Golden Ratio The Golden Ratio or Golden Mean is when something is perfectly proportioned. It is a special number approximately 1.618 (decimals continue on in a random sequence), it is also known as phi. For example: a rectangle that is perfectly proportioned is when the length is divided by the width, it will equal phi (length / width = phi OR width x phi = length). A golden rectangle will also enable that when you take a square out, another rectangle will from and so on.

This creates a spiral that appears

in other perfectly proportioned

objects (Golden Spiral). The Golden Ratio In Relation To The Great Pyramids The Golden Ratio is found in the triangle made by the height, sloping height and half base of The Great Pyramid.

The triangle shown is perfectly proportioned by the rules of the golden ratio.

Did the Egyptians know about the Golden Ratio? A perfectly proportioned rectangle is found by:

length / width = phi

OR

width x phi = l,

so a perfectly proportioned triangle would be found by:

base / perpendicular height = phi

OR

(base of pyramid / 2) x phi = sloping height (apothem).

OR

sloping height / base of pyramid / 2 = phi One of the special things about phi, is that

phi + 1 = squared phi The simplest equation to find phi is

(1 + squared 5) / 2 = phi Were they capable of the complicated equations to create a perfectly proportioned pyramid? The dimensions of The Great Pyramid are (approximately) :

base = 230.37 metres

sloping height = 186.54 metres

perpendicular height = 146.73 metres

phi = 1.618 (approximately)

so:

230.37 / 146.73 = 1.570 (approx)

base / perpendicular height = phi

OR

(230.37 / 2) x phi = 186.373

(base of pyramid / 2) x phi = sloping height

OR

186.54 / (230.37/ 2) = 1.619

sloping height / base of pyramid / 2 = phi All these answers are very close to what they are meant to be. The pyramid was therefore built by the Egyptians to perfect proportion, whether they knew it or not. Note: measurements in this picture are in feet. By Nia Bickham The Golden Ratio, Golden Spiral and Fibonacci Number System appear in many forms of nature. Maybe this is why nature is so beautiful? Many people believe that nature uses these things as a perfect growth pattern, but others believe that nature is purely just perfectly proportioned.

Golden Ratio, Golden Spiral or Fibonacci Number Sequence in:

- pine cones

- sunflower seed pattern

- pineapples

- snail shells

- nautilus shells

- flowers

- plant leaf formations

- Brassica Romanesco (broccoli/cauliflower vegetable) Golden Ratio, Golden Spiral & Fibonacci Number Sequence

In relation to nature The Fibonacci Number Sequence is a infinite string of numbers where the next number is found by the sum of the previous two numbers.

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

It relates to the Golden Ratio and Golden Spiral in many ways and appears in many different forms in nature.

For example, each square in the Golden Rectangle, the area is a Fibonacci number.

Also, the ratio to a Fibonacci number and the one previous is approximately 1.618 - the Golden Ratio. Fibonacci Number Sequence Pine Cones Pine cones grow in spirals starting from the base, growing around till they reach the top. A set of spirals grow in each direction (2 sets of spirals). The number of spirals in the sets will be pairs of consecutive Fibonacci numbers. This occurs in all pine cones because the ratio of pairs of consecutive Fibonacci numbers is approximately the golden ratio. Plants use the golden ratio for growing patterns because it is a irrational number, so the parts in the plants will always fit in and never overlap. Sunflowers also use the Fibonacci numbers for their seed formations. Hundreds of seeds are in the middle of the flower with none overlapping and they all fit. This is for the same reason as the pine cones. If you look closely you will also notice the Fibonacci spirals reflected in the seeds. Sunflowers Pineapples Pineapple have three sets of spirals, with the number of arms in each being consecutive Fibonacci numbers. Snail & Nautilus Shells Snail and nautilus shells are pretty much in the perfect shape of the Golden Spiral. Flowers Many flowers have a Fibonacci number as their petal count: 8 2 5 Even petal layers found in roses have consecutive Fibonacci number petal count. You can also find the golden spiral in roses. Plant Leaf Formations And many other random things with the Golden Spiral. Plants use the Golden Ratio to align their leaves up in a way that none of them block the sunlight from each other. In this plant each leaf is approximately 1.618 of a turn away from each other. Also all the leafs numbered with a Fibonacci number all on the same side. In this plant, if you count clockwise starting at the bottom, you can number the leaves (shown). Starting at 1, count following the leaf numbers until you reach a leaf on top of the one you started with, noting how many times you reach a leaf on the same side as leaf 1 (3 times). Then count the number of leaves you passed (5 leaves). Count from the leaf you found directly on top of leaf 1 anti-clockwise, noting how many times you reach a leaf on the same side as the leaf you just started with (2 times). These numbers 2, 3, 5 are consecutive Fibonacci numbers. This patterns occurs in more plants than you would think including the African Violet (shown to the left; computer generated). Many other aspects of nature have the Fibonacci Spiral Brassica Romanesco The total number of every single floret on this vegetable is a Fibonacci number. This occurs in all Brassica Romanescos. They are a Italian cross between cauliflower and broccoli. A top view of this shell will a Fibonacci spiral going around and down. In these succulent the parts are spiraling out in Fibonacci Spirals but also in 5 arms - Fibonacci number. Succulents In this succulent the leaves in the different layers are consecutive Fibonacci numbers. In all the succulents shown the number of spirals are Fibonacci numbers. Whirlpools, the galaxies and cyclones are made in the pattern of Fibonacci Spirals. Reflection & Evaluation I think the Egyptians did not know about the Golden Ratio but rather had an eye for Maths and Proportion. In the Egyptian culture there are many aspects that have to be done perfectly or in a particular way (e.g. funerary practices). This attribute may have helped the Egyptians build their pyramid. I believe it is not possible for them to solve the complicated equations of the Golden Ratio while using symbols as numbers and hands as units of measurement. I don't think it was a coincidence either, but they had just worked out the almost perfect proportion in a different method that is not used today. A simpler, but less precise method was used=, using the unique Egyptian tools. The modern man has now discovered the Golden Ratio present in the pyramids, years after the Egyptians had already found their perfect proportion. The Ancient Egyptians Maths knowledge was astonishing, but they simply did not have the skills to apply the modern Golden Ratio.

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# Golden Ratio

In relation to the Great Pyramid

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