Golden number
Can you imagine that mathematics makes things beautiful?
There are certain geometric shapes and ratios that crop up again and again in art and nature. Artists and architects use these mathematical elements to make their work pleasing to the eye, while nature presumably has her own reasons for including it. One idea that has endured since ancient times is the Golden Ratio, also known as Divine Proportion.
It's really golden thinking...isn't it?
The Golden ratio (symbol: Φ , the Greek letter "phi" )
is a special number equal to 1.618(approximately).
It is an irrational number that is a solution of the quadratic equation:
x²-x-1=0;
with a value of: φ =(1+√5)/2=1.618033989...
The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio , medial section , divine proportion ,divine section , golden proportion , golden cut and Golden number.
The idea behind the Golden ratio:
To find the golden ratio , we divide a line into two parts such that:
the long part divided by the short part ; which is also equal to
the whole length divided by the long part.
Here we take a straight line of length a+b; where a is the long part (red shaded portion) and b is the short part (blue shaded portion).Then the Golden ratio is represented algebrically as:
a+b/b=a/b=φ
These numbers can be applied to the proportions of a rectangle, called the Golden rectangle. This is known as one of the most visually satisfying of all geometric forms ; hence, the appearance of the Golden ratio in art. The Golden rectangle is also related to the Golden spiral, which is created by making adjacent squares of Fibonacci dimensions.
The Golden ratio also appears in all forms of nature and science. Some intersting places include:
Face beauty:
It appears around us in our daily lives, even in our aesthetic views. Studies have shown that when test subjects view random faces, the ones they deem most attractive are those with solid parallels to the Golden ratio. Faces judged as the most attractive show Golden ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows. The test subjects weren't mathematicians or physicists familiar with " phi" they were just average people, and the Golden ratio elicited an instinctual reaction.
Flower petals: The number of petals on some flowers follows the Fibonacci sequence. It is believed that in the Darwinian processes, each petal is placed to allow for the best possible exposure to sunlight and other factors.
Seed heads: The seeds of a flower are often produced at the center and migrate outward to fill the space.
As for example : sunflowers follow this pattern.
Pinecones: The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers.
Tree branches: The way tree branches form or split is an example of the Fibonacci sequence. Root systems and algae exhibit this formation pattern.
Shells: Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.
Spiral galaxies: The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of approximately 12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy.
Hurricanes: Much like shells, hurricanes often display the Golden spiral.
Fingers: The length of our fingers, each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.
Animal bodies: The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion.
DNA molecules: A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers.
Golden ratio myths & facs:
There are many misconceptions and misrepresentations about the Golden ratio. Some look too strongly for patterns and say it exists where it really doesn’t. Some whose goal is to spread golden ratio myth say it doesn’t exist where it really does, missing the obvious and often not stating what proportions appear instead. People on both sides often just repeat what they’ve heard rather than personally performing the analysis required to support their conclusions. Intelligence and education are not always factors in getting to the truth, as even Ph.D.’s in mathematics sometimes get it wrong. . Let’s look at some of the common points of confusion and debate, covering beauty, the Parthenon, the UN Secretariat Building, the Great Pyramid, Nautilus shell, use by famous artists (Da Vinci, Botticelli, Seurat, etc.) and other topics.
If you find out any incorrect information or know anything more about this , please write it in the comment section!
Can you imagine that mathematics makes things beautiful?
There are certain geometric shapes and ratios that crop up again and again in art and nature. Artists and architects use these mathematical elements to make their work pleasing to the eye, while nature presumably has her own reasons for including it. One idea that has endured since ancient times is the Golden Ratio, also known as Divine Proportion.
It's really golden thinking...isn't it?
The Golden ratio (symbol: Φ , the Greek letter "phi" )
is a special number equal to 1.618(approximately).
It is an irrational number that is a solution of the quadratic equation:
x²-x-1=0;
The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio , medial section , divine proportion ,divine section , golden proportion , golden cut and Golden number.
The idea behind the Golden ratio:
To find the golden ratio , we divide a line into two parts such that:
the long part divided by the short part ; which is also equal to
the whole length divided by the long part.
Here we take a straight line of length a+b; where a is the long part (red shaded portion) and b is the short part (blue shaded portion).Then the Golden ratio is represented algebrically as:
a+b/b=a/b=φ
These numbers can be applied to the proportions of a rectangle, called the Golden rectangle. This is known as one of the most visually satisfying of all geometric forms ; hence, the appearance of the Golden ratio in art. The Golden rectangle is also related to the Golden spiral, which is created by making adjacent squares of Fibonacci dimensions.
The Golden ratio also appears in all forms of nature and science. Some intersting places include:
Face beauty:
It appears around us in our daily lives, even in our aesthetic views. Studies have shown that when test subjects view random faces, the ones they deem most attractive are those with solid parallels to the Golden ratio. Faces judged as the most attractive show Golden ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows. The test subjects weren't mathematicians or physicists familiar with " phi" they were just average people, and the Golden ratio elicited an instinctual reaction.
Flower petals: The number of petals on some flowers follows the Fibonacci sequence. It is believed that in the Darwinian processes, each petal is placed to allow for the best possible exposure to sunlight and other factors.
Seed heads: The seeds of a flower are often produced at the center and migrate outward to fill the space.
As for example : sunflowers follow this pattern.
Pinecones: The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers.
Tree branches: The way tree branches form or split is an example of the Fibonacci sequence. Root systems and algae exhibit this formation pattern.
Shells: Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.
Spiral galaxies: The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of approximately 12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy.
Hurricanes: Much like shells, hurricanes often display the Golden spiral.
Fingers: The length of our fingers, each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.
Animal bodies: The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion.
DNA molecules: A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers.
Golden ratio myths & facs:
There are many misconceptions and misrepresentations about the Golden ratio. Some look too strongly for patterns and say it exists where it really doesn’t. Some whose goal is to spread golden ratio myth say it doesn’t exist where it really does, missing the obvious and often not stating what proportions appear instead. People on both sides often just repeat what they’ve heard rather than personally performing the analysis required to support their conclusions. Intelligence and education are not always factors in getting to the truth, as even Ph.D.’s in mathematics sometimes get it wrong. . Let’s look at some of the common points of confusion and debate, covering beauty, the Parthenon, the UN Secretariat Building, the Great Pyramid, Nautilus shell, use by famous artists (Da Vinci, Botticelli, Seurat, etc.) and other topics.
If you find out any incorrect information or know anything more about this , please write it in the comment section!
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