Tag: exploremaths

Example of the Golden Ratio in architecture

1)    The Parthenon, an ancient temple located in Greece. The Golden Ratio can clearly be seen, dividing it  into 4 parts.

 

 

 

 

 

2)    Notre Dame de Paris, a historic Catholic cathedral, clearly demonstrates the Golden Rectangle, with rectangles going both vertical and horizontal.

 

 

 

 

 

 

 

 

 

3)     The Taj Mahal, a white marbled mausoleum located in India, shows off multiple Golden Rectangles, cutting the Taj Mahal into 16 rectangles. But factoring in the middle archway, cutting it into a further 12 sector.

 

 

Examples of the Golden Ratio in nature

1) Snail and nautilus shells have the same appearance of the golden spiral that forms from making continuous smaller rectangles out of larger ones. As shown in the picture.

 

 

 

 

 

 

 

 

 

 
2) Spiral galaxies also follow the Fibonacci sequence, where each spiral is a result of the ratio of the rectangle before it.

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Examples of Golden Rectangle in Architecture

1)     
Taj Mahal

 

The Taj Mahal, located in India, has the Golden Rectangle
located on the front of the building. Three rectangles are visible at the front which are all in the ratio of phi, which approximates 1.618. The three rectangles are visible in the picture above. 

 

 

 

 

 

 

2)     
Parthenon

The Golden Ratio is visible on the
Parthenon in the columns above of the Parthenon. Situated in Greece, the
Parthenon was later said by historians, that the architects of the Parthenon
had anything to do with the Golden Ratio.

 

3)     
Notre Dame

The Notre Dame, situated in Paris,
can be listed as one of the architectural buildings which demonstrates the
Golden Rectangle.  These rectangles can
be seen going vertically up and down the building.

 

 

 

 

 

 

                               

 

Example of Golden Spiral in Nature

1)      Hurricanes

 

The eye of the storm in a hurricane, is possibly similar to
saying the smallest part of the Golden Spiral. A hurricane is strongest in the
centre just like how the Golden Spiral spirals outwards and gest larger and
weaker.

 

 

 

 

2)     
DNA Molecules

         
Though an uncanny example, the full helix rotation of a DNA molecule
approximates 1.618, which we learnt is phi.

 

3)     
Spiral Galaxies

     Spiral
Galaxies, such as the Milky Way, have spiraling arms which equate to 12
degrees. These galaxies follow the Fibonacci sequence in which each spiral line
is a ratio of the one before it.  

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Phi in Architecture

The UN Tower

The west face of the UN Secretariat consists of three main window panels. It may not seem so close up, but when viewed from afar, it is evident that each panel is the Golden Rectangle.

This design was formed by Le Corbusier using his ‘Modulor’ system in 1943 and presented to the US in 1946, a year prior to the construction of the UN Secretariat.

The Pyramids of Giza

Currently the oldest monument with the use of phi in its architecture, the Pyramids of Giza incorporate the golden ratio correct to the fifth decimal place. The ratio of the slant of the pyramid to the distance from ground centre is 1.61804… The name given to such triangles is the Egyptian Triangle.

Toronto’s CN Tower

Phi can be found in the CN Tower in nothing more than its height. The full height of the tower (553.33m) to the height of the observation deck (342m) gives the golden ratio.

Phi in Nature

Bodies

Though it may not seem like it, the proportions of several areas of the body bring the result of phi. The ratio of the height of your entire body to the height of your naval to your head is, in fact, the golden ratio. Even animals reveal the golden ratio; each section of an ant in relation to another brings out phi.

Reproductive dynamics

Within a honey bee colony, when the number of females is divided by the number of males, the quotient is often very close to 1.618. Additionally, the family tree of any given bee will represent the Fibonacci sequence (which, of course, has a close relationship with the golden ratio). Males have one, female, parent, and females have both male and female parents. And so, when bees are asked to draw out their family trees, the number of bees they would receive would be 2, 3, 5, 8, 11 etc. respectively.

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Golden Ratio – Nature

1. Pinecones- The seed pods on a pinecone are in golden
ratio as each pair of spirals are in the cone, spiral upwards in different
directions, taking steps which will match a pair of consecutive Fibonacci
sequence. 

2. Tree Branches – the golden ration is shown through the
way tree branches split. When the tree grows old enough to grow branches, it
will split into two, then one of the two will split again, while the other is
to remain dormant.

3. Spiral Galaxies – the shape of the galaxies is
following the golden ratio as each of the spiral arms has a logarithmic spiral
of about 12 degrees. This relates to the golden ratio as it logarithmic spirals
are golden ratio spirals which appear in nature.

 

Golden Ratio – Architecture

1. Mobius Strip Temple – it is a Buddhist temple made out of
unique geometric shapes that has no orientation.

2. Tetrehedral Shaped Church – a complex pyramid in the
shape of a Tetrehedral, which is a convex polyhedron with four triangular
faces.

3. A mathematically- inclined cucumber in the sky – it is
a building in a shape of a cucumber with 41 floors and is 591 feet tall. In order
to create this tower, many mathematical equations and formulas were used.

The golden ratio can be seen almost every where around the world, if the golden ratio didn’t exist, it would be affecting many both in nature and architecture.

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​

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, built in 1648 by Ustad Ahmad Lahore for Mughal Emperor Shahjahan and his wife, is also known to form a golden rectangle from a frontal view

A more modern example of the golden ratio in architecture, the UN Building has 4 levels with clear glass, forming 3 golden rectangles.

​Pinecones display many golden spirals from the centre of seed outwards.

​

It is surprising, personally, to see that the golden ratio is present in inanimate, abiotic objects. The Milky Way, our home galaxy, has several golden spirals extending from the centre. 

​Even at a miniscule level, the golden ratio is known to exist. Each DNA molecule measures 34 angstroms long by 21 angstroms wide and as learnt in class, the higher the consecutive fibonacci numbers are, the closer and closer the ratio of these 2 numbers gets to phi.

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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!

 

 

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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!

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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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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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