By Darcy Wilson-Burgess 7G The Golden Ratio What is the Golden Ratio? • The Greek letter above is called “phi” and it represents the Golden Ratio.

• The Golden Ratio is an Irrational Number. This is because it has an infinite number of decimal places and it never repeats itself.

• The Golden Ratio is equal to: 1.61803398874989484820... (etc.) but we normally round this to 1.618.

• It is believed that the Golden Ratio makes a perfect shape that is very pleasing to the eye.

• Ancient civilizations discovered the Golden Ratio and because it makes appealing shapes it has been used for centuries in art, architecture and many other objects e.g. computer screens, music and picture frames etc.

• The Golden Ratio is also found naturally in the world e.g. in the shape of a person’s face and body, plants, the solar system, DNA and shells etc.

• The Golden Ratio is also called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion. How to work out the Golden Ratio (phi) •Phi is the ratio of the line segments that result when a line is divided in a certain way.

•The longer line a+b is divided into two parts a and b so that:

The ratio of the length of the larger line segment a to the length of the smaller line segment b

Equals

The ratio of the length of the entire line a+b to the length of the larger line segment a

So: a = a+b = 1.618… = phi

b a

This only happens when:

a+b is 1.618 times bigger than a, and a is 1.618 times bigger than b

Or

b is 0.618 smaller than a, and a is 0.618 smaller than a+b Phi in the Fibonacci Series •This is a numerical series that was discovered by Leonardo Fibonacci in the 12th century.

How it works:

Start with 0 and 1 and then each new number in the series is the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . .

The ratio of each pair of numbers in the series is approximately equal to phi (1.618. . .)

For example:

5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.

After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .

Phi appears in many geometric designs.

For example:

3 lines

• Take 3 equal lines.

• Lay the 2nd line against the midpoint of the 1st line.

• Lay the 3rd line against the midpoint of the 2nd line.

• The ratio of AG to AB is Phi.

Equilateral Triangle (3 sides)

• Insert an equilateral triangle inside a circle.

• Add a line at the midpoint of the two sides and extend that line to the circle.

• The ratio of AG to AB is Phi.

Square (4 sides)

• Insert a square inside a semi-circle.

• The ratio of AG to AB is Phi.

Pentagon (5 sides)

• Insert a pentagon inside a circle.

• Connect three of the five points to cut one line into three sections.

• The ratio of AG to AB is Phi. Phi in Geometry Golden Rectangles

• Using the Golden Ratio, you can make a perfect rectangle and it is called a Golden Rectangle:

a = a+b = 1.618…= phi

b a

• If you take a square out of a Golden Rectangle you are left with another smaller Golden Rectangle.

Golden Spirals

•Using Golden Rectangles you can make a perfect spiral and it is called a Golden Spiral.

3D Geometric Designs

•3 Golden Rectangles have been put together to make a 3D Geometric Design that has a golden ratio. Golden Shapes 1. Draw a square.

2. Extend two parallel sides.

3. Draw a line from the midpoint of one side of the square to the opposite corner.

4. Using the line you just drew as a radius, draw an arc between the two parallel lines.

5. Draw a perpendicular line from between the parallel lines from the intersection of the arc on the bottom line.

6. You now have a golden rectangle. This is how you can draw a Golden Rectangle: The Golden Ratio and

The Great Pyramid of Giza • There is evidence that when the Egyptians designed The Great Pyramid of Giza they used phi and the Golden Ratio.

• It is difficult to be accurate with the measurements because the outer smooth layer of the pyramid is no longer there.

However, there are many examples where the Golden Ratio can be seen:

• In the diagram below you can see how there is a phi ratio between the distance from the top to the middle of one side to the length of one half of the base. Therefore, the basic cross - section of the structure is a Golden Section.

•The angle of inclination of the pyramid face on its base is 89/55 = 1.61818 = phi.

•The pyramid is made up of lots of right-angle triangles that calculate to phi. This is because they are right-angle triangles with dimensions a, b and c, where c/b = b/a.

