 2007, Hartley Hyde

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The Golden Ratio We are going to ask the ClassPad to solve the equation

ϕ2−ϕ−1=0

In the Status Bar at the very bottom of your screen make sure that you are in Decimal Mode. If you can’t see the word Decimal, then tap on Standard. Press the k button and choose the 0 keyboard. Type the word “solve(”. Choose the Greek M keyboard. Look for

.

By moving between the 0 keyboard and the 9 keyboard finish typing the command line “solve(ϕ^2−ϕ−1=0,

ϕ)” and then press E.

Because you have solved a quadratic equation, there are two solutions. For the moment we will ignore the negative solution. By tapping on the arrow to the right of the solution

ϕ =1.6 you will see that the second solution is actually ϕ =1.618033989 …

For simplicity we will round the decimal value back to

ϕ =1.618, because like π,

these solutions are irrational and however many decimal places we use, we would still not have an accurate value. But then, who said we have to work in decimals? The Exact Solutions In the Status Bar at the very bottom of your screen tap on the word Decimal and it will change to the word Standard. Drag your stylus across the command line “Solve(ϕ^2−ϕ−1=0,

ϕ)” and drag the command

to the prompt square

. Press E.

Now you can see the two solutions accurately represented in their surd form as

– 5 1 + 2 2

Copyright 2007, Hartley Hyde

and

5 1 + 2 2

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The Golden Ratio In a regular algebra lesson we would now choose to let

1 2

α = +

5 2

= 1.618 …

1 2

β = –

and

5 2

=

– 0.618 …

However, the traditions of the Golden Ratio consistently require that we use

ϕ

instead of α for the positive solution. There is considerable disagreement about

what symbol to use for the second solution. Some writers change the sign of β so that β = +0.618 …, which can be confusing for students who are learning about the sum and products of quadratic roots. Some writers use an upper case φ for the positive solution and a lower case ϕ for the negative solution, however it is fairly popular to reverse this and use

phi

=ϕ

=

5 1 + 2 2

= 1.618 …

and

Phi =

φ

=

5 1 – 2 2

=

+ 0.618 …

Notice that Phi is defined as –β so that its value is positive. Even though ClassPad has shown that

ϕ

=

5 1 + 2 2

it has not yet stored the solutions as a variable called ϕ. To do this we must use the assignment command S from the 9 keyboard. Do not confuse the assignment command S with the implication symbol

⇒.

ClassPad copes better with the exact solutions if we use the simplify command. At the prompt

, type:

simplify(

Now use your stylus to highlight and drag the positive solution into place directly after the “(”, as shown. Then close the simplify command with “)”. From the 9 keyboard, look in the top row between the “,” and the “x” and you will find the assignment command S and then from the Greek M keyboard tap the variable name ϕ which, on the ClassPad, looks like this At the prompt

, type:

.

simplify(–(

Use your stylus to highlight and drag the negative solution into place directly after the second “(”. Then close the simplify command with “))”. Finish by asg this as shown on the next page.

Copyright 2007, Hartley Hyde

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The Golden Ratio From the 9 keyboard, choose the assignment command S and then from the Greek M keyboard tap the caps icon which is just above the EXE icon. From here you can choose the variable name

φ which, on the ClassPad, looks like this

.

Golden Ratio Arithmetic Now that ClassPad has been told what we mean by ϕ and φ we can start using these numbers to do arithmetic.

At the prompt

type ϕ + φ and press E.

You will find that ClassPad does some nice graphics shuffling but you won’t get many marks if you give that answer in a test.

To get ClassPad to actually work out a neat solution you have to type: simplify(ϕ + φ) and press E. Then it is more likely to do what you expect. We started with the equation

ϕ2−ϕ−1=0

From which we get a=1, b=–1 and c=–1. When we come to work out ϕ

– φ we should recall that ϕ = α and – φ = β.

In this space use the values of a, b and c to help you evaluate (ϕ−φ) and ϕφ

Copyright 2007, Hartley Hyde

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The Golden Ratio ing a Number Fact 2

Suppose you need to the equation φ = 1

– φ.

Then you substitute into each side, evaluate each side of the equation separately and check that you get the same answer for each side. 2

RHS = 1

LHS = φ

=  5 −1 5 −1 

 

 

2

2

 

=  5 − 2 5 +1

 

4

 

=  3− 5     2 

–φ

= 1−  5 −1  





2

= 2 − ( 5 −1)

2

  = 3− 5  

2

 

Thus LHS = RHS and the equation is verified. Use this space to these equations and use your ClassPad to check your solutions at each step.

