0.1 The Golden Ratio

0.1.1 Finding the Golden Ratio

In this unit, we will continue to study the relation between art and geometry. During therenaissance period, many artist, architects, and sculptors used ratios of distance in theirwork. An example of one of these special ratios is called the Golden ratio. The Golden ratiocan be found by dividing a segment into two pieces where the ratio of the longer piece tothe shorter is the same as the ratio entire line segment to the longer piece. (See illustrationbelow.)

ab =

a+ba

a = Shorter Piece of the line segmentb = Longer Piece of the line segmenta+ b = Entire line segment

In the next passage, we will find the value of the Golden ratio by solving above proportionfor a.

ab =

a+ba

a2 = b(a+ b)a2 = ab+ b2

a2 − ab− b2 = 0

Now, use the quadratic formula to solve for a

a =b+

√b2+4b2

2

a =b+

√5b2

2

a =b+b

√5

2Now, substitute back into the original ratio to get the Golden ratio.

ϕ = ab =

b+b√5

2

b = b+b√5

2 · 1b =

b(1+√5)

2 · 1b =

1+√5

2 = 1.61803......

It turns out that the Golden ratio is an irrational number meaning that the number does

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not repeat or terminate. Recall that the number π ”pi” is also an irrational number.

The Golden ratio was referred to as the divina proportione meaning divine proportion byLuca Parcioli in 1509. The famous physicist Johann Kelper later called it the sectio divinain 1610 which translates to divine section. The term Golden ratio or Golden section cameinto use around 1640. Many artists and composers have since used the Golden ratio in theirwork including famous artist Leonardo da Vinci and composer Bela Barlok.

0.1.2 The Parthenon and the Golden Rectangle

The Golden rectangle is a special rectangle where the ratio between the length and the widthis the Golden ratio ϕ

1.621 = 1.62 = ϕ

Typically, the value 1.62 is used for the Golden ratio to make computations easier. TheGreeks believe that Golden rectangles where more pleasing to the eye than other rectangles.The Golden Rectangle can be found in the measurement in many structures of ancientGreece. In Athens, buildings such as the Parthenon on the Arcopolis contains several Goldenrectangles.

Using the picture of the parthenon above, we a can draw a rectangle around the perimeter ofthe parthenon as shown in the next diagram. After making this rectangle, we will measurethe length and width of the rectangle.

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Now, let’s find the ratio between the length and the width of the rectangle drawn over theperimeter of the Parthenon. This value will be very close to the Golden ratio.

LengthofRectangleWidthofRectangle

= 7.54.6

= 1.6304

The next diagram illustrates how several Golden rectangles can be found in the Parthenon.You will later discover that this pattern of Golden rectangles is very similar to the NautilusSpiral.

0.1.3 The Golden Cross

A Golden cross is a cross that is constructed using two special ratios. We start by defininglength of the upper portion of the cross as T and the length of the lower portion as B. Wewill also define the overall height of the cross as T+B and the width of the cross as W. (Seeillustration below:)

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A cross with the measurements shown above is a Golden cross if the following ratios equalthe Golden ratio.

BT= ϕ = 1.62, B+T

B= ϕ = 1.62, and B+T

W= ϕ = 1.62

In the next example, we check to see if the cross is a Golden cross.

Example 1

Is the following cross shown below a Golden cross?

Solution:

Check all three ratios to see they are equal to the Golden ratio.

BT= 4.86cm

3cm= 1.62

B+TB

= 7.86cm4.85cm

= 1.62

B+TW

= 7.86cm4.85cm

= 1.62

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Example 2

Find the values of T, H, and W that will make the cross a Golden cross. (Round your answerto the nearest hundredth.)

Solution:

First find the value of T by setting the following ratio equal to the Golden ratio.8T= 1.62

T ( 8T) = 1.62(T )

8 = 1.62TT = 8

1.62T = 4.9in

Now, find H by using the fact that H = T + B. Therefore, H = 8 + 4.9 = 12.9 inches

Now, set the following ratio equal to the Golden ratio (1.62) to find W.12.9W

= 1.62

W (12.9W

) = 1.62(W )12.9 = 1.62WW = 12.9

1.62W = 8.0in

Example 3

Given the value of the top segment of a cross is T = 10 in Find the value of H and W thatwill make the cross a Golden cross. (Round your answer to the nearest hundredth.)

Solution:

First find the value of B by setting the following ratio equal to the Golden ratio.B10

= 1.62

10( B10) = 1.62(10)

B = 16.2 in.

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Now, find H by using the fact that H = T + B. Therefore, H = 10 + 16.72 = 26.72 inches

Now, set the following ratio equal to the Golden ratio (1.62) to find W.26.72W

= 1.62

W (26.72W

) = 1.62(W )26.72 = 1.62WW = 26.72

1.62W = 16.5 inches

0.1.4 The Golden Box

The Golden box is a rectangle solid whose length (L), width (W), and height (H) satisfy theGolden ratio. These Golden Ratios are define as:

WH

= LW

= ϕ = 1.62

Example 4

Find the width and length to the nearest tenth of a Golden box that has a height of 3centimeters. (Round your answer to the nearest hundredth.)

Solution:

First, find the value of the width by setting up a ratio using the Golden ratio.

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W3in

= 1.62

3in(W3) = 1.62(3in)

W = 4.86 in

Now, find the length of the box.L

4.9in= 1.62

4.9in( L4.9in

) = 1.62(4.9in)L = 7.9in

0.1.5 Art and the Golden Ratio

The Golden ratio can be found in the artwork of many famous artist. In particular, theGolden ratio can be in some of Leonardo Da Vinci works. The next picture is a photo ofLeonardo self portrait.

In the next picture, it is shown how a Golden rectangle can be constructed around Leonardoface in his self portrait. Notice that the width of the rectangle is 2.49 cm and the height ofthe rectangle of 4.02 cm.

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If you divide the height of the rectangle by the width of the rectangle, you will get theGolden ratio.

HeigthWidth

= 4.022.49

= 1.614

Other Examples of objects that use or contain the Golden Ratio or GoldenRectangles

Another classic example of an object that contains the Golden ratio or the Golden rectangleis the Nautilus Spiral. The Nautilus is a seashell that can be divided into several goldenrectangles. The illustration below shows how the Nautilus Spiral can be divided into at least8 golden rectangles.

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0.1.6 Exercises

1. Using a straightedge or a ruler, try to draw a Golden rectangle aroundthe Mona Lisa’s face. Confirm your results by dividing the longer sideby the shorter side.

2. Find the width and length to the nearest tenth of a Golden box that hasa height of 8 centimeters.

3. Find the height and length to the nearest tenth of a Golden box that hasa height of 6 centimeters.

4. Find the width and length to the nearest tenth of a Golden box that hasa height of 12 centimeters.

5. Find the value of T, H, and W that will make the cross a Golden cross.

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6. Find the value of T, H, and W that will make the cross a Golden cross.

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