•The image below shows how the Golden Spiral fits inside the pyramid. What is amazing is how the beginning of the spiral is on top of the chambers. This is another great example of how phi and the Golden Ratio were used by the Egyptians in the design.

•There is also a Golden Spiral when you look at the position ratio of the Great Pyramid in comparison to the other pyramids: The Golden Ratio in Architecture Since ancient times, beauty and balance in architecture has been created by using The Golden Ratio.

3000 BCE

• Egyptians used the golden ratio when designing the Great Pyramid of Giza and today it is considered one of the Seven Ancient Wonders of the World.

Between 447 and 472BCE

• The ancient Greeks also used the golden ratio when building the Greek Parthenon.

• In the diagram you can see that by placing the Golden Rectangle over features of the design many of its design elements form a Golden Ratio.

Between 12th and 14th centuries

• Renaissance artists used the Golden Ratio in the design of the Cathedral of Notre Dame in Paris. The picture shows some of these Golden Ratios.

Modern Architecture

• Many buildings today are built using the Golden Ratio because it is so pleasing to the eye.

• This is the United Nations Building and once again the Golden Rectangles prove the Golden Ratio.

• The CN Tower in Toronto contains the golden ratio in its design. The ratio of the observation deck at 342 meters to the total height of 553.33 is 0.618 = phi. Bibliography

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[Accessed 03 09 2012].

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[Accessed 05 09 2012].

Au, K., 1999-2012. What is the golden ratio. [Online]

Available at: http://www.ehow.com/video_4979262_golden-ratio_.html

[Accessed 01 09 2012].

Deif, A., 27 Mar 2008 . Pi, Phi and the Great Pyramid. [Online]

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[Accessed 04 09 2012].

EmptyEasel, 2006-2012. A Guide to the Golden Ratio (AKA Golden Section or Golden Mean) for Artists. [Online]

Available at: http://emptyeasel.com/2009/01/20/a-guide-to-the-golden-ratio-aka-golden-section-or-golden-mean-for-artists/

[Accessed 05 09 2012].

Mathsisfun, 2012. Golden Ratio. [Online]

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[Accessed 01 09 2012].

Meisner, G., 2012. What is Phi?. [Online]

Available at: http://www.goldennumber.net/what-is-phi/

[Accessed 02 09 2012].

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[Accessed 01 09 2012].

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Available at: http://garyosborn.moonfruit.com/#/phi-at-giza-part-2/4516091048

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PhiPoint Solutions, 2012. Phi and the Golden Section in Architecture. [Online]

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PhiPoint Solutions, 2012. Phi, Pi and the Great Pyramid of Egypt at Giza. [Online]

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PhiPoint Solutions, 2012. What is Fibonacci?. [Online]

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R.McKenty, 2000 . DIMENSIONS AND MATHEMATICS. [Online]

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Wikipedia, 2012. Great Pyramid of Giza. [Online]

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Wikipedia, n.d. File:SimilarGoldenRectangles.svg. [Online]

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[Accessed 03 09 2012]. The Golden Spiral can also show the Golden Ratio in Architecture:

I do believe that the Egyptians used the golden ratio in and around the Great Pyramid of Giza. There are many ways that the Golden Ratio has been found:

•The basic cross-section of the structure is a Golden Section.

•The angle of inclination of the pyramid face = phi.

•The pyramid is made up of lots of right-angle triangles that calculate to phi.

•The Golden Spiral fits inside the pyramid.

•There is a Golden Spiral when you look at the position ratio of the Great Pyramid in comparison to the other pyramids.

However, the Egyptians used simple forms of measurement. It was more than likely that they did not know what the golden ratio was; but just found the design shape pleasing to the eye and maybe easy to build. Therefore, if you agree with this argument the Egyptians were just lucky in their design choices.

Unfortunately, for various reasons nobody will ever know the true answer to this question. The surface of the pyramid has worn away and is no longer smooth; this makes it impossible to measure it accurately and to have a real calculation of phi. Also, there are no hieroglyphs/documents to show us that the Egyptians knew about The Golden Ratio.

But I can’t help but think, that considering how many ways and places it has been found they must have known something.

Reflection and Evaluation

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# The Golden Ratio

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