Category: Submissions

Billiards and Mathematics (Eric Sun)

At first glance, the game billiards doesn’t have any
resemblance to mathematics. Many would think that the diamonds around the table
are there for no reason and that it would require luck. However, throughout the
course of the video, I realise that is has little to do with luck. The lines
around the table are there for a reason and striking the ball at different
angles changes how the Q ball acts, like how striking the Q ball low so it will
spin backwards. I had always thought that it was based on luck and years of
practise But after watching the video, I now know that it has little to do with
luck and more to do with how you strike the ball and how the diamonds are there
for a reason and not there for aesthetic purposes.

#exploremaths

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The Golden Ratio in nature and
architecture (by Eric Sun)

Examples of
the golden ratio in architecture

1: Toronto’s CN Tower

The CN Tower in Toronto is the tallest,
freestanding structure in the world and contains the golden ratio in its
design. The ratio of the observation desk at 343 metres to the total height of
533.33 is 0.618 or phi.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2: The UN Building

The current headquarters for the United
Nations was constructed on an 18 acre piece of land in the east side on
Manhattan. The lead architect was not known for using the golden ratio in his
designs, however, a French architect on the team was known frequently to use
the golden ratio in his designs. When constructing the United Nations
headquarters, the team decided to use the ratio in a few different ways.

 

 

 

 

 

 

3: The Notre Dame

Phi and the use of the golden ratio are found in the design
of Notre Dame in Paris, France. The west façade of the church was completed
around the year 1200, and it is here where the presence of the use of the
golden ratios is visible.

 

 

 

 

 

 

 

 

 

 

Examples
of the golden ratio in nature

1:
Spiral Galaxies

Spiral
Galaxies also follow the Fibonnaci Pattern. The Milky Way has several spiral arms, each of them a
logarithmic spiral of about 12 degrees.

 

 

 

 

 

 

 

 

 

 

 

 

2: Pinecones

Similarly, the
seed pods on a pinecone are arranged in a spiral pattern. Each
cone consists of a pair of spirals, each one spiralling upwards in opposing direction

 

#exploremaths

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Symmetry and Tessellation (Eric Sun)

Today in class we learnt about the properties of symmetry
where we tested different methods such
as rotation, reflectional, scale symmetry and translational symmetry. Tessellation
can be found within games. A chess board consisting of 64 squares, with 2
alternating colours (usually a dark colour and
a light colour) feature a tessellation of squares, unlike a beehive, with its hexagons
and relatively same colours. It was amazing to see and create the scale
symmetry with my own shape, which could be made smaller or bigger depending on
the scale chosen on my ruler. The completed
shape with the spirals joined, although what started off as something simple,
turned out to produce something complex.

#exploremaths

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Fractals
(Eric Sun)

Fractals are everywhere in nature. They are never ending.
They go on forever until the human eye can’t see it anymore. Fractals can be seen
almost everywhere in our environment such as shorelines where they keep splitting
up into various individual rivers and the dividing branches on trees. Fractals
can be enlarged and part of it still resembles the original shape.

Favourite Fractal                                                                                                                                        My favourite fractal is the salt flats
in San Francisco and Bolivia. They may seem normal but a closer look will
reveal that they are full of fractals. It looks normal at first but look closer
and they look amazing but it’s subtle.

#exploremaths 

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#exploremaths
In this lesson I learnt many things such as Donald duck is cool and how maths is cool

I the first video that we watched was all about Pythagoras and his discoveries. The main ideas that I took away from this were: the ratio of an octave being 2:1, how pentagons are really, really, really, cool and finally the overwhelmingly awe-inspiring, inconceivable fact that Mr. Woo just so happened to mention under his breath: Pythagoras suggested that the universe is made of numbers. Now, that may have seemed relatively anti-climactic after all of those big words but when you think about it; the universe being made of numbers is pretty darn amazing. This is especially true as if numbers, on their own, are just things that we conceive, as they are no linger adjectives for nouns, then the entirety of the universe is something that we conceive. So in that sense numbers and the universe are just like infinity; there is no infinity but it is still real even though we can not see it or ever have an infinite amount of anything.

The other things that I mentioned earlier were the pentagram and the octave. These are not as amazing as the conceived universe but they are still pretty exceptional. The pentagram consists completely of fractals, the golden ratio, spiral, rectangle and angle. Either that or it consists completely of illuminati, the devil and that angry goat thing.

