What Other Mathematical Significance Does The Golden Ratio Have.

What makes the Golden Ratio special?

How is the Golden Ratio found in The Great Pyramid

Calculation of Golden Ratio within Great Pyramid

The secrets of the Golden Ratio

within the Great Pyramid and

architecture in general

**The**

**Golden**

**Ratio**

The Golden Ratio is also known as the Golden Mean, The Golden Proportion or The Divine Proportion. This ratio is 1:1.618. This ratio originates from Leonardo Fibonacci's discovery of the Fibonacci Numbers, 800 years ago.

What are the Fibonacci Numbers?

The Golden Ratio has very peculiar properties with certain shapes. For example take a pentogram (star).

The vast majority of people prefer rectangles that have the Golden Ratio's proportions. The Golden Ratio was heavily used by The Greeks in their architecture for its pleasing proportions. A unique property of Golden Rectangles is that if you remove a square from a golden rectangle you are left with a Golden Rectangle. However the Golden Ratio is not just limited to rectangles, it can just compare 2 lines and it can be found in natural spirals and triangles.

How Do We Do It ?

This can be worked out using Pythagoras' theorem for right-angled triangles (a²+b²=c²). Pythagoras' theorem clearly states that if lengths of 2 sides are known, the third (sloping) side length can be found, but this only works for right angled triangles not the triangles on the Great Pyramid. As the diagram below shows the right-angled triangle is found by running a line from the centre of one of edges to the centre of the base, another line from there up to the apex and another line that connects the other 2 along the triangular face

What Other Buildings Have The Golden Ratio

b

c

a

The Great Pyramid is not the only structure, modern or ancient, to have Golden Ratio's proportions. The Golden Ratio has been found in buildings dates from 2500 years ago to the modern day, and in every era in between. Some of these buildings contain Golden Rectangles, other contain the pure ratio, in relationships between different parts of the building or a combination of both.

Taj Mahal

Notre Dame

The Parthenon (432BCE, Greece)

In order to find out if a square based pyramid has the Golden Ratio you require 2 key measurements, the length of one side of the base and the sloping height along one of its faces. No one has ever measured the sloping height of the Great Pyramid, but there is a way around this.

The equation from the previous slide can be easily translated to the Great Pyramid's dimension. The base length has to divided by 2 from 230.33m to 115m to form the pronumeral a. a²(115m²) + b²(146.59m²) = c²(186.41782459m²). If you take sloping height (represented by pronumeral c) and half the base (represented as pronumeral a) and divide c by a it equals 1.61870207606. This is the Golden Ratio correct to 3 decimal places

(1345, France)

(1653, India)

The Parthenon's designer incorporated the Golden Ratio in all rectangles in the structure, as well as the ratio of the roof to the frieze and the ratio of the width of the columns and the gaps between them

Notre Dame's architects incorporated the Golden Ratio in the ratio of the height of the first section/floor to the second and third. As well the ratio of the tower to the tower and the gap between the two has Golden Ratio porportions

The Taj Mahal incorporates the Golden Ratio in the ratio of the height of the arches to the width of the arches.

United Nations HQ

The ratio of the red line to the green is the golden ratio, as well as the ratio of the blue line to the red and purple line to blue.

The Golden Ratio is called this because shapes that have this ratio are the most pleasing to look at. An example of this is when a person is given a selection of rectangles and asked which one they find the most pleasing.

Reflection & Evaluation

The part of the Great Pyramid containing the Golden Ratio, could have only been designed using Pythagoras' theorem, but Pythagoras was born 1000 years after completion of the pyramid. This begs the question did the Egyptians knowingly create the Great Pyramid knowing of or using the Golden Ratio?

Reflection & Evaluation

I am of the opinion that the Ancient Egyptians did not knowingly incorporate the Golden Ratio into the design of The Great Pyramid. The Golden Ratio has not been found in any of the records relating to pyramid design, known as the "Pyramid Texts". This has lead many, including myself, to believe that the Ancient Egyptians were unaware of the Golden Ratio. Many different pyramid designs were trialled by the Ancient Egyptians and The Great Pyramid was the culmination of these trials. It happens to be that the pyramid design with the best looking proportions (which happened to contain the Golden Ratio) was applied to the Great Pyramid. This is justified as the Golden Ratio is known to have proportions that appeal to the majority of people. As well, the theories that would have enabled the Ancient Egyptians to knowingly incorporate the Golden Ratio would only be discovered and documented thousands of years later. This means that the only conclusion that can be drawn from this information is that the Golden Ratio, being found as accurately as it in the Great Pyramid, is a fluke, in terms of the mathematics associated.

The architect of UN HQ, Charles-Édouard Jeanneret (better known as Le Corbusier) was very interested in the Golden Ratio in architecture and incorporated it in the ratio of the length and width of the tower and the 3 "blocks" (rectangles) that make the tower are golden rectangles.