Month: February 2015

Exploring Mathematics: The Golden Raito in architecture and nature ( By Kevin Xu)

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Exploring Mathematics: The Golden Ratio in architecture
and nature

The Golden Ratio in architecture can be found in many famous
buildings around the world. These buildings include:

·        
The Parthenon in Greece

·        
Notre Dame in Paris

·        
Taj Mahal in India

The Parthenon in Greece uses the Golden Ratio in the height
of the columns, the dividing line of the root support beam and the width of the
columns. The golden ratio of the columns is the structural beam on top of the
columns which is in proportion of the golden ratio to the height of the
columns. The dividing line of the root support beam is in proportion to the
golden ratio as the structural beam on top of the columns has a horizontal
dividing line. The width of the columns is in proportion formed by the distance
from the center line of the columns to the exterior columns.

The Notre Dame in Paris have three major architecture lines.
They are commonly known as the Red, Blue and green lines. The red line is the
vertical line height of the base at ground level in ration with the top of the
first and second level. The blue line is the vertical height of the base at
ground level proportion to the top of the second and third floor. The green
line is horizontal width outside of the left top section proportion the inside
of the top right section and the outside of the top right section.

The Taj Mahal in India is another example of the golden
ratio. It itself displays the ratio proportion in the width of the grand
central arch to its actual width and also the height of the windows inside the
arch to the height of the main section under the domes.

The Golden Ratio in nature can be found in various types of
plants. Examples of them are:

·        
Romanesque Broccoli

·        
Hurricanes

·        
Sunflowers                        

The Romanesque Broccoli has always had a spiral starting
from the center and all the florets are formed around the spiral thus creating
the Fibonacci spiral which consist of a series of smaller arcs with a radii
that will imitate the Fibonacci series.

A Hurricane is also an
example of the golden ratio as it has the golden ratio spiral formed inside the
hurricane from the center.

The sunflower is the
most common example of the golden ratio. The spirals in the sunflower use the Fibonacci
series to create their seed heads.Those seeds should come out golden!

 

 

 #exploremaths

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5 Unique Examples of the Golden Ratio in Life (Josh Luong)

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1. Research 2-3 examples (architecture)

The Taj Mahal: The Taj Mahal is located in India and is a
Mausoleum made from white marble. The design of the building is shown to use
the golden ratio in the width of the grand central arch and the heights of the
windows inside the arch shown in the picture below.

Notre Dame: The Notre Dame is located in Paris, France and
is a historic Catholic cathedral. The golden ratio is shown in the height of
the building and also seen in the structure of the columns at the top. Each of
the 3 golden ratio used is difference in proportion to each other.

The Parthenon: The Parthenon is
located in Athens, Greece and was a temple, which was dedicated to a Greek
Goddess, Athena. Even if the Parthenon was constructed with the golden ratio
proportions, people argue whether if the ratio was intended or whether it a
mere coincidence.  The golden ratio
can be seen across all the parallel pillars making 1 golden rectangle.

2. Research 2-3 examples (nature)

Spiral Galaxies: The Golden ratio has no limits and
can see seen in the spiral of the galaxies, which include our milky way. 

The Human Face: Even if
everyone’s face is unique, the average human face tends to be close to the
golden ratio. The eyes, lips and forehead all resemble the golden ratio.  Apparently the closer our faces look
like “phi” the more attractive we look. Very faceinating!

#exploremaths

Examples of the Golden Ratio in nature and architecture

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1) Golden rectangles in architecture
Notre Dame

The Notre Dame in Paris expresses the Golden ratio in the design of the structure. The building is quite asymmetrical which makes it difficult measure the ratio.

The United Nations Secretaria

the United Nations building conveys the idea of the Golden ratio and was completed in 1952. the structure was constructed by Oscar Niemeyer and Le Corbusier.

Taj Mahal

The Taj Mahal was completed in 1648 and was recognise as the Divine Proportion. This was because of the use of the Golden Ratio thorugh out the many features of the building. 

2) Golden spiral in nature

Spiral Galaxyies

In 1925, astronomers realized that the spiral in the milky way was in respect of the Golden Ratio. 

Hurricanes 

Pine cones 

The spiral arrangement of the seed pods on a pine cones portrays the Golden Spiral one after another. 

#exploremaths 

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Examples of Golden Rectangle #exploremaths

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#exploremaths

1. Research 2-3 examples (architecture):

The Parthenon uses the golden ratio to make up height and length of the building, including the columns that hold it up. Although it is assumed to use the golden rectangle’s proportions, it is debated that The Parthenon was not built based upon the golden rectangle or spiral.

The Notre Dame is known to have a number of golden ratio proportions in it’s main designs. The bottom is the biggest golden rectangle, and then the sizes gradually become smaller golden rectangles. The Notre Dame is a very clear example of the use of the golden rectangle.

The UN Secretariat Building, like The Parthenon, is also debated whether or not the golden rectangle was used. Although the clear rectangular shape of the building makes it highly likely to be built based on the golden ratio, it is also assumed to have used the proportions of the human body (based on Leonardo da Vinci’s Vitruvian Man).

2. Research 2-3 examples (nature):

The human body has a divine proportion. the entire height of a person (from head to toes) can have golden ratios. The golden section of the entire height would reach from the head to the fingertips. Then the golden ratio of that would be from the head to the navel etc. and it keeps going.

Sea shells also have the same dimensions, measuring from the outermost spiral growth, and gradually getter smaller as it closes into the innermost spiral growth.

In a dolphin’s body and fins, they also carry a similar dimension to the previous two examples. The length from the dolphin’s tail to the head corresponds to the section’s of the dolphin’s tail.

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The Presence of the Golden Ratio and the Golden Spiral throughout the World

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#exploremaths

When looked at from the front, the shape of the Parthenon appears to resemble a golden rectangle. This is interesting as construction of it had completed in 438 B.C.E but it wasn’t documented by the Greeks until 300 B.C.E. Many theories for this include that, as it is present in nature, the golden ratio is aesthatically pleasing to the human eye.

 

As seen above, the Taj Mahal is also known to form a golden rectangle.

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