Anyway all the things it is made of basically constitute to maths and here is a neat example of a supposedly infinite amount of pentagrams inside a pentagram.

 

Personally I think that there must be a pattersn or rule that tells you how much golden ration tht there is in a shape and that the pentagram is just the most prominent example of the golden ratio in a shape.

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#exploremaths

Fractals are scattered every in nature, from the lightning
that strikes the Earth to our very own lungs. As a line can never really be
straight, they show how the world can be perfect through it’s imperfections.
Many things throughout the world have this property and it can also be known as
iteration or scale symmetry. Personally, I find the fractals found in rivers to
be mesmerising as you can look at them from satellite imagery or through a
microscope to find the same basic shape. 

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#exploremaths

 

In the lesson, we went over some older memories from primary
school and talked about the different types of symmetry. Although we drew some
examples of rotational and reflectional symmetries, the main points of discussion
were scale and translation symmetry. Scale symmetry, as we went on to learn
about this in future lessons, was essentially describing the self-similarity of
shapes. We attempted to show this type of symmetry in our books too, by drawing
a square with many rotating squares inside of it. The remaining type,
translational symmetry or tessellation, was how one shape could be used to fill
an area without leaving any spaces. One example would be, as we discussed in
class, the hexagons of a beehive. Another naturally occurring tessellation is the pineapple as it is, like the beehive, filled with many hexagons. Though
these are irregular shapes, it is still quite interesting to see how nature’s
beauty is based on mathematics. 

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Fractals are never ending, goes on forever, until the human eyes can’t see it anymore. it is one of the most beautiful type of shape that I have ever seen. Fractals can be enlarged and enlarged, part of it still resembling the original shape. Although it sounds unreal and magical, it exists all around us, like the coastline, fern leaves, rivers and tree branches. All of these are wonderful and awesome, but my favorite has got to be the Koch Fractals.

Favorite Fractal

The Koch Fractal is so simple to draw but extremely complex to our eyes. Its hexagons continue on spreading. it is extremely intriguing to our eyes, as all the fractals are. However, what makes this different to others, to me, is the fact that it resembles snowflakes. it shows the beauty of snowflakes, crafted in precise conditions in order to make such beautiful and intricate shape.

#exploremaths

These are the architectures and beings of nature that have become my favorite Golden Ratios.

Golden Ratio in Architecture:

 

1) Parthenon

The Parthenon in Greece follow the golden ratio in different parts of its structure. The entire front face of the Parthenon closely resembles the golden rectangle. Also, the structural beam on top of the supporting poles are proportional to each other, in golden ratio.

2) Notre Dame

Notre Dame in Paris, which was built in between 1163 and 1250 appears to have golden ratio proportions in a number of its key proportions of design. Although it is inaccurate to measure by a photographic source, it is possible to notice the golden ration implanted into the structure in various places.

3)Toronto’s CN Tower

The Toronto’s CN Tower, although modern, contains the golden ratio in its design. the ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or phi, the reciprocal of phi.  

Golden Ratio in nature

The Fibonacci Sequence or Series has a relatioship to the Golden Ratio. The Fibonacci Series shows up in the number of leaves on a plant and The Fibonaccie Wequence or Series has a relationship to the Golden Ratio.

1)The Shell

Shells are the results of a famous shape made utilizing the Golden Ratio called Golden Spirals. The Shells closely resemble this shape as seen in this picture.

Fractals in Nature

 Fractals are everywherein nature, never ending, only as far as our eyes can see. Rivers, with theirwinding channels and weird small side ones, all relatively follow the shape asthe big one, as seen in the diagram shown in class, where taking a closer lookon the small ones, show a similar shape to the main. In a video shown, therewas conclusively no way to measure the coastline, simply because having smallermeasuring equipment yield different results from larger ones. Fractals are usedto describe this coastline paradox, making it impossible to accurately measurethe actual length.

 

Favourite Fractal

My favourite fractal would have to be in trees. I find treesintriguing, just spending time to watch them sway to the wind as well as wonderhow each branch is capable to grow from the trunk, and have further branches.The leaves too, seeing the patterns on them, and being able to hold ontobranches and then just fall off.

#exploremaths