ϕ=φ+1

ϕ2 = ϕ+1

Checkpoint

Copyright 2007, Hartley Hyde

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The Golden Ratio The Ancient Method The quickest way to show you how the ancients laid out the foundation shape for an ancient building is to use the ClassPad Geometry package G If your page 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. From the Draw Menu tap on Line Segment. Tap at point A and then at point B. But don’t waste time getting it straight. That will happen soon. From the View Menu tap Select. Use the Selection Arrow G to highlight the line segment AB. Click on the arrow u at the right of the Tool Bar to “go around the corner”. In the Measurement Bar, select the slope icon Q and write a zero over the top of whatever number appears in the window. Tap the tick R. The line should become horizontal and stay that way. In the Measurement Bar, select the distance icon and overwrite whatever appears in the window with the number 1. Tap the tick R. Nothing moves. You have simply told ClassPad that you are going to call the length of AB, 1 unit.

Before using the Selection Arrow to choose a new construct, always clear any old selections you don’t want to include by tapping in free space. Use the Selection Arrow G to highlight the line segment AB and the point A. From the Draw Menu select Construction and tap on Perpendicular. Put another perpendicular line through B. From the Draw Menu choose Line Segment, tap on the point B and tap on another point further up the perpendicular from B. This point will be called C.

Copyright 2007, Hartley Hyde

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The Golden Ratio Use the Selection Arrow G to highlight AB and BC. Click on the arrow u at the extreme right of the Tool Bar to “go around the corner”. In the Measurement Bar, select the congruency icon e. Tap the Tick R. The line segments should become equal. Adjust AD to the same length as AB using the same method. DC.

Use the Selection Arrow G to highlight BC. Take care not to select the perpendicular line. If several line segments overlap, try to tap at the middle of the one you want. From the Draw Menu select Construct and tap on Midpoint. The new point is E. From the Draw Menu select Circle. Tap on E and then on A. Imagine the ancient Greeks using a length of rope as a large com. Use the Selection Arrow G to highlight the perpendicular line through B and C. Avoid selecting BC by tapping higher than C. Then tap on the circle. From the Draw Menu choose Construct and tap on Intersection. This will define a new point F where the circle intersects BC produced. From the Draw Menu choose Line Segment. CF. BF. Use the Selection Arrow G to highlight the perpendicular line through B and C. From the Edit Menu choose Properties and tap on Hide. Do the same to the perpendicular through A. Use the Selection Arrow G to highlight AB. Click on the arrow u at the extreme right of the Tool Bar to “go around the corner”. In the Measurement Bar, select the length icon x. Check that AB is still 1 unit long. Then highlight the line segment BF. I think you have seen that number before.

Copyright 2007, Hartley Hyde

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The Golden Ratio How it Works We know that this is the method that ancient architects and artists used to find a Golden Ratio. Euclid described it in The Elements. You can still see the construction marks designers left on their sheets of papyrus. Some artists built the construction lines into the framework of their design and then painted features of the art-work along the construction lines. Here is a copy of your construction. You should be able to work these answers out from the diagram, but use your ClassPad to check your answers.

How long is the line segment BE? _____________ the line ED. In the right-angled ECD, How long is the line segment EC? ______________

How long is the line segment CD? ______________ Use the Pythagoras Theorem to find the length of ED.

Keeping your answer in surd form how long is the line segment ED? ______________

How long is the line segment EF? ______________

How long is the line segment BF? ______________ Checkpoint

Copyright 2007, Hartley Hyde

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The Golden Ratio Checkpoints Checkpoint 1 Evaluate (ϕ − φ)

Evaluate ϕφ

=α+β

= − αβ

= – b/a

= – c/a

= – (–1/1)

= – (–1/1)

=1

=1

ϕ=φ+1 RHS

ϕ=φ+1

=φ+1

LHS

RHS

=ϕ

5 1 = + 2 2

=1−β = α + β −β =α =ϕ

=

5 −1 + +1 2 2

=

5 1 + 2 2

= LHS

= LHS

Thus LHS = RHS

ϕ2 = ϕ+1 RHS

= φ+1

= ϕ+1

ϕ2 = ϕ+1 LHS

= α – αβ = α(1−β) = α2 = ϕ2

 5 1  5 1  =  +  +   2 2  2 2 

RHS

 5 1 =  +  +1  2 2  5 1 2 = + +   2 2 2  3+ 5  =   2 

6 + 2 5  =   4   3+ 5  =   2 

Thus LHS = RHS

= LHS

Checkpoint 2

1 2 1 CE = 2 BE =

CD = 1

Copyright 2007, Hartley Hyde

5 2 5 EF = 2 5 1 BF = + =ϕ 2 2 EF =

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