The Magic of Symmetry (Terrence Wong)

The Magic of Symmetry

Today in our class, we studied the different types of
symmetry and how we can apply it through our irregular drawings. These
different types of symmetry included: rotational symmetry, reflectional
symmetry, scale symmetry and translational symmetry which is better known as a tessellation.

The most common type of tessellation which can be found in
nature is the beehive. But as an additional piece of work, we were asked to
find another example. My research has led to the tessellation of the pineapple.
The outer skin of the pineapple is constructed by pentagonal shapes. Very much
like the beehive which is constructed of the shape hexagon. This lesson has
made me realise how mathematics is all around us, especially in places as
normal as in nature. Many times we overlook the simplicity of nature and don’t
realise how maths can tie in with all of this. An example was shown in class. A
simple 10 by 10 square can be dissected into quarters and each time, a scale of
the same shape appeared and was symmetrical to the previous shape. As you
complete the shape, you can connect the halfway points to the centre and get a
golden spiral. This just shows how simple something can be but yet consists of
something so complex. This lesson has made me really think about what is around
us and how ‘beautiful’ it is.

#exploremaths

The Golden Ratio (by Erik Willison)

#exploremaths
The golden ratio is found throughout architecture and nature. Examples of this are the Notre-Dame Cathedral and the Great pyramids. In nature we can see it in sunflowers, plants with spiral leaves and even in spiral galaxies.

Firstly architecture, The Notre-Dame Cathedral was built way back in 1163 and was designed by some random guy that was of great importance to somebody; anyway, this spectacular work of art has golden rectangles scattered throughout it. the three primary examples are:

1. The ratio of the height of the first floor to the height of the second floor (starting from the top of the first floor) is the golden ratio, or pretty close to it (outlined in red)

2. The height of the second floor to the height of the third floor is a golden ratio (outlined in blue)

3. The ratio of the length of the top left box including the space between it and the top right one to the length of the right one by itself is another golden ratio.

My second example of the golden spiral in architecture is the Great Pyramids of Giza. Each of these pyramids use the golden section. The diagram below is so good that it is basically self-explanatory, but for those who are lost I shall explain. Inside this pyramid you can see a triangle reaching from the centre of one edge to the centre of the base and up to the very top of the pyramid. The Egyptians built this pyramid with dimensions that just so happen to be able to be simplified into the diagram below where Phi is the hypotenuse. The dimensions that the Egyptians used were:

– A base of 230.4m

– An estimated original height of 146.5m

The ratio between these two numbers is 0.636. Now I know that 0.636 is not the golden ratio and may seem quite far off 0.618etc.; but the height only needed to be 0.0376m taller and the ratio between these dimensions would fit the golden ratio to those three significant figures.

Onto nature. Nature proves to be mathematical in every sense and likes to use the golden spiral.

Starting off with the sunflower; the seeds of a sunflower grow in a particular pattern although they may seem to appear scattered randomly. A sunflower needs to decide how far it should turn around before it grows a new seed from its centre, the angle that it turns will determine how much of the flower’s circular centre will be filled with seeds, and obviously the aim is to optimise this number. Through micro-evolution, or something like that, the sunflower came to a decision: it will have each new seed that grows from its centre 137.5 degrees  (or close enough to that) appart. This value seems random but it is infact the corresponding angle to the golden ratio, also know as the golden angle. and that is where the sunflower uses golden mathematics, optimizing its seed growth.

Secondly, in nature, we have plants with spiral leaves. These types of plants simply use the golden angle to grow their leaves so that no new leaves block out the sun of exiusting leaves on the plant, also this leaf formation helps dew and rain that fall on the plant flow directly towards its roots.

Finally, on the biggest scale in the universe, we have galaxies. A spiral galaxy, like our own, has arms that are meant to wrap around it as the galaxy turns from the centre. For some unknow reason, the arms of a spiral galaxy seem to defy this, and with that newtonian physics, as the stars on the extremities of the arms move at a higher velocity; thus maintaining the golden spiral in our galaxy.

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Reflection on Mathematical Ideas (Josh Luong)

The Relation Between Music and Maths

The concept learnt after watching the various videos in
class have extended my knowledge on how much closer music and maths really is.
Being made to do mathematics all my life, it was definitely interesting seeing
that simple fractions and equations could be so similar to the octaves of notes
or triads played. After learning to play my own instrument, I can’t promise
that every time I play a note, several equations will be visualized and used to
my advantage like the genius musician Beethoven most likely could but it’s definitely
something that I can think about in the future.

The Philosophy of Pythagoras

only seen Pythagoras as a mathematician who had been someone who has given the
rule to find sides in a right angled triangle which has only made me grieve
over my maths homework but after viewing the video I had come to understand
that he was not only someone who had discovered maths formulas but his philosophy
tells that everything in the world is  made
of numbers which is not entirely true. It was interesting seeing a cartoon representation
of Pythagoras having to secretly discover these new concepts of mathematics alongside
the Pythagoreans which only makes me appreciate that we get to relearn and
maybe improve on his findings from thousands of years ago. #exploremaths

Music and Maths: How does it Connect?

Through this lesson, we viewed
three mathematically entertaining videos which connected music and maths. We
explored the concept of the Pentagram in which we learnt that you could keep
adding a pentagram in the middle of the previous one to form an infinite
pattern. This allowed us to acknowledge the fact that your mind is the only
method of thinking infinitely. Something that surprised me this lesson, was the
fact of how mathematics is so connected with music. We saw that calculating
wavelengths of musical notes, you could make a sound which was pleasant to the
ear. This is called consonance. Apart from the pentagram and how music is made
with maths, we learnt that Pythagoras’, an ancient Greek Philosopher, was the
father of maths and music. Although often arguing with Democritus about what
the world revolved around, Pythagoras was the main reason we have our music
today, not through only musical practices, but from a mathematical viewpoint,
i.e. Pythagoras discovered an octave, which is 8 keys, had a frequency of which
equalled 2:1.

The questions I have that are
left unanswered from today’s lesson, reflecting on the basis of maths and
music, is how can you construct a piece or musical score using maths. How can
you make a song which combines both influences of maths and music into one
thing?

#exploremaths

Mathematics in Music and Games (by Soha Rizvi)

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)

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.

#exploremaths

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

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.

#exploremaths

#exploremaths

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

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.

#exploremaths

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

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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