Month: February 2015

Mathematics in Music and Games (by Soha Rizvi)

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During the videos I learnt that mathematics was used to discover music
and is used to play music. I also learned that many games are played
on geometric shapes. This relates to things I already know because I
knew that chess is a game of mathematical strategy. It was surprising
to find out that mathematics is the music of the mind and music is the
mathematics of the heart. Also I found it surprising to find out that
all optical instruments were created through maths. It was difficult
to understand what it means when it said that a keyboard is divided
into 12 equal parts and that is linear algebra. It was also difficult
to understand what it meant when it said that Pythagoras discovered
that the octave had a ration of 2:1. #exploremaths

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Examples of the Golden Ratio in Nature & Architecture (by Alex Ho)

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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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The Golden Ratio appearing around the whole world, wherever you are (Terrence Wong)

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Examples of Golden Rectangle in Architecture

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. 








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.


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





DNA Molecules

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


Spiral Galaxies

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.  


Examples of the Golden Ratio – architecture and nature [Adam Tan]

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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


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.


#exploremaths Examples of the Golden Ratio in Nature and Architecture (By Michelle So)

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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

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

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

